annotate scripts/special-matrix/gallery.m @ 24534:194eb4bd202b

maint: Update punctuation for GPL v3 license text. * COPYING, Makefile.am, README, bootstrap, bootstrap.conf, OctJavaQry.java, changelog.tmpl, check-subst-vars.in.sh, find-defun-files.sh, find-files-with-tests.sh, get-source-mtime.sh, mk-hg-id.sh, mk-octave-config-h.sh, mk-opts.pl, stl_algo.h-fixed, subst-config-vals.in.sh, subst-cross-config-vals.in.sh, subst-default-vals.in.sh, subst-script-vals.in.sh, configure.ac, Doxyfile.in, arith.txi, audio.txi, basics.txi, bugs.txi, config-images.sh, container.txi, cp-idx.txi, data.txi, debug.txi, diagperm.txi, diffeq.txi, add_to_aspell_dict, mk_undocumented_list, spellcheck, errors.txi, eval.txi, expr.txi, external.txi, fn-idx.txi, func.txi, genpropdoc.m, geometry.txi, geometryimages.m, grammar.txi, gui.txi, image.txi, images.awk, install.txi, interp.txi, interpimages.m, intro.txi, io.txi, linalg.txi, macros.texi, matrix.txi, mk-doc-cache.pl, mkcontrib.awk, mkoctfile.1, munge-texi.pl, nonlin.txi, numbers.txi, obsolete.txi, octave-cli.1, octave-config.1, octave.1, octave.css, octave.texi, oop.txi, op-idx.txi, optim.txi, package.txi, plot.txi, plotimages.m, poly.txi, preface.txi, quad.txi, set.txi, signal.txi, sparse.txi, sparseimages.m, splineimages.m, stats.txi, stmt.txi, strings.txi, system.txi, testfun.txi, var.txi, vectorize.txi, array.texi, bugs.texi, cp-idx.texi, dae.texi, diffeq.texi, error.texi, factor.texi, fn-idx.texi, gpl.texi, install.texi, intro.texi, liboctave.texi, matvec.texi, nleqn.texi, nlfunc.texi, ode.texi, optim.texi, preface.texi, quad.texi, range.texi, refcard-a4.tex, refcard-legal.tex, refcard-letter.tex, refcard.tex, HACKING.md, octave.appdata.xml.in, Backend.cc, Backend.h, BaseControl.cc, BaseControl.h, ButtonControl.cc, ButtonControl.h, ButtonGroup.cc, ButtonGroup.h, Canvas.cc, Canvas.h, CheckBoxControl.cc, CheckBoxControl.h, Container.cc, Container.h, ContextMenu.cc, ContextMenu.h, EditControl.cc, EditControl.h, Figure.cc, Figure.h, FigureWindow.cc, FigureWindow.h, GLCanvas.cc, GLCanvas.h, GenericEventNotify.h, KeyMap.cc, KeyMap.h, ListBoxControl.cc, ListBoxControl.h, Logger.cc, Logger.h, Menu.cc, Menu.h, MenuContainer.h, MouseModeActionGroup.cc, MouseModeActionGroup.h, Object.cc, Object.h, ObjectFactory.cc, ObjectFactory.h, ObjectProxy.cc, ObjectProxy.h, Panel.cc, Panel.h, PopupMenuControl.cc, PopupMenuControl.h, PushButtonControl.cc, PushButtonControl.h, PushTool.cc, PushTool.h, QtHandlesUtils.cc, QtHandlesUtils.h, RadioButtonControl.cc, RadioButtonControl.h, SliderControl.cc, SliderControl.h, TextControl.cc, TextControl.h, TextEdit.cc, TextEdit.h, ToggleButtonControl.cc, ToggleButtonControl.h, ToggleTool.cc, ToggleTool.h, ToolBar.cc, ToolBar.h, ToolBarButton.cc, ToolBarButton.h, __init_qt__.cc, __init_qt__.h, annotation-dialog.cc, annotation-dialog.h, gl-select.cc, gl-select.h, liboctgui-build-info.h, liboctgui-build-info.in.cc, mk-default-qt-settings.in.sh, QTerminal.cc, QTerminal.h, BlockArray.cpp, BlockArray.h, Character.h, CharacterColor.h, Emulation.cpp, Emulation.h, Filter.cpp, Filter.h, History.cpp, History.h, KeyboardTranslator.cpp, KeyboardTranslator.h, QUnixTerminalImpl.cpp, QUnixTerminalImpl.h, Screen.cpp, Screen.h, ScreenWindow.cpp, ScreenWindow.h, SelfListener.cpp, SelfListener.h, TerminalCharacterDecoder.cpp, TerminalCharacterDecoder.h, TerminalModel.cpp, TerminalModel.h, TerminalView.cpp, TerminalView.h, Vt102Emulation.cpp, Vt102Emulation.h, kpty.cpp, kpty.h, kpty_p.h, QTerminalColors.cpp, QTerminalColors.h, QWinTerminalImpl.cpp, QWinTerminalImpl.h, main.cpp, color-picker.cc, color-picker.h, dialog.cc, dialog.h, documentation-dock-widget.cc, documentation-dock-widget.h, external-editor-interface.cc, external-editor-interface.h, files-dock-widget.cc, files-dock-widget.h, find-files-dialog.cc, find-files-dialog.h, find-files-model.cc, find-files-model.h, history-dock-widget.cc, history-dock-widget.h, file-editor-interface.h, file-editor-tab.cc, file-editor-tab.h, file-editor.cc, file-editor.h, find-dialog.cc, find-dialog.h, marker.cc, marker.h, octave-qscintilla.cc, octave-qscintilla.h, octave-txt-lexer.cc, octave-txt-lexer.h, main-window.cc, main-window.h, octave-cmd.cc, octave-cmd.h, octave-dock-widget.cc, octave-dock-widget.h, octave-gui.cc, octave-gui.h, octave-qt-link.cc, octave-qt-link.h, octave-settings.h, texinfo-parser.cc, texinfo-parser.h, webinfo.cc, webinfo.h, resource-manager.cc, resource-manager.h, settings-dialog.cc, settings-dialog.h, shortcut-manager.cc, shortcut-manager.h, terminal-dock-widget.cc, terminal-dock-widget.h, thread-manager.cc, thread-manager.h, variable-editor-model.cc, variable-editor-model.h, variable-editor.cc, variable-editor.h, welcome-wizard.cc, welcome-wizard.h, workspace-model.cc, workspace-model.h, workspace-view.cc, workspace-view.h, build-env.h, build-env.in.cc, builtins.h, Cell.cc, Cell.h, __contourc__.cc, __dsearchn__.cc, __ichol__.cc, __ilu__.cc, __lin_interpn__.cc, __luinc__.cc, __magick_read__.cc, __pchip_deriv__.cc, __qp__.cc, balance.cc, base-text-renderer.h, besselj.cc, betainc.cc, bitfcns.cc, bsxfun.cc, c-file-ptr-stream.cc, c-file-ptr-stream.h, call-stack.cc, call-stack.h, cdisplay.c, cdisplay.h, cellfun.cc, coct-hdf5-types.c, colloc.cc, conv2.cc, daspk.cc, dasrt.cc, dassl.cc, data.cc, data.h, debug.cc, default-defs.in.h, defaults.cc, defaults.h, defun-dld.h, defun-int.h, defun.cc, defun.h, det.cc, dirfns.cc, dirfns.h, display.cc, display.h, dlmread.cc, dot.cc, dynamic-ld.cc, dynamic-ld.h, eig.cc, ellipj.cc, environment.cc, environment.h, error.cc, error.h, errwarn.cc, errwarn.h, event-queue.cc, event-queue.h, fcn-info.cc, fcn-info.h, fft.cc, fft2.cc, fftn.cc, file-io.cc, file-io.h, filter.cc, find.cc, ft-text-renderer.cc, ft-text-renderer.h, gammainc.cc, gcd.cc, genprops.awk, getgrent.cc, getpwent.cc, getrusage.cc, givens.cc, gl-render.cc, gl-render.h, gl2ps-print.cc, gl2ps-print.h, graphics-handle.h, graphics-toolkit.cc, graphics-toolkit.h, graphics.cc, graphics.in.h, gripes.cc, gripes.h, gsvd.cc, gtk-manager.cc, gtk-manager.h, hash.cc, help.cc, help.h, hess.cc, hex2num.cc, hook-fcn.cc, hook-fcn.h, input.cc, input.h, interpreter-private.cc, interpreter-private.h, interpreter.cc, interpreter.h, inv.cc, kron.cc, load-path.cc, load-path.h, load-save.cc, load-save.h, lookup.cc, ls-ascii-helper.cc, ls-ascii-helper.h, ls-hdf5.cc, ls-hdf5.h, ls-mat-ascii.cc, ls-mat-ascii.h, ls-mat4.cc, ls-mat4.h, ls-mat5.cc, ls-mat5.h, ls-oct-binary.cc, ls-oct-binary.h, ls-oct-text.cc, ls-oct-text.h, ls-utils.cc, ls-utils.h, lsode.cc, lu.cc, mappers.cc, matrix_type.cc, max.cc, mex.cc, mex.h, mexproto.h, mgorth.cc, mk-errno-list.sh, mk-mxarray-h.in.sh, mxarray.in.h, nproc.cc, oct-errno.h, oct-errno.in.cc, oct-fstrm.cc, oct-fstrm.h, oct-handle.h, oct-hdf5-types.cc, oct-hdf5-types.h, oct-hdf5.h, oct-hist.cc, oct-hist.h, oct-iostrm.cc, oct-iostrm.h, oct-map.cc, oct-map.h, oct-obj.h, oct-opengl.h, oct-prcstrm.cc, oct-prcstrm.h, oct-procbuf.cc, oct-procbuf.h, oct-stdstrm.h, oct-stream.cc, oct-stream.h, oct-strstrm.cc, oct-strstrm.h, oct-tex-lexer.in.ll, oct-tex-parser.in.yy, oct.h, octave-default-image.h, octave-link.cc, octave-link.h, ordschur.cc, pager.cc, pager.h, pinv.cc, pr-output.cc, pr-output.h, procstream.cc, procstream.h, psi.cc, quad.cc, quadcc.cc, qz.cc, rand.cc, rcond.cc, regexp.cc, schur.cc, sighandlers.cc, sighandlers.h, sparse-xdiv.cc, sparse-xdiv.h, sparse-xpow.cc, sparse-xpow.h, sparse.cc, spparms.cc, sqrtm.cc, str2double.cc, strfind.cc, strfns.cc, sub2ind.cc, svd.cc, sylvester.cc, symrec.cc, symrec.h, symscope.cc, symscope.h, symtab.cc, symtab.h, syscalls.cc, sysdep.cc, sysdep.h, text-renderer.cc, text-renderer.h, time.cc, toplev.cc, toplev.h, tril.cc, tsearch.cc, txt-eng.cc, txt-eng.h, typecast.cc, url-handle-manager.cc, url-handle-manager.h, urlwrite.cc, utils.cc, utils.h, variables.cc, variables.h, workspace-element.h, xdiv.cc, xdiv.h, xnorm.cc, xnorm.h, xpow.cc, xpow.h, zfstream.cc, zfstream.h, deprecated-config.h, __delaunayn__.cc, __eigs__.cc, __fltk_uigetfile__.cc, __glpk__.cc, __init_fltk__.cc, __init_gnuplot__.cc, __ode15__.cc, __osmesa_print__.cc, __voronoi__.cc, amd.cc, audiodevinfo.cc, audioread.cc, ccolamd.cc, chol.cc, colamd.cc, config-module.awk, config-module.sh, convhulln.cc, dmperm.cc, fftw.cc, gzip.cc, oct-qhull.h, qr.cc, symbfact.cc, symrcm.cc, liboctinterp-build-info.h, liboctinterp-build-info.in.cc, mk-build-env-features.sh, mk-builtins.pl, mk-doc.pl, mk-pkg-add.sh, mk-version-h.in.sh, ov-base-diag.cc, ov-base-diag.h, ov-base-int.cc, ov-base-int.h, ov-base-mat.cc, ov-base-mat.h, ov-base-scalar.cc, ov-base-scalar.h, ov-base-sparse.cc, ov-base-sparse.h, ov-base.cc, ov-base.h, ov-bool-mat.cc, ov-bool-mat.h, ov-bool-sparse.cc, ov-bool-sparse.h, ov-bool.cc, ov-bool.h, ov-builtin.cc, ov-builtin.h, ov-cell.cc, ov-cell.h, ov-ch-mat.cc, ov-ch-mat.h, ov-class.cc, ov-class.h, ov-classdef.cc, ov-classdef.h, ov-colon.cc, ov-colon.h, ov-complex.cc, ov-complex.h, ov-cs-list.cc, ov-cs-list.h, ov-cx-diag.cc, ov-cx-diag.h, ov-cx-mat.cc, ov-cx-mat.h, 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ov-usr-fcn.cc, ov-usr-fcn.h, ov.cc, ov.h, ovl.cc, ovl.h, octave.cc, octave.h, op-kw-docs, mk-ops.sh, op-b-b.cc, op-b-bm.cc, op-b-sbm.cc, op-bm-b.cc, op-bm-bm.cc, op-bm-sbm.cc, op-cdm-cdm.cc, op-cdm-cm.cc, op-cdm-cs.cc, op-cdm-dm.cc, op-cdm-m.cc, op-cdm-s.cc, op-cell.cc, op-chm.cc, op-class.cc, op-cm-cdm.cc, op-cm-cm.cc, op-cm-cs.cc, op-cm-dm.cc, op-cm-m.cc, op-cm-pm.cc, op-cm-s.cc, op-cm-scm.cc, op-cm-sm.cc, op-cs-cm.cc, op-cs-cs.cc, op-cs-m.cc, op-cs-s.cc, op-cs-scm.cc, op-cs-sm.cc, op-dm-cdm.cc, op-dm-cm.cc, op-dm-cs.cc, op-dm-dm.cc, op-dm-m.cc, op-dm-s.cc, op-dm-scm.cc, op-dm-sm.cc, op-dm-template.cc, op-dms-template.cc, op-fcdm-fcdm.cc, op-fcdm-fcm.cc, op-fcdm-fcs.cc, op-fcdm-fdm.cc, op-fcdm-fm.cc, op-fcdm-fs.cc, op-fcm-fcdm.cc, op-fcm-fcm.cc, op-fcm-fcs.cc, op-fcm-fdm.cc, op-fcm-fm.cc, op-fcm-fs.cc, op-fcm-pm.cc, op-fcn.cc, op-fcs-fcm.cc, op-fcs-fcs.cc, op-fcs-fm.cc, op-fcs-fs.cc, op-fdm-fcdm.cc, op-fdm-fcm.cc, op-fdm-fcs.cc, op-fdm-fdm.cc, op-fdm-fm.cc, op-fdm-fs.cc, op-fm-fcdm.cc, op-fm-fcm.cc, op-fm-fcs.cc, op-fm-fdm.cc, op-fm-fm.cc, op-fm-fs.cc, op-fm-pm.cc, op-fs-fcm.cc, op-fs-fcs.cc, op-fs-fm.cc, op-fs-fs.cc, op-i16-i16.cc, op-i32-i32.cc, op-i64-i64.cc, op-i8-i8.cc, op-int-concat.cc, op-int.h, op-m-cdm.cc, op-m-cm.cc, op-m-cs.cc, op-m-dm.cc, op-m-m.cc, op-m-pm.cc, op-m-s.cc, op-m-scm.cc, op-m-sm.cc, op-pm-cm.cc, op-pm-fcm.cc, op-pm-fm.cc, op-pm-m.cc, op-pm-pm.cc, op-pm-scm.cc, op-pm-sm.cc, op-pm-template.cc, op-range.cc, op-s-cm.cc, op-s-cs.cc, op-s-m.cc, op-s-s.cc, op-s-scm.cc, op-s-sm.cc, op-sbm-b.cc, op-sbm-bm.cc, op-sbm-sbm.cc, op-scm-cm.cc, op-scm-cs.cc, op-scm-m.cc, op-scm-s.cc, op-scm-scm.cc, op-scm-sm.cc, op-sm-cm.cc, op-sm-cs.cc, op-sm-m.cc, op-sm-s.cc, op-sm-scm.cc, op-sm-sm.cc, op-str-m.cc, op-str-s.cc, op-str-str.cc, op-struct.cc, op-ui16-ui16.cc, op-ui32-ui32.cc, op-ui64-ui64.cc, op-ui8-ui8.cc, ops.h, options-usage.h, bp-table.cc, bp-table.h, comment-list.cc, comment-list.h, jit-ir.cc, jit-ir.h, jit-typeinfo.cc, jit-typeinfo.h, jit-util.cc, jit-util.h, lex.h, lex.ll, oct-lvalue.cc, oct-lvalue.h, oct-parse.in.yy, octave.gperf, parse.h, profiler.cc, profiler.h, pt-all.h, pt-arg-list.cc, pt-arg-list.h, pt-array-list.cc, pt-array-list.h, pt-assign.cc, pt-assign.h, pt-binop.cc, pt-binop.h, pt-bp.cc, pt-bp.h, pt-cbinop.cc, pt-cbinop.h, pt-cell.cc, pt-cell.h, pt-check.cc, pt-check.h, pt-classdef.cc, pt-classdef.h, pt-cmd.h, pt-colon.cc, pt-colon.h, pt-const.cc, pt-const.h, pt-decl.cc, pt-decl.h, pt-eval.cc, pt-eval.h, pt-except.cc, pt-except.h, pt-exp.cc, pt-exp.h, pt-fcn-handle.cc, pt-fcn-handle.h, pt-funcall.cc, pt-funcall.h, pt-id.cc, pt-id.h, pt-idx.cc, pt-idx.h, pt-jit.cc, pt-jit.h, pt-jump.cc, pt-jump.h, pt-loop.cc, pt-loop.h, pt-mat.cc, pt-mat.h, pt-misc.cc, pt-misc.h, pt-pr-code.cc, pt-pr-code.h, pt-select.cc, pt-select.h, pt-stmt.cc, pt-stmt.h, pt-tm-const.cc, pt-tm-const.h, pt-unop.cc, pt-unop.h, pt-walk.cc, pt-walk.h, pt.cc, pt.h, token.cc, token.h, Array-jit.cc, Array-tc.cc, version.cc, version.in.h, Array-C.cc, Array-b.cc, Array-ch.cc, Array-d.cc, Array-f.cc, Array-fC.cc, Array-i.cc, Array-idx-vec.cc, Array-s.cc, Array-str.cc, Array-util.cc, Array-util.h, Array-voidp.cc, Array.cc, Array.h, CColVector.cc, CColVector.h, CDiagMatrix.cc, CDiagMatrix.h, CMatrix.cc, CMatrix.h, CNDArray.cc, CNDArray.h, CRowVector.cc, CRowVector.h, CSparse.cc, CSparse.h, DiagArray2.cc, DiagArray2.h, MArray-C.cc, MArray-d.cc, MArray-f.cc, MArray-fC.cc, MArray-i.cc, MArray-s.cc, MArray.cc, MArray.h, MDiagArray2.cc, MDiagArray2.h, MSparse-C.cc, MSparse-d.cc, MSparse.cc, MSparse.h, Matrix.h, MatrixType.cc, MatrixType.h, PermMatrix.cc, PermMatrix.h, Range.cc, Range.h, Sparse-C.cc, Sparse-b.cc, Sparse-d.cc, Sparse.cc, Sparse.h, boolMatrix.cc, boolMatrix.h, boolNDArray.cc, boolNDArray.h, boolSparse.cc, boolSparse.h, chMatrix.cc, chMatrix.h, chNDArray.cc, chNDArray.h, dColVector.cc, dColVector.h, dDiagMatrix.cc, dDiagMatrix.h, dMatrix.cc, dMatrix.h, dNDArray.cc, dNDArray.h, dRowVector.cc, dRowVector.h, dSparse.cc, dSparse.h, dim-vector.cc, dim-vector.h, fCColVector.cc, fCColVector.h, fCDiagMatrix.cc, fCDiagMatrix.h, fCMatrix.cc, fCMatrix.h, fCNDArray.cc, fCNDArray.h, fCRowVector.cc, fCRowVector.h, fColVector.cc, fColVector.h, fDiagMatrix.cc, fDiagMatrix.h, fMatrix.cc, fMatrix.h, fNDArray.cc, fNDArray.h, fRowVector.cc, fRowVector.h, idx-vector.cc, idx-vector.h, int16NDArray.cc, int16NDArray.h, int32NDArray.cc, int32NDArray.h, int64NDArray.cc, int64NDArray.h, int8NDArray.cc, int8NDArray.h, intNDArray.cc, intNDArray.h, uint16NDArray.cc, uint16NDArray.h, uint32NDArray.cc, uint32NDArray.h, uint64NDArray.cc, uint64NDArray.h, uint8NDArray.cc, uint8NDArray.h, cconv2.f, cdotc3.f, cmatm3.f, csconv2.f, dconv2.f, ddot3.f, dmatm3.f, sconv2.f, sdot3.f, smatm3.f, zconv2.f, zdconv2.f, zdotc3.f, zmatm3.f, crsf2csf.f, zrsf2csf.f, mk-f77-def.in.sh, liboctave-build-info.h, liboctave-build-info.in.cc, CollocWt.cc, CollocWt.h, DAE.h, DAEFunc.h, DAERT.h, DAERTFunc.h, DASPK-opts.in, DASPK.cc, DASPK.h, 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compare_plot_demos.m, dump_demos.m, html_compare_plot_demos.m, rundemos.m, runtests.m, speed.m, test.m, addtodate.m, asctime.m, calendar.m, clock.m, ctime.m, date.m, datenum.m, datestr.m, datevec.m, eomday.m, etime.m, is_leap_year.m, now.m, weekday.m, display-available.c, display-available.h, main-cli.cc, main-gui.cc, main.in.cc, mkoctfile.in.cc, octave-build-info.h, octave-build-info.in.cc, octave-config.in.cc, shared-fcns.h, args.tst, bug-31371.tst, bug-35448.tst, bug-35881.tst, bug-36025.tst, bug-38236.tst, bug-38565.tst, bug-38576.tst, bug-38691.tst, bug-41723.tst, bug-44940.tst, bug-46330.tst, bug-46660.tst, bug-50014.tst, bug-50035.tst, bug-50716.tst, bug-51192.tst, bug-51532.tst, bug-51534.tst, bug-51599.tst, bug-52075.tst, class-concat.tst, classdef-multiple-inheritance.tst, classdef.tst, classes.tst, colormaps.tst, command.tst, complex.tst, ctor-vs-method.tst, deprecate-props.tst, diag-perm.tst, error.tst, eval-catch.tst, fcn-handle-derived-resolution.tst, fntests.m, for.tst, func.tst, global.tst, if.tst, index.tst, io.tst, jit.tst, leftdiv.tst, line-continue.tst, logical-index.tst, mk-bc-overloads-tst.sh, mk-conv-tst.sh, mk-sparse-tst.sh, nest.tst, null-assign.tst, parser.tst, prefer.tst, publish.tst, range.tst, recursion.tst, return.tst, show-failures.awk, single-index.tst, slice.tst, struct.tst, switch.tst, system.tst, transpose.tst, try.tst, unwind.tst, while.tst: Changed punctuation of GPL license text to match that suggested by FSF.
author Rik <rik@octave.org>
date Sat, 06 Jan 2018 07:57:19 -0800
parents d20b385d8c99
children 2365c2661b3c
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rev   line source
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1 ## Copyright (C) 1989-1995 Nicholas .J. Higham
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2 ## Copyright (C) 2013-2016 Carnë Draug
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3 ##
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4 ## This file is part of Octave.
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5 ##
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6 ## Octave is free software: you can redistribute it and/or modify it
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7 ## under the terms of the GNU General Public License as published by
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8 ## the Free Software Foundation, either version 3 of the License, or
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9 ## (at your option) any later version.
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10 ##
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11 ## Octave is distributed in the hope that it will be useful, but
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12 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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14 ## GNU General Public License for more details.
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15 ##
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16 ## You should have received a copy of the GNU General Public License
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17 ## along with Octave; see the file COPYING. If not, see
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18 ## <https://www.gnu.org/licenses/>.
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19
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20 ## -*- texinfo -*-
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21 ## @deftypefn {} {} gallery (@var{name})
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22 ## @deftypefnx {} {} gallery (@var{name}, @var{args})
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23 ## Create interesting matrices for testing.
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24 ##
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25 ## @end deftypefn
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26 ##
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27 ## @deftypefn {} {@var{c} =} gallery ("cauchy", @var{x})
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28 ## @deftypefnx {} {@var{c} =} gallery ("cauchy", @var{x}, @var{y})
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29 ## Create a Cauchy matrix.
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30 ##
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31 ## @end deftypefn
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32 ##
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33 ## @deftypefn {} {@var{c} =} gallery ("chebspec", @var{n})
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34 ## @deftypefnx {} {@var{c} =} gallery ("chebspec", @var{n}, @var{k})
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35 ## Create a Chebyshev spectral differentiation matrix.
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36 ##
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37 ## @end deftypefn
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38 ##
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39 ## @deftypefn {} {@var{c} =} gallery ("chebvand", @var{p})
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40 ## @deftypefnx {} {@var{c} =} gallery ("chebvand", @var{m}, @var{p})
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41 ## Create a Vandermonde-like matrix for the Chebyshev polynomials.
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42 ##
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43 ## @end deftypefn
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44 ##
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45 ## @deftypefn {} {@var{a} =} gallery ("chow", @var{n})
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46 ## @deftypefnx {} {@var{a} =} gallery ("chow", @var{n}, @var{alpha})
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47 ## @deftypefnx {} {@var{a} =} gallery ("chow", @var{n}, @var{alpha}, @var{delta})
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48 ## Create a Chow matrix -- a singular Toeplitz lower Hessenberg matrix.
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49 ##
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50 ## @end deftypefn
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51 ##
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52 ## @deftypefn {} {@var{c} =} gallery ("circul", @var{v})
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53 ## Create a circulant matrix.
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54 ##
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55 ## @end deftypefn
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56 ##
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57 ## @deftypefn {} {@var{a} =} gallery ("clement", @var{n})
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58 ## @deftypefnx {} {@var{a} =} gallery ("clement", @var{n}, @var{k})
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59 ## Create a tridiagonal matrix with zero diagonal entries.
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60 ##
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61 ## @end deftypefn
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62 ##
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63 ## @deftypefn {} {@var{c} =} gallery ("compar", @var{a})
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64 ## @deftypefnx {} {@var{c} =} gallery ("compar", @var{a}, @var{k})
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65 ## Create a comparison matrix.
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66 ##
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67 ## @end deftypefn
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68 ##
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69 ## @deftypefn {} {@var{a} =} gallery ("condex", @var{n})
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70 ## @deftypefnx {} {@var{a} =} gallery ("condex", @var{n}, @var{k})
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71 ## @deftypefnx {} {@var{a} =} gallery ("condex", @var{n}, @var{k}, @var{theta})
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72 ## Create a `counterexample' matrix to a condition estimator.
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73 ##
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74 ## @end deftypefn
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75 ##
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76 ## @deftypefn {} {@var{a} =} gallery ("cycol", [@var{m} @var{n}])
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77 ## @deftypefnx {} {@var{a} =} gallery ("cycol", @var{n})
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78 ## @deftypefnx {} {@var{a} =} gallery (@dots{}, @var{k})
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79 ## Create a matrix whose columns repeat cyclically.
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80 ##
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81 ## @end deftypefn
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82 ##
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83 ## @deftypefn {} {[@var{c}, @var{d}, @var{e}] =} gallery ("dorr", @var{n})
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84 ## @deftypefnx {} {[@var{c}, @var{d}, @var{e}] =} gallery ("dorr", @var{n}, @var{theta})
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85 ## @deftypefnx {} {@var{a} =} gallery ("dorr", @dots{})
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86 ## Create a diagonally dominant, ill-conditioned, tridiagonal matrix.
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87 ##
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88 ## @end deftypefn
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89 ##
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90 ## @deftypefn {} {@var{a} =} gallery ("dramadah", @var{n})
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91 ## @deftypefnx {} {@var{a} =} gallery ("dramadah", @var{n}, @var{k})
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92 ## Create a (0, 1) matrix whose inverse has large integer entries.
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93 ##
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94 ## @end deftypefn
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95 ##
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96 ## @deftypefn {} {@var{a} =} gallery ("fiedler", @var{c})
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97 ## Create a symmetric @nospell{Fiedler} matrix.
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98 ##
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99 ## @end deftypefn
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100 ##
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101 ## @deftypefn {} {@var{a} =} gallery ("forsythe", @var{n})
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102 ## @deftypefnx {} {@var{a} =} gallery ("forsythe", @var{n}, @var{alpha})
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103 ## @deftypefnx {} {@var{a} =} gallery ("forsythe", @var{n}, @var{alpha}, @var{lambda})
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104 ## Create a @nospell{Forsythe} matrix (a perturbed Jordan block).
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105 ##
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106 ## @end deftypefn
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107 ##
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108 ## @deftypefn {} {@var{f} =} gallery ("frank", @var{n})
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109 ## @deftypefnx {} {@var{f} =} gallery ("frank", @var{n}, @var{k})
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110 ## Create a Frank matrix (ill-conditioned eigenvalues).
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111 ##
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112 ## @end deftypefn
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113 ##
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114 ## @deftypefn {} {@var{c} =} gallery ("gcdmat", @var{n})
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115 ## Create a greatest common divisor matrix.
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116 ##
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117 ## @var{c} is an @var{n}-by-@var{n} matrix whose values correspond to the
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118 ## greatest common divisor of its coordinate values, i.e., @var{c}(i,j)
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119 ## correspond @code{gcd (i, j)}.
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120 ## @end deftypefn
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121 ##
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122 ## @deftypefn {} {@var{a} =} gallery ("gearmat", @var{n})
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123 ## @deftypefnx {} {@var{a} =} gallery ("gearmat", @var{n}, @var{i})
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124 ## @deftypefnx {} {@var{a} =} gallery ("gearmat", @var{n}, @var{i}, @var{j})
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125 ## Create a Gear matrix.
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126 ##
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127 ## @end deftypefn
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128 ##
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129 ## @deftypefn {} {@var{g} =} gallery ("grcar", @var{n})
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130 ## @deftypefnx {} {@var{g} =} gallery ("grcar", @var{n}, @var{k})
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131 ## Create a Toeplitz matrix with sensitive eigenvalues.
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132 ##
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133 ## @end deftypefn
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134 ##
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135 ## @deftypefn {} {@var{a} =} gallery ("hanowa", @var{n})
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136 ## @deftypefnx {} {@var{a} =} gallery ("hanowa", @var{n}, @var{d})
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137 ## Create a matrix whose eigenvalues lie on a vertical line in the complex
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138 ## plane.
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139 ##
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140 ## @end deftypefn
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141 ##
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142 ## @deftypefn {} {@var{v} =} gallery ("house", @var{x})
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143 ## @deftypefnx {} {[@var{v}, @var{beta}] =} gallery ("house", @var{x})
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144 ## Create a householder matrix.
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145 ##
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146 ## @end deftypefn
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147 ##
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148 ## @deftypefn {} {@var{a} =} gallery ("integerdata", @var{imax}, [@var{M} @var{N} @dots{}], @var{j})
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149 ## @deftypefnx {} {@var{a} =} gallery ("integerdata", @var{imax}, @var{M}, @var{N}, @dots{}, @var{j})
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150 ## @deftypefnx {} {@var{a} =} gallery ("integerdata", [@var{imin}, @var{imax}], [@var{M} @var{N} @dots{}], @var{j})
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151 ## @deftypefnx {} {@var{a} =} gallery ("integerdata", [@var{imin}, @var{imax}], @var{M}, @var{N}, @dots{}, @var{j})
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152 ## @deftypefnx {} {@var{a} =} gallery ("integerdata", @dots{}, "@var{class}")
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153 ## Create a matrix with random integers in the range [1, @var{imax}].
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154 ## If @var{imin} is given then the integers are in the range
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155 ## [@var{imin}, @var{imax}].
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156 ##
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157 ## The second input is a matrix of dimensions describing the size of the
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158 ## output. The dimensions can also be input as comma-separated arguments.
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159 ##
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160 ## The input @var{j} is an integer index in the range [0, 2^32-1]. The values
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161 ## of the output matrix are always exactly the same (reproducibility) for a
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162 ## given size input and @var{j} index.
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163 ##
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164 ## The final optional argument determines the class of the resulting matrix.
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165 ## Possible values for @var{class}: @qcode{"uint8"}, @qcode{"uint16"},
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166 ## @qcode{"uint32"}, @qcode{"int8"}, @qcode{"int16"}, int32", @qcode{"single"},
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167 ## @qcode{"double"}. The default is @qcode{"double"}.
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168 ##
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169 ## @end deftypefn
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170 ##
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171 ## @deftypefn {} {@var{a} =} gallery ("invhess", @var{x})
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172 ## @deftypefnx {} {@var{a} =} gallery ("invhess", @var{x}, @var{y})
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173 ## Create the inverse of an upper Hessenberg matrix.
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174 ##
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175 ## @end deftypefn
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176 ##
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177 ## @deftypefn {} {@var{a} =} gallery ("invol", @var{n})
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178 ## Create an involutory matrix.
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179 ##
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180 ## @end deftypefn
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181 ##
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182 ## @deftypefn {} {@var{a} =} gallery ("ipjfact", @var{n})
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183 ## @deftypefnx {} {@var{a} =} gallery ("ipjfact", @var{n}, @var{k})
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184 ## Create a Hankel matrix with factorial elements.
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185 ##
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186 ## @end deftypefn
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187 ##
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188 ## @deftypefn {} {@var{a} =} gallery ("jordbloc", @var{n})
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189 ## @deftypefnx {} {@var{a} =} gallery ("jordbloc", @var{n}, @var{lambda})
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190 ## Create a Jordan block.
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191 ##
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192 ## @end deftypefn
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193 ##
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194 ## @deftypefn {} {@var{u} =} gallery ("kahan", @var{n})
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195 ## @deftypefnx {} {@var{u} =} gallery ("kahan", @var{n}, @var{theta})
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196 ## @deftypefnx {} {@var{u} =} gallery ("kahan", @var{n}, @var{theta}, @var{pert})
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197 ## Create a @nospell{Kahan} matrix (upper trapezoidal).
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198 ##
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199 ## @end deftypefn
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200 ##
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201 ## @deftypefn {} {@var{a} =} gallery ("kms", @var{n})
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202 ## @deftypefnx {} {@var{a} =} gallery ("kms", @var{n}, @var{rho})
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203 ## Create a @nospell{Kac-Murdock-Szego} Toeplitz matrix.
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204 ##
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205 ## @end deftypefn
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206 ##
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207 ## @deftypefn {} {@var{b} =} gallery ("krylov", @var{a})
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208 ## @deftypefnx {} {@var{b} =} gallery ("krylov", @var{a}, @var{x})
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209 ## @deftypefnx {} {@var{b} =} gallery ("krylov", @var{a}, @var{x}, @var{j})
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210 ## Create a Krylov matrix.
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211 ##
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212 ## @end deftypefn
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213 ##
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214 ## @deftypefn {} {@var{a} =} gallery ("lauchli", @var{n})
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215 ## @deftypefnx {} {@var{a} =} gallery ("lauchli", @var{n}, @var{mu})
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216 ## Create a @nospell{Lauchli} matrix (rectangular).
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217 ##
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218 ## @end deftypefn
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219 ##
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220 ## @deftypefn {} {@var{a} =} gallery ("lehmer", @var{n})
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221 ## Create a @nospell{Lehmer} matrix (symmetric positive definite).
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222 ##
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223 ## @end deftypefn
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224 ##
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225 ## @deftypefn {} {@var{t} =} gallery ("lesp", @var{n})
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226 ## Create a tridiagonal matrix with real, sensitive eigenvalues.
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227 ##
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228 ## @end deftypefn
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229 ##
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230 ## @deftypefn {} {@var{a} =} gallery ("lotkin", @var{n})
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231 ## Create a @nospell{Lotkin} matrix.
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232 ##
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233 ## @end deftypefn
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234 ##
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235 ## @deftypefn {} {@var{a} =} gallery ("minij", @var{n})
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236 ## Create a symmetric positive definite matrix MIN(i,j).
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237 ##
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238 ## @end deftypefn
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239 ##
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240 ## @deftypefn {} {@var{a} =} gallery ("moler", @var{n})
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241 ## @deftypefnx {} {@var{a} =} gallery ("moler", @var{n}, @var{alpha})
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242 ## Create a @nospell{Moler} matrix (symmetric positive definite).
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243 ##
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244 ## @end deftypefn
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245 ##
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246 ## @deftypefn {} {[@var{a}, @var{t}] =} gallery ("neumann", @var{n})
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247 ## Create a singular matrix from the discrete Neumann problem (sparse).
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248 ##
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249 ## @end deftypefn
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250 ##
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251 ## @deftypefn {} {@var{a} =} gallery ("normaldata", [@var{M} @var{N} @dots{}], @var{j})
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252 ## @deftypefnx {} {@var{a} =} gallery ("normaldata", @var{M}, @var{N}, @dots{}, @var{j})
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253 ## @deftypefnx {} {@var{a} =} gallery ("normaldata", @dots{}, "@var{class}")
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254 ## Create a matrix with random samples from the standard normal distribution
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255 ## (mean = 0, std = 1).
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256 ##
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257 ## The first input is a matrix of dimensions describing the size of the output.
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258 ## The dimensions can also be input as comma-separated arguments.
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259 ##
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260 ## The input @var{j} is an integer index in the range [0, 2^32-1]. The values
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261 ## of the output matrix are always exactly the same (reproducibility) for a
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262 ## given size input and @var{j} index.
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263 ##
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264 ## The final optional argument determines the class of the resulting matrix.
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265 ## Possible values for @var{class}: @qcode{"single"}, @qcode{"double"}.
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266 ## The default is @qcode{"double"}.
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267 ##
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268 ## @end deftypefn
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269 ##
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270 ## @deftypefn {} {@var{q} =} gallery ("orthog", @var{n})
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271 ## @deftypefnx {} {@var{q} =} gallery ("orthog", @var{n}, @var{k})
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272 ## Create orthogonal and nearly orthogonal matrices.
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273 ##
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274 ## @end deftypefn
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275 ##
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276 ## @deftypefn {} {@var{a} =} gallery ("parter", @var{n})
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277 ## Create a @nospell{Parter} matrix (a Toeplitz matrix with singular values
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278 ## near pi).
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279 ##
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280 ## @end deftypefn
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281 ##
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282 ## @deftypefn {} {@var{p} =} gallery ("pei", @var{n})
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283 ## @deftypefnx {} {@var{p} =} gallery ("pei", @var{n}, @var{alpha})
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284 ## Create a Pei matrix.
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285 ##
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286 ## @end deftypefn
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287 ##
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288 ## @deftypefn {} {@var{a} =} gallery ("Poisson", @var{n})
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289 ## Create a block tridiagonal matrix from Poisson's equation (sparse).
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290 ##
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291 ## @end deftypefn
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292 ##
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293 ## @deftypefn {} {@var{a} =} gallery ("prolate", @var{n})
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294 ## @deftypefnx {} {@var{a} =} gallery ("prolate", @var{n}, @var{w})
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295 ## Create a prolate matrix (symmetric, ill-conditioned Toeplitz matrix).
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296 ##
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297 ## @end deftypefn
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298 ##
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299 ## @deftypefn {} {@var{h} =} gallery ("randhess", @var{x})
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300 ## Create a random, orthogonal upper Hessenberg matrix.
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301 ##
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302 ## @end deftypefn
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303 ##
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304 ## @deftypefn {} {@var{a} =} gallery ("rando", @var{n})
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305 ## @deftypefnx {} {@var{a} =} gallery ("rando", @var{n}, @var{k})
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306 ## Create a random matrix with elements -1, 0 or 1.
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307 ##
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308 ## @end deftypefn
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309 ##
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310 ## @deftypefn {} {@var{a} =} gallery ("randsvd", @var{n})
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311 ## @deftypefnx {} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa})
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312 ## @deftypefnx {} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa}, @var{mode})
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313 ## @deftypefnx {} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa}, @var{mode}, @var{kl})
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314 ## @deftypefnx {} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa}, @var{mode}, @var{kl}, @var{ku})
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315 ## Create a random matrix with pre-assigned singular values.
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316 ##
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317 ## @end deftypefn
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318 ##
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319 ## @deftypefn {} {@var{a} =} gallery ("redheff", @var{n})
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320 ## Create a zero and ones matrix of @nospell{Redheffer} associated with the
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321 ## Riemann hypothesis.
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322 ##
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323 ## @end deftypefn
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324 ##
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325 ## @deftypefn {} {@var{a} =} gallery ("riemann", @var{n})
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326 ## Create a matrix associated with the Riemann hypothesis.
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327 ##
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328 ## @end deftypefn
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329 ##
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330 ## @deftypefn {} {@var{a} =} gallery ("ris", @var{n})
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331 ## Create a symmetric Hankel matrix.
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332 ##
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333 ## @end deftypefn
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334 ##
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335 ## @deftypefn {} {@var{a} =} gallery ("smoke", @var{n})
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336 ## @deftypefnx {} {@var{a} =} gallery ("smoke", @var{n}, @var{k})
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337 ## Create a complex matrix, with a `smoke ring' pseudospectrum.
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338 ##
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339 ## @end deftypefn
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340 ##
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341 ## @deftypefn {} {@var{t} =} gallery ("toeppd", @var{n})
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342 ## @deftypefnx {} {@var{t} =} gallery ("toeppd", @var{n}, @var{m})
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343 ## @deftypefnx {} {@var{t} =} gallery ("toeppd", @var{n}, @var{m}, @var{w})
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344 ## @deftypefnx {} {@var{t} =} gallery ("toeppd", @var{n}, @var{m}, @var{w}, @var{theta})
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345 ## Create a symmetric positive definite Toeplitz matrix.
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346 ##
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347 ## @end deftypefn
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348 ##
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349 ## @deftypefn {} {@var{p} =} gallery ("toeppen", @var{n})
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350 ## @deftypefnx {} {@var{p} =} gallery ("toeppen", @var{n}, @var{a})
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351 ## @deftypefnx {} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b})
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352 ## @deftypefnx {} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b}, @var{c})
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353 ## @deftypefnx {} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b}, @var{c}, @var{d})
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354 ## @deftypefnx {} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b}, @var{c}, @var{d}, @var{e})
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355 ## Create a pentadiagonal Toeplitz matrix (sparse).
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356 ##
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357 ## @end deftypefn
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358 ##
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359 ## @deftypefn {} {@var{a} =} gallery ("tridiag", @var{x}, @var{y}, @var{z})
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360 ## @deftypefnx {} {@var{a} =} gallery ("tridiag", @var{n})
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361 ## @deftypefnx {} {@var{a} =} gallery ("tridiag", @var{n}, @var{c}, @var{d}, @var{e})
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362 ## Create a tridiagonal matrix (sparse).
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363 ##
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364 ## @end deftypefn
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365 ##
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366 ## @deftypefn {} {@var{t} =} gallery ("triw", @var{n})
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367 ## @deftypefnx {} {@var{t} =} gallery ("triw", @var{n}, @var{alpha})
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368 ## @deftypefnx {} {@var{t} =} gallery ("triw", @var{n}, @var{alpha}, @var{k})
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369 ## Create an upper triangular matrix discussed by
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370 ## @nospell{Kahan, Golub, and Wilkinson}.
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371 ##
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372 ## @end deftypefn
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373 ##
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374 ## @deftypefn {} {@var{a} =} gallery ("uniformdata", [@var{M} @var{N} @dots{}], @var{j})
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375 ## @deftypefnx {} {@var{a} =} gallery ("uniformdata", @var{M}, @var{N}, @dots{}, @var{j})
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376 ## @deftypefnx {} {@var{a} =} gallery ("uniformdata", @dots{}, "@var{class}")
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377 ## Create a matrix with random samples from the standard uniform distribution
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378 ## (range [0,1]).
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379 ##
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380 ## The first input is a matrix of dimensions describing the size of the output.
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381 ## The dimensions can also be input as comma-separated arguments.
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382 ##
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383 ## The input @var{j} is an integer index in the range [0, 2^32-1]. The values
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384 ## of the output matrix are always exactly the same (reproducibility) for a
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385 ## given size input and @var{j} index.
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386 ##
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387 ## The final optional argument determines the class of the resulting matrix.
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388 ## Possible values for @var{class}: @qcode{"single"}, @qcode{"double"}.
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389 ## The default is @qcode{"double"}.
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390 ##
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391 ## @end deftypefn
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392 ##
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393 ## @deftypefn {} {@var{a} =} gallery ("wathen", @var{nx}, @var{ny})
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394 ## @deftypefnx {} {@var{a} =} gallery ("wathen", @var{nx}, @var{ny}, @var{k})
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395 ## Create the @nospell{Wathen} matrix.
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396 ##
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397 ## @end deftypefn
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398 ##
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399 ## @deftypefn {} {[@var{a}, @var{b}] =} gallery ("wilk", @var{n})
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400 ## Create various specific matrices devised/discussed by Wilkinson.
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401 ##
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402 ## @end deftypefn
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403
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404 ## Code for most of the individual matrices (except binomial, gcdmat,
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405 ## integerdata, leslie, normaldata, randcolu, randcorr, randjorth, sampling,
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406 ## uniformdata) by Nicholas J. Higham <Nicholas.J.Higham@manchester.ac.uk>
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407 ## Adapted for Octave and into single gallery function by Carnë Draug
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408
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409 function [varargout] = gallery (name, varargin)
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410
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411 if (nargin < 1)
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412 print_usage ();
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413 elseif (! ischar (name))
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414 error ("gallery: NAME must be a string.");
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415 endif
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416
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417 ## NOTE: there isn't a lot of input check in the individual functions
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418 ## that actually build the functions. This is by design. The original
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419 ## code by Higham did not perform it and was propagated to Matlab, so
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420 ## for compatibility, we also don't make it. For example, arguments
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421 ## that behave as switches, and in theory accepting a value of 0 or 1,
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422 ## will use a value of 0, for any value other than 1 (only check made
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Carnë Draug <carandraug@octave.org>
parents:
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423 ## is if the value is equal to 1). It will often also accept string
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424 ## values instead of numeric. Only input check added was where it
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425 ## would be causing an error anyway.
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426
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427 ## we will always want to return at least 1 output
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428 n_out = nargout;
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429 if (n_out == 0)
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430 n_out = 1;
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431 endif
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432
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433 switch (tolower (name))
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434 case "binomial"
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435 error ("gallery: matrix %s not implemented.", name);
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436 case "cauchy" , [varargout{1:n_out}] = cauchy (varargin{:});
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437 case "chebspec" , [varargout{1:n_out}] = chebspec (varargin{:});
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438 case "chebvand" , [varargout{1:n_out}] = chebvand (varargin{:});
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439 case "chow" , [varargout{1:n_out}] = chow (varargin{:});
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440 case "circul" , [varargout{1:n_out}] = circul (varargin{:});
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441 case "clement" , [varargout{1:n_out}] = clement (varargin{:});
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442 case "compar" , [varargout{1:n_out}] = compar (varargin{:});
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443 case "condex" , [varargout{1:n_out}] = condex (varargin{:});
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444 case "cycol" , [varargout{1:n_out}] = cycol (varargin{:});
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445 case "dorr" , [varargout{1:n_out}] = dorr (varargin{:});
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446 case "dramadah" , [varargout{1:n_out}] = dramadah (varargin{:});
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447 case "fiedler" , [varargout{1:n_out}] = fiedler (varargin{:});
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448 case "forsythe" , [varargout{1:n_out}] = forsythe (varargin{:});
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449 case "frank" , [varargout{1:n_out}] = frank (varargin{:});
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450 case "gearmat" , [varargout{1:n_out}] = gearmat (varargin{:});
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451 case "gcdmat" , [varargout{1:n_out}] = gcdmat (varargin{:});
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452 case "grcar" , [varargout{1:n_out}] = grcar (varargin{:});
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453 case "hanowa" , [varargout{1:n_out}] = hanowa (varargin{:});
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454 case "house" , [varargout{1:n_out}] = house (varargin{:});
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455 case "integerdata", [varargout{1:n_out}] = integerdata (varargin{:});
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456 case "invhess" , [varargout{1:n_out}] = invhess (varargin{:});
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457 case "invol" , [varargout{1:n_out}] = invol (varargin{:});
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458 case "ipjfact" , [varargout{1:n_out}] = ipjfact (varargin{:});
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459 case "jordbloc" , [varargout{1:n_out}] = jordbloc (varargin{:});
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460 case "kahan" , [varargout{1:n_out}] = kahan (varargin{:});
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461 case "kms" , [varargout{1:n_out}] = kms (varargin{:});
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462 case "krylov" , [varargout{1:n_out}] = krylov (varargin{:});
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463 case "lauchli" , [varargout{1:n_out}] = lauchli (varargin{:});
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464 case "lehmer" , [varargout{1:n_out}] = lehmer (varargin{:});
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465 case "leslie"
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466 error ("gallery: matrix %s not implemented.", name);
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467 case "lesp" , [varargout{1:n_out}] = lesp (varargin{:});
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468 case "lotkin" , [varargout{1:n_out}] = lotkin (varargin{:});
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469 case "minij" , [varargout{1:n_out}] = minij (varargin{:});
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470 case "moler" , [varargout{1:n_out}] = moler (varargin{:});
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471 case "neumann" , [varargout{1:n_out}] = neumann (varargin{:});
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472 case "normaldata" , [varargout{1:n_out}] = normaldata (varargin{:});
16634
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473 case "orthog" , [varargout{1:n_out}] = orthog (varargin{:});
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474 case "parter" , [varargout{1:n_out}] = parter (varargin{:});
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475 case "pei" , [varargout{1:n_out}] = pei (varargin{:});
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476 case "poisson" , [varargout{1:n_out}] = poisson (varargin{:});
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477 case "prolate" , [varargout{1:n_out}] = prolate (varargin{:});
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478 case "randcolu"
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479 error ("gallery: matrix %s not implemented.", name);
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480 case "randcorr"
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481 error ("gallery: matrix %s not implemented.", name);
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482 case "randhess" , [varargout{1:n_out}] = randhess (varargin{:});
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diff changeset
483 case "randjorth"
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diff changeset
484 error ("gallery: matrix %s not implemented.", name);
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485 case "rando" , [varargout{1:n_out}] = rando (varargin{:});
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486 case "randsvd" , [varargout{1:n_out}] = randsvd (varargin{:});
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487 case "redheff" , [varargout{1:n_out}] = redheff (varargin{:});
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488 case "riemann" , [varargout{1:n_out}] = riemann (varargin{:});
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489 case "ris" , [varargout{1:n_out}] = ris (varargin{:});
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diff changeset
490 case "sampling"
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parents:
diff changeset
491 error ("gallery: matrix %s not implemented.", name);
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diff changeset
492 case "smoke" , [varargout{1:n_out}] = smoke (varargin{:});
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parents:
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493 case "toeppd" , [varargout{1:n_out}] = toeppd (varargin{:});
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494 case "toeppen" , [varargout{1:n_out}] = toeppen (varargin{:});
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495 case "tridiag" , [varargout{1:n_out}] = tridiag (varargin{:});
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496 case "triw" , [varargout{1:n_out}] = triw (varargin{:});
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parents: 16933
diff changeset
497 case "uniformdata" , [varargout{1:n_out}] = uniformdata (varargin{:});
16634
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498 case "wathen" , [varargout{1:n_out}] = wathen (varargin{:});
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parents:
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499 case "wilk" , [varargout{1:n_out}] = wilk (varargin{:});
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parents:
diff changeset
500 otherwise
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diff changeset
501 error ("gallery: unknown matrix with NAME %s", name);
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parents:
diff changeset
502 endswitch
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parents:
diff changeset
503
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parents:
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504 endfunction
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parents:
diff changeset
505
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parents:
diff changeset
506 function C = cauchy (x, y)
21562
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
507 ## CAUCHY Cauchy matrix.
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
508 ## C = CAUCHY(X, Y), where X, Y are N-vectors, is the N-by-N matrix
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
509 ## with C(i,j) = 1/(X(i)+Y(j)). By default, Y = X.
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
510 ## Special case: if X is a scalar CAUCHY(X) is the same as CAUCHY(1:X).
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
511 ## Explicit formulas are known for DET(C) (which is nonzero if X and Y
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
512 ## both have distinct elements) and the elements of INV(C).
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
513 ## C is totally positive if 0 < X(1) < ... < X(N) and
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
514 ## 0 < Y(1) < ... < Y(N).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
515 ##
21562
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
516 ## References:
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
517 ## N.J. Higham, Accuracy and Stability of Numerical Algorithms,
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
518 ## Society for Industrial and Applied Mathematics, Philadelphia, PA,
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
519 ## USA, 1996; sec. 26.1.
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
520 ## D.E. Knuth, The Art of Computer Programming, Volume 1,
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
521 ## Fundamental Algorithms, second edition, Addison-Wesley, Reading,
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
522 ## Massachusetts, 1973, p. 36.
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
523 ## E.E. Tyrtyshnikov, Cauchy-Toeplitz matrices and some applications,
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
524 ## Linear Algebra and Appl., 149 (1991), pp. 1-18.
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
525 ## O. Taussky and M. Marcus, Eigenvalues of finite matrices, in
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
526 ## Survey of Numerical Analysis, J. Todd, ed., McGraw-Hill, New York,
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21546
diff changeset
527 ## pp. 279-313, 1962. (States the totally positive property on p. 295.)
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
528
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
529 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
530 error ("gallery: 1 or 2 arguments are required for cauchy matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
531 elseif (! isnumeric (x))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
532 error ("gallery: X must be numeric for cauchy matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
533 elseif (nargin == 2 && ! isnumeric (y))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
534 error ("gallery: Y must be numeric for cauchy matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
535 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
536
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
537 n = numel (x);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
538 if (isscalar (x) && fix (x) == x)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
539 n = x;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
540 x = 1:n;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
541 elseif (n > 1 && isvector (x))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
542 ## do nothing
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
543 else
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
544 error ("gallery: X be an integer or a vector for cauchy matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
545 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
546
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
547 if (nargin == 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
548 y = x;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
549 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
550
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
551 ## Ensure x and y are column vectors
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
552 x = x(:);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
553 y = y(:);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
554 if (numel (x) != numel (y))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
555 error ("gallery: X and Y must be vectors of same length for cauchy matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
556 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
557
23381
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
558 C = 1 ./ (x .+ y.');
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
559 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
560
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
561 function C = chebspec (n, k = 0)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
562 ## CHEBSPEC Chebyshev spectral differentiation matrix.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
563 ## C = CHEBSPEC(N, K) is a Chebyshev spectral differentiation
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
564 ## matrix of order N. K = 0 (the default) or 1.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
565 ## For K = 0 (`no boundary conditions'), C is nilpotent, with
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
566 ## C^N = 0 and it has the null vector ONES(N,1).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
567 ## C is similar to a Jordan block of size N with eigenvalue zero.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
568 ## For K = 1, C is nonsingular and well-conditioned, and its eigenvalues
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
569 ## have negative real parts.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
570 ## For both K, the computed eigenvector matrix X from EIG is
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
571 ## ill-conditioned (MESH(REAL(X)) is interesting).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
572 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
573 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
574 ## C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
575 ## Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988; p. 69.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
576 ## L.N. Trefethen and M.R. Trummer, An instability phenomenon in
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
577 ## spectral methods, SIAM J. Numer. Anal., 24 (1987), pp. 1008-1023.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
578 ## D. Funaro, Computing the inverse of the Chebyshev collocation
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
579 ## derivative, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 1050-1057.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
580
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
581 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
582 error ("gallery: 1 to 2 arguments are required for chebspec matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
583 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
584 error ("gallery: N must be an integer for chebspec matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
585 elseif (! isnumeric (k) || ! isscalar (k))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
586 error ("gallery: K must be a scalar for chebspec matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
587 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
588
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
589 ## k = 1 case obtained from k = 0 case with one bigger n.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
590 switch (k)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
591 case (0), # do nothing
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
592 case (1), n = n + 1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
593 otherwise
20294
2102c4582e32 gallery: fix chebspec, cycol, gearmat, hanowa, lauchli, and pei (bug #45466)
Massimiliano Fasi <massimiliano.fasi@gmail.com>
parents: 20231
diff changeset
594 error ("gallery: K should be either 0 or 1 for chebspec matrix.");
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
595 endswitch
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
596
20231
83792dd9bcc1 Use in-place operators in m-files where possible.
Rik <rik@octave.org>
parents: 20162
diff changeset
597 n -= 1;
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
598 C = zeros (n+1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
599
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
600 one = ones (n+1, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
601 x = cos ((0:n)' * (pi/n));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
602 d = ones (n+1, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
603 d(1) = 2;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
604 d(n+1) = 2;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
605
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
606 ## eye(size(C)) on next line avoids div by zero.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
607 C = (d * (one./d)') ./ (x*one'-one*x' + eye (size (C)));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
608
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
609 ## Now fix diagonal and signs.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
610 C(1,1) = (2*n^2+1)/6;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
611 for i = 2:n+1
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
612 if (rem (i, 2) == 0)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
613 C(:,i) = -C(:,i);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
614 C(i,:) = -C(i,:);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
615 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
616 if (i < n+1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
617 C(i,i) = -x(i)/(2*(1-x(i)^2));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
618 else
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
619 C(n+1,n+1) = -C(1,1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
620 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
621 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
622
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
623 if (k == 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
624 C = C(2:n+1,2:n+1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
625 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
626 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
627
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
628 function C = chebvand (m, p)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
629 ## CHEBVAND Vandermonde-like matrix for the Chebyshev polynomials.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
630 ## C = CHEBVAND(P), where P is a vector, produces the (primal)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
631 ## Chebyshev Vandermonde matrix based on the points P,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
632 ## i.e., C(i,j) = T_{i-1}(P(j)), where T_{i-1} is the Chebyshev
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
633 ## polynomial of degree i-1.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
634 ## CHEBVAND(M,P) is a rectangular version of CHEBVAND(P) with M rows.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
635 ## Special case: If P is a scalar then P equally spaced points on
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
636 ## [0,1] are used.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
637 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
638 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
639 ## N.J. Higham, Stability analysis of algorithms for solving confluent
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
640 ## Vandermonde-like systems, SIAM J. Matrix Anal. Appl., 11 (1990),
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
641 ## pp. 23-41.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
642
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
643 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
644 error ("gallery: 1 or 2 arguments are required for chebvand matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
645 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
646
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
647 ## because the order of the arguments changes if nargin is 1 or 2 ...
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
648
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
649 if (nargin == 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
650 p = m;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
651 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
652
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
653 n = numel (p);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
654 if (! isnumeric (p))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
655 error ("gallery: P must be numeric for chebvand matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
656 elseif (isscalar (p) && fix (p) == p)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
657 n = p;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
658 p = linspace (0, 1, n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
659 elseif (n > 1 && isvector (p))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
660 ## do nothing
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
661 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
662 p = p(:).'; # Ensure p is a row vector.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
663
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
664 if (nargin == 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
665 m = n;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
666 elseif (! isnumeric (m) || ! isscalar (m))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
667 error ("gallery: M must be a scalar for chebvand matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
668 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
669
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
670 C = ones (m, n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
671 if (m != 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
672 C(2,:) = p;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
673 ## Use Chebyshev polynomial recurrence.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
674 for i = 3:m
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
675 C(i,:) = 2.*p.*C(i-1,:) - C(i-2,:);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
676 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
677 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
678 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
679
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
680 function A = chow (n, alpha = 1, delta = 0)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
681 ## CHOW Chow matrix - a singular Toeplitz lower Hessenberg matrix.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
682 ## A = CHOW(N, ALPHA, DELTA) is a Toeplitz lower Hessenberg matrix
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
683 ## A = H(ALPHA) + DELTA*EYE, where H(i,j) = ALPHA^(i-j+1).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
684 ## H(ALPHA) has p = FLOOR(N/2) zero eigenvalues, the rest being
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
685 ## 4*ALPHA*COS( k*PI/(N+2) )^2, k=1:N-p.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
686 ## Defaults: ALPHA = 1, DELTA = 0.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
687 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
688 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
689 ## T.S. Chow, A class of Hessenberg matrices with known
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
690 ## eigenvalues and inverses, SIAM Review, 11 (1969), pp. 391-395.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
691 ## G. Fairweather, On the eigenvalues and eigenvectors of a class of
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
692 ## Hessenberg matrices, SIAM Review, 13 (1971), pp. 220-221.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
693
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
694 if (nargin < 1 || nargin > 3)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
695 error ("gallery: 1 to 3 arguments are required for chow matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
696 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
697 error ("gallery: N must be an integer for chow matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
698 elseif (! isnumeric (alpha) || ! isscalar (alpha))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
699 error ("gallery: ALPHA must be a scalar for chow matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
700 elseif (! isnumeric (delta) || ! isscalar (delta))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
701 error ("gallery: DELTA must be a scalar for chow matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
702 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
703
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
704 A = toeplitz (alpha.^(1:n), [alpha 1 zeros(1, n-2)]) + delta * eye (n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
705 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
706
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
707 function C = circul (v)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
708 ## CIRCUL Circulant matrix.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
709 ## C = CIRCUL(V) is the circulant matrix whose first row is V.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
710 ## (A circulant matrix has the property that each row is obtained
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
711 ## from the previous one by cyclically permuting the entries one step
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
712 ## forward; it is a special Toeplitz matrix in which the diagonals
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
713 ## `wrap round'.)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
714 ## Special case: if V is a scalar then C = CIRCUL(1:V).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
715 ## The eigensystem of C (N-by-N) is known explicitly. If t is an Nth
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
716 ## root of unity, then the inner product of V with W = [1 t t^2 ... t^N]
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
717 ## is an eigenvalue of C, and W(N:-1:1) is an eigenvector of C.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
718 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
719 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
720 ## P.J. Davis, Circulant Matrices, John Wiley, 1977.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
721
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
722 if (nargin != 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
723 error ("gallery: 1 argument is required for circul matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
724 elseif (! isnumeric (v))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
725 error ("gallery: V must be numeric for circul matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
726 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
727
16734
67b67fc0969a gallery: fix bug from typo in variable name
Carnë Draug <carandraug@octave.org>
parents: 16634
diff changeset
728 n = numel (v);
67b67fc0969a gallery: fix bug from typo in variable name
Carnë Draug <carandraug@octave.org>
parents: 16634
diff changeset
729 if (isscalar (v) && fix (v) == v)
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
730 n = v;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
731 v = 1:n;
16734
67b67fc0969a gallery: fix bug from typo in variable name
Carnë Draug <carandraug@octave.org>
parents: 16634
diff changeset
732 elseif (n > 1 && isvector (v))
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
733 ## do nothing
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
734 else
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
735 error ("gallery: X must be a scalar or a vector for circul matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
736 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
737
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
738 v = v(:).'; # Make sure v is a row vector
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
739 C = toeplitz ([v(1) v(n:-1:2)], v);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
740 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
741
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
742 function A = clement (n, k = 0)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
743 ## CLEMENT Clement matrix - tridiagonal with zero diagonal entries.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
744 ## CLEMENT(N, K) is a tridiagonal matrix with zero diagonal entries
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
745 ## and known eigenvalues. It is singular if N is odd. About 64
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
746 ## percent of the entries of the inverse are zero. The eigenvalues
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
747 ## are plus and minus the numbers N-1, N-3, N-5, ..., (1 or 0).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
748 ## For K = 0 (the default) the matrix is unsymmetric, while for
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
749 ## K = 1 it is symmetric.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
750 ## CLEMENT(N, 1) is diagonally similar to CLEMENT(N).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
751 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
752 ## Similar properties hold for TRIDIAG(X,Y,Z) where Y = ZEROS(N,1).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
753 ## The eigenvalues still come in plus/minus pairs but they are not
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
754 ## known explicitly.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
755 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
756 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
757 ## P.A. Clement, A class of triple-diagonal matrices for test
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
758 ## purposes, SIAM Review, 1 (1959), pp. 50-52.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
759 ## A. Edelman and E. Kostlan, The road from Kac's matrix to Kac's
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
760 ## random polynomials. In John~G. Lewis, editor, Proceedings of
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
761 ## the Fifth SIAM Conference on Applied Linear Algebra Society
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
762 ## for Industrial and Applied Mathematics, Philadelphia, 1994,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
763 ## pp. 503-507.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
764 ## O. Taussky and J. Todd, Another look at a matrix of Mark Kac,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
765 ## Linear Algebra and Appl., 150 (1991), pp. 341-360.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
766
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
767 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
768 error ("gallery: 1 or 2 arguments are required for clement matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
769 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
770 error ("gallery: N must be an integer for clement matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
771 elseif (! isnumeric (k) || ! isscalar (k))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
772 error ("gallery: K must be a numeric scalar for clement matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
773 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
774
20231
83792dd9bcc1 Use in-place operators in m-files where possible.
Rik <rik@octave.org>
parents: 20162
diff changeset
775 n -= 1;
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
776 x = n:-1:1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
777 z = 1:n;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
778
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
779 if (k == 0)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
780 A = diag (x, -1) + diag (z, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
781 elseif (k == 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
782 y = sqrt (x.*z);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
783 A = diag (y, -1) + diag (y, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
784 else
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
785 error ("gallery: K must have a value of 0 or 1 for clement matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
786 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
787 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
788
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
789 function C = compar (A, k = 0)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
790 ## COMP Comparison matrices.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
791 ## COMP(A) is DIAG(B) - TRIL(B,-1) - TRIU(B,1), where B = ABS(A).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
792 ## COMP(A, 1) is A with each diagonal element replaced by its
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
793 ## absolute value, and each off-diagonal element replaced by minus
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
794 ## the absolute value of the largest element in absolute value in
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
795 ## its row. However, if A is triangular COMP(A, 1) is too.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
796 ## COMP(A, 0) is the same as COMP(A).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
797 ## COMP(A) is often denoted by M(A) in the literature.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
798 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
799 ## Reference (e.g.):
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
800 ## N.J. Higham, A survey of condition number estimation for
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
801 ## triangular matrices, SIAM Review, 29 (1987), pp. 575-596.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
802
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
803 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
804 error ("gallery: 1 or 2 arguments are required for compar matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
805 elseif (! isnumeric (A) || ndims (A) != 2)
17199
9deb214ae9d5 Use 2-D, not 2D, in error messages.
Rik <rik@octave.org>
parents: 16982
diff changeset
806 error ("gallery: A must be a 2-D matrix for compar matrix.");
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
807 elseif (! isnumeric (k) || ! isscalar (k))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
808 error ("gallery: K must be a numeric scalar for compar matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
809 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
810
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
811 [m, n] = size (A);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
812 p = min (m, n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
813
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
814 if (k == 0)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
815 ## This code uses less temporary storage than
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
816 ## the `high level' definition above.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
817 C = -abs (A);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
818 for j = 1:p
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
819 C(j,j) = abs (A(j,j));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
820 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
821
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
822 elseif (k == 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
823 C = A';
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
824 for j = 1:p
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
825 C(k,k) = 0;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
826 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
827 mx = max (abs (C));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
828 C = -mx'*ones (1, n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
829 for j = 1:p
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
830 C(j,j) = abs (A(j,j));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
831 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
832 if (all (A == tril (A))), C = tril (C); endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
833 if (all (A == triu (A))), C = triu (C); endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
834
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
835 else
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
836 error ("gallery: K must have a value of 0 or 1 for compar matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
837 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
838
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
839 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
840
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
841 function A = condex (n, k = 4, theta = 100)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
842 ## CONDEX `Counterexamples' to matrix condition number estimators.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
843 ## CONDEX(N, K, THETA) is a `counterexample' matrix to a condition
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
844 ## estimator. It has order N and scalar parameter THETA (default 100).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
845 ## If N is not equal to the `natural' size of the matrix then
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
846 ## the matrix is padded out with an identity matrix to order N.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
847 ## The matrix, its natural size, and the estimator to which it applies
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
848 ## are specified by K (default K = 4) as follows:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
849 ## K = 1: 4-by-4, LINPACK (RCOND)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
850 ## K = 2: 3-by-3, LINPACK (RCOND)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
851 ## K = 3: arbitrary, LINPACK (RCOND) (independent of THETA)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
852 ## K = 4: N >= 4, SONEST (Higham 1988)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
853 ## (Note that in practice the K = 4 matrix is not usually a
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
854 ## counterexample because of the rounding errors in forming it.)
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
855 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
856 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
857 ## A.K. Cline and R.K. Rew, A set of counter-examples to three
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
858 ## condition number estimators, SIAM J. Sci. Stat. Comput.,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
859 ## 4 (1983), pp. 602-611.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
860 ## N.J. Higham, FORTRAN codes for estimating the one-norm of a real or
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
861 ## complex matrix, with applications to condition estimation
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
862 ## (Algorithm 674), ACM Trans. Math. Soft., 14 (1988), pp. 381-396.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
863
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
864 if (nargin < 1 || nargin > 3)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
865 error ("gallery: 1 to 3 arguments are required for condex matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
866 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
867 error ("gallery: N must be an integer for condex matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
868 elseif (! isnumeric (k) || ! isscalar (k))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
869 error ("gallery: K must be a numeric scalar for condex matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
870 elseif (! isnumeric (theta) || ! isscalar (theta))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
871 error ("gallery: THETA must be a numeric scalar for condex matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
872 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
873
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
874 if (k == 1) # Cline and Rew (1983), Example B.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
875 A = [1 -1 -2*theta 0
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
876 0 1 theta -theta
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
877 0 1 1+theta -(theta+1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
878 0 0 0 theta];
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
879
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
880 elseif (k == 2) # Cline and Rew (1983), Example C.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
881 A = [1 1-2/theta^2 -2
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
882 0 1/theta -1/theta
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
883 0 0 1];
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
884
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
885 elseif (k == 3) # Cline and Rew (1983), Example D.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
886 A = gallery ("triw", n, -1)';
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
887 A(n,n) = -1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
888
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
889 elseif (k == 4) # Higham (1988), p. 390.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
890 x = ones (n, 3); # First col is e
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
891 x(2:n,2) = zeros (n-1, 1); # Second col is e(1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
892
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
893 ## Third col is special vector b in SONEST
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
894 x(:, 3) = (-1).^[0:n-1]' .* ( 1 + [0:n-1]'/(n-1) );
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
895
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
896 Q = orth (x); # Q*Q' is now the orthogonal projector onto span(e(1),e,b)).
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
897 P = eye (n) - Q*Q';
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
898 A = eye (n) + theta*P;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
899
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
900 else
16766
7268845c0a1e avoid backquote in error messages, some uses in doc strings
John W. Eaton <jwe@octave.org>
parents: 16734
diff changeset
901 error ("gallery: unknown estimator K '%d' for condex matrix.", k);
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
902 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
903
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
904 ## Pad out with identity as necessary.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
905 m = columns (A);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
906 if (m < n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
907 for i = n:-1:m+1
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
908 A(i,i) = 1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
909 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
910 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
911 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
912
20298
ababbe103048 gallery: allow N to be a 2 element vectors for cycol matrices.
Carnë Draug <carandraug@octave.org>
parents: 20297
diff changeset
913 function A = cycol (n, k = max (round (n(end)/4), 1))
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
914 ## CYCOL Matrix whose columns repeat cyclically.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
915 ## A = CYCOL([M N], K) is an M-by-N matrix of the form A = B(1:M,1:N)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
916 ## where B = [C C C...] and C = RANDN(M, K). Thus A's columns repeat
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
917 ## cyclically, and A has rank at most K. K need not divide N.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
918 ## K defaults to ROUND(N/4).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
919 ## CYCOL(N, K), where N is a scalar, is the same as CYCOL([N N], K).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
920 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
921 ## This type of matrix can lead to underflow problems for Gaussian
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
922 ## elimination: see NA Digest Volume 89, Issue 3 (January 22, 1989).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
923
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
924 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
925 error ("gallery: 1 or 2 arguments are required for cycol matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
926 elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
927 error ("gallery: N must be a 1 or 2 element integer for cycol matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
928 elseif (! isnumeric (k) || ! isscalar (k))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
929 error ("gallery: K must be a scalar for cycol matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
930 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
931
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
932 ## Parameter n specifies dimension: m-by-n
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
933 m = n(1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
934 n = n(end);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
935
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
936 A = randn (m, k);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
937 for i = 2:ceil (n/k)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
938 A = [A A(:,1:k)];
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
939 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
940 A = A(:,1:n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
941 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
942
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
943 function [c, d, e] = dorr (n, theta = 0.01)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
944 ## DORR Dorr matrix - diagonally dominant, ill conditioned, tridiagonal.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
945 ## [C, D, E] = DORR(N, THETA) returns the vectors defining a row diagonally
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
946 ## dominant, tridiagonal M-matrix that is ill conditioned for small
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
947 ## values of the parameter THETA >= 0.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
948 ## If only one output parameter is supplied then
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
949 ## C = FULL(TRIDIAG(C,D,E)), i.e., the matrix iself is returned.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
950 ## The columns of INV(C) vary greatly in norm. THETA defaults to 0.01.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
951 ## The amount of diagonal dominance is given by (ignoring rounding errors):
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
952 ## COMP(C)*ONES(N,1) = THETA*(N+1)^2 * [1 0 0 ... 0 1]'.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
953 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
954 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
955 ## F.W. Dorr, An example of ill-conditioning in the numerical
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
956 ## solution of singular perturbation problems, Math. Comp., 25 (1971),
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
957 ## pp. 271-283.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
958
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
959 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
960 error ("gallery: 1 or 2 arguments are required for dorr matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
961 elseif (! isscalar (n) || ! isnumeric (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
962 error ("gallery: N must be an integer for dorr matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
963 elseif (! isscalar (theta) || ! isnumeric (theta))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
964 error ("gallery: THETA must be a numeric scalar for dorr matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
965 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
966
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
967 c = zeros (n, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
968 e = c;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
969 d = c;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
970 ## All length n for convenience. Make c, e of length n-1 later.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
971
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
972 h = 1/(n+1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
973 m = floor ((n+1)/2);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
974 term = theta/h^2;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
975
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
976 i = (1:m)';
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
977 c(i) = -term * ones (m, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
978 e(i) = c(i) - (0.5-i*h)/h;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
979 d(i) = -(c(i) + e(i));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
980
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
981 i = (m+1:n)';
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
982 e(i) = -term * ones (n-m, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
983 c(i) = e(i) + (0.5-i*h)/h;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
984 d(i) = -(c(i) + e(i));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
985
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
986 c = c(2:n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
987 e = e(1:n-1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
988
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
989 if (nargout <= 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
990 c = tridiag (c, d, e);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
991 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
992 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
993
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
994 function A = dramadah (n, k = 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
995 ## DRAMADAH A (0,1) matrix whose inverse has large integer entries.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
996 ## An anti-Hadamard matrix A is a matrix with elements 0 or 1 for
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
997 ## which MU(A) := NORM(INV(A),'FRO') is maximal.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
998 ## A = DRAMADAH(N, K) is an N-by-N (0,1) matrix for which MU(A) is
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
999 ## relatively large, although not necessarily maximal.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1000 ## Available types (the default is K = 1):
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1001 ## K = 1: A is Toeplitz, with ABS(DET(A)) = 1, and MU(A) > c(1.75)^N,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1002 ## where c is a constant.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1003 ## K = 2: A is upper triangular and Toeplitz.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1004 ## The inverses of both types have integer entries.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1005 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1006 ## Another interesting (0,1) matrix:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1007 ## K = 3: A has maximal determinant among (0,1) lower Hessenberg
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1008 ## matrices: det(A) = the n'th Fibonacci number. A is Toeplitz.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1009 ## The eigenvalues have an interesting distribution in the complex
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1010 ## plane.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1011 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1012 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1013 ## R.L. Graham and N.J.A. Sloane, Anti-Hadamard matrices,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1014 ## Linear Algebra and Appl., 62 (1984), pp. 113-137.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1015 ## L. Ching, The maximum determinant of an nxn lower Hessenberg
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1016 ## (0,1) matrix, Linear Algebra and Appl., 183 (1993), pp. 147-153.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1017
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1018 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1019 error ("gallery: 1 to 2 arguments are required for dramadah matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1020 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1021 error ("gallery: N must be an integer for dramadah matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1022 elseif (! isnumeric (k) || ! isscalar (k))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1023 error ("gallery: K must be a numeric scalar for dramadah matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1024 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1025
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1026 switch (k)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1027 case (1) # Toeplitz
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1028 c = ones (n, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1029 for i = 2:4:n
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1030 m = min (1, n-i);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1031 c(i:i+m) = zeros (m+1, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1032 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1033 r = zeros (n, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1034 r(1:4) = [1 1 0 1];
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1035 if (n < 4)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1036 r = r(1:n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1037 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1038 A = toeplitz (c, r);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1039
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1040 case (2) # Upper triangular and Toeplitz
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1041 c = zeros (n, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1042 c(1) = 1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1043 r = ones (n, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1044 for i= 3:2:n
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1045 r(i) = 0;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1046 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1047 A = toeplitz (c, r);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1048
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1049 case (3) # Lower Hessenberg
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1050 c = ones (n, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1051 for i= 2:2:n
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1052 c(i) = 0;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1053 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1054 A = toeplitz (c, [1 1 zeros(1,n-2)]);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1055
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1056 otherwise
16766
7268845c0a1e avoid backquote in error messages, some uses in doc strings
John W. Eaton <jwe@octave.org>
parents: 16734
diff changeset
1057 error ("gallery: unknown K '%d' for dramadah matrix.", k);
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1058 endswitch
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1059 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1060
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1061 function A = fiedler (c)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1062 ## FIEDLER Fiedler matrix - symmetric.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1063 ## FIEDLER(C), where C is an n-vector, is the n-by-n symmetric
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1064 ## matrix with elements ABS(C(i)-C(j)).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1065 ## Special case: if C is a scalar, then A = FIEDLER(1:C)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1066 ## (i.e. A(i,j) = ABS(i-j)).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1067 ## Properties:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1068 ## FIEDLER(N) has a dominant positive eigenvalue and all the other
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1069 ## eigenvalues are negative (Szego, 1936).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1070 ## Explicit formulas for INV(A) and DET(A) are given by Todd (1977)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1071 ## and attributed to Fiedler. These indicate that INV(A) is
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1072 ## tridiagonal except for nonzero (1,n) and (n,1) elements.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1073 ## [I think these formulas are valid only if the elements of
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1074 ## C are in increasing or decreasing order---NJH.]
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1075 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1076 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1077 ## G. Szego, Solution to problem 3705, Amer. Math. Monthly,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1078 ## 43 (1936), pp. 246-259.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1079 ## J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1080 ## Birkhauser, Basel, and Academic Press, New York, 1977, p. 159.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1081
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1082 if (nargin != 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1083 error ("gallery: 1 argument is required for fiedler matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1084 elseif (! isnumeric (c))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1085 error ("gallery: C must be numeric for fiedler matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1086 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1087
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1088 n = numel (c);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1089 if (isscalar (c) && fix (c) == c)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1090 n = c;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1091 c = 1:n;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1092 elseif (n > 1 && isvector (c))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1093 ## do nothing
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1094 else
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1095 error ("gallery: C must be an integer or a vector for fiedler matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1096 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1097 c = c(:).'; # Ensure c is a row vector.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1098
23381
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
1099 A = abs (c - c.');
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1100 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1101
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1102 function A = forsythe (n, alpha = sqrt (eps), lambda = 0)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1103 ## FORSYTHE Forsythe matrix - a perturbed Jordan block.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1104 ## FORSYTHE(N, ALPHA, LAMBDA) is the N-by-N matrix equal to
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1105 ## JORDBLOC(N, LAMBDA) except it has an ALPHA in the (N,1) position.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1106 ## It has the characteristic polynomial
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1107 ## DET(A-t*EYE) = (LAMBDA-t)^N - (-1)^N ALPHA.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1108 ## ALPHA defaults to SQRT(EPS) and LAMBDA to 0.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1109
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1110 if (nargin < 1 || nargin > 3)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1111 error ("gallery: 1 to 3 arguments are required for forsythe matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1112 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1113 error ("gallery: N must be an integer for forsythe matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1114 elseif (! isnumeric (alpha) || ! isscalar (alpha))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1115 error ("gallery: ALPHA must be a numeric scalar for forsythe matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1116 elseif (! isnumeric (lambda) || ! isscalar (lambda))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1117 error ("gallery: LAMBDA must be a numeric scalar for forsythe matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1118 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1119
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1120 A = jordbloc (n, lambda);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1121 A(n,1) = alpha;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1122 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1123
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1124 function F = frank (n, k = 0)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1125 ## FRANK Frank matrix---ill conditioned eigenvalues.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1126 ## F = FRANK(N, K) is the Frank matrix of order N. It is upper
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1127 ## Hessenberg with determinant 1. K = 0 is the default; if K = 1 the
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1128 ## elements are reflected about the anti-diagonal (1,N)--(N,1).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1129 ## F has all positive eigenvalues and they occur in reciprocal pairs
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1130 ## (so that 1 is an eigenvalue if N is odd).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1131 ## The eigenvalues of F may be obtained in terms of the zeros of the
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1132 ## Hermite polynomials.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1133 ## The FLOOR(N/2) smallest eigenvalues of F are ill conditioned,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1134 ## the more so for bigger N.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1135 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1136 ## DET(FRANK(N)') comes out far from 1 for large N---see Frank (1958)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1137 ## and Wilkinson (1960) for discussions.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1138 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1139 ## This version incorporates improvements suggested by W. Kahan.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1140 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1141 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1142 ## W.L. Frank, Computing eigenvalues of complex matrices by determinant
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1143 ## evaluation and by methods of Danilewski and Wielandt, J. Soc.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1144 ## Indust. Appl. Math., 6 (1958), pp. 378-392 (see pp. 385, 388).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1145 ## G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1146 ## computation of the Jordan canonical form, SIAM Review, 18 (1976),
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1147 ## pp. 578-619 (Section 13).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1148 ## H. Rutishauser, On test matrices, Programmation en Mathematiques
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1149 ## Numeriques, Editions Centre Nat. Recherche Sci., Paris, 165,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1150 ## 1966, pp. 349-365. Section 9.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1151 ## J.H. Wilkinson, Error analysis of floating-point computation,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1152 ## Numer. Math., 2 (1960), pp. 319-340 (Section 8).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1153 ## J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1154 ## Press, 1965 (pp. 92-93).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1155 ## The next two references give details of the eigensystem, as does
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1156 ## Rutishauser (see above).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1157 ## P.J. Eberlein, A note on the matrices denoted by B_n, SIAM J. Appl.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1158 ## Math., 20 (1971), pp. 87-92.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1159 ## J.M. Varah, A generalization of the Frank matrix, SIAM J. Sci. Stat.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1160 ## Comput., 7 (1986), pp. 835-839.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1161
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1162 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1163 error ("gallery: 1 to 2 arguments are required for frank matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1164 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1165 error ("gallery: N must be an integer for frank matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1166 elseif (! isnumeric (k) || ! isscalar (k))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1167 error ("gallery: K must be a numeric scalar for frank matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1168 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1169
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1170 p = n:-1:1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1171 F = triu (p(ones (n, 1), :) - diag (ones (n-1, 1), -1), -1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1172
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1173 switch (k)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1174 case (0), # do nothing
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1175 case (1), F = F(p,p)';
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1176 otherwise
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1177 error ("gallery: K must have a value of 0 or 1 for frank matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1178 endswitch
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1179 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1180
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1181 function c = gcdmat (n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1182 if (nargin != 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1183 error ("gallery: 1 argument is required for gcdmat matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1184 elseif (! isscalar (n) || ! isnumeric (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1185 error ("gallery: N must be an integer for gcdmat matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1186 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1187 c = gcd (repmat ((1:n)', [1 n]), repmat (1:n, [n 1]));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1188 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1189
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1190 function A = gearmat (n, i = n, j = -n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1191 ## NOTE: this function was named gearm in the original Test Matrix Toolbox
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1192 ## GEARMAT Gear matrix.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1193 ## A = GEARMAT(N,I,J) is the N-by-N matrix with ones on the sub- and
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1194 ## super-diagonals, SIGN(I) in the (1,ABS(I)) position, SIGN(J)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1195 ## in the (N,N+1-ABS(J)) position, and zeros everywhere else.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1196 ## Defaults: I = N, j = -N.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1197 ## All eigenvalues are of the form 2*COS(a) and the eigenvectors
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1198 ## are of the form [SIN(w+a), SIN(w+2a), ..., SIN(w+Na)].
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1199 ## The values of a and w are given in the reference below.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1200 ## A can have double and triple eigenvalues and can be defective.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1201 ## GEARMAT(N) is singular.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1202 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1203 ## (GEAR is a Simulink function, hence GEARMAT for Gear matrix.)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1204 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1205 ## C.W. Gear, A simple set of test matrices for eigenvalue programs,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1206 ## Math. Comp., 23 (1969), pp. 119-125.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1207
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1208 if (nargin < 1 || nargin > 3)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1209 error ("gallery: 1 to 3 arguments are required for gearmat matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1210 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1211 error ("gallery: N must be an integer for gearmat matrix.");
20294
2102c4582e32 gallery: fix chebspec, cycol, gearmat, hanowa, lauchli, and pei (bug #45466)
Massimiliano Fasi <massimiliano.fasi@gmail.com>
parents: 20231
diff changeset
1212 elseif (! isnumeric (i) || ! isscalar (i) || i == 0 || abs (i) > n)
18812
9ac2357f19bc doc: Replace "non-zero" with "nonzero" to match existing usage.
Rik <rik@octave.org>
parents: 17386
diff changeset
1213 error ("gallery: I must be a nonzero scalar, and abs (I) <= N for gearmat matrix.");
20294
2102c4582e32 gallery: fix chebspec, cycol, gearmat, hanowa, lauchli, and pei (bug #45466)
Massimiliano Fasi <massimiliano.fasi@gmail.com>
parents: 20231
diff changeset
1214 elseif (! isnumeric (j) || ! isscalar (j) || i == 0 || abs (j) > n)
18812
9ac2357f19bc doc: Replace "non-zero" with "nonzero" to match existing usage.
Rik <rik@octave.org>
parents: 17386
diff changeset
1215 error ("gallery: J must be a nonzero scalar, and abs (J) <= N for gearmat matrix.");
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1216 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1217
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1218 A = diag (ones (n-1, 1), -1) + diag (ones (n-1, 1), 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1219 A(1, abs (i)) = sign (i);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1220 A(n, n+1 - abs (j)) = sign (j);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1221 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1222
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1223 function G = grcar (n, k = 3)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1224 ## GRCAR Grcar matrix - a Toeplitz matrix with sensitive eigenvalues.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1225 ## GRCAR(N, K) is an N-by-N matrix with -1s on the
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1226 ## subdiagonal, 1s on the diagonal, and K superdiagonals of 1s.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1227 ## The default is K = 3. The eigenvalues of this matrix form an
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1228 ## interesting pattern in the complex plane (try PS(GRCAR(32))).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1229 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1230 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1231 ## J.F. Grcar, Operator coefficient methods for linear equations,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1232 ## Report SAND89-8691, Sandia National Laboratories, Albuquerque,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1233 ## New Mexico, 1989 (Appendix 2).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1234 ## N.M. Nachtigal, L. Reichel and L.N. Trefethen, A hybrid GMRES
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1235 ## algorithm for nonsymmetric linear systems, SIAM J. Matrix Anal.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1236 ## Appl., 13 (1992), pp. 796-825.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1237
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1238 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1239 error ("gallery: 1 to 2 arguments are required for grcar matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1240 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1241 error ("gallery: N must be an integer for grcar matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1242 elseif (! isnumeric (k) || ! isscalar (k))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1243 error ("gallery: K must be a numeric scalar for grcar matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1244 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1245
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1246 G = tril (triu (ones (n)), k) - diag (ones (n-1, 1), -1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1247 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1248
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1249 function A = hanowa (n, d = -1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1250 ## HANOWA A matrix whose eigenvalues lie on a vertical line in the complex plane.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1251 ## HANOWA(N, d) is the N-by-N block 2x2 matrix (thus N = 2M must be even)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1252 ## [d*EYE(M) -DIAG(1:M)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1253 ## DIAG(1:M) d*EYE(M)]
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1254 ## It has complex eigenvalues lambda(k) = d +/- k*i (1 <= k <= M).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1255 ## Parameter d defaults to -1.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1256 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1257 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1258 ## E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1259 ## Differential Equations I: Nonstiff Problems, Springer-Verlag,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1260 ## Berlin, 1987. (pp. 86-87)
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1261
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1262 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1263 error ("gallery: 1 to 2 arguments are required for hanowa matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1264 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1265 error ("gallery: N must be an integer for hanowa matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1266 elseif (rem (n, 2) != 0)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1267 error ("gallery: N must be even for hanowa matrix.");
20294
2102c4582e32 gallery: fix chebspec, cycol, gearmat, hanowa, lauchli, and pei (bug #45466)
Massimiliano Fasi <massimiliano.fasi@gmail.com>
parents: 20231
diff changeset
1268 elseif (! isnumeric (d) || ! isscalar (d))
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1269 error ("gallery: D must be a numeric scalar for hanowa matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1270 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1271
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1272 m = n/2;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1273 A = [ d*eye(m) -diag(1:m)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1274 diag(1:m) d*eye(m) ];
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1275 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1276
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1277 function [v, beta] = house (x)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1278 ## HOUSE Householder matrix.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1279 ## If [v, beta] = HOUSE(x) then H = EYE - beta*v*v' is a Householder
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1280 ## matrix such that Hx = -sign(x(1))*norm(x)*e_1.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1281 ## NB: If x = 0 then v = 0, beta = 1 is returned.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1282 ## x can be real or complex.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1283 ## sign(x) := exp(i*arg(x)) ( = x./abs(x) when x ~= 0).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1284 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1285 ## Theory: (textbook references Golub & Van Loan 1989, 38-43;
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1286 ## Stewart 1973, 231-234, 262; Wilkinson 1965, 48-50).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1287 ## Hx = y: (I - beta*v*v')x = -s*e_1.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1288 ## Must have |s| = norm(x), v = x+s*e_1, and
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1289 ## x'y = x'Hx =(x'Hx)' real => arg(s) = arg(x(1)).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1290 ## So take s = sign(x(1))*norm(x) (which avoids cancellation).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1291 ## v'v = (x(1)+s)^2 + x(2)^2 + ... + x(n)^2
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1292 ## = 2*norm(x)*(norm(x) + |x(1)|).
19593
446c46af4b42 strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents: 17386
diff changeset
1293 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1294 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1295 ## G.H. Golub and C.F. Van Loan, Matrix Computations, second edition,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1296 ## Johns Hopkins University Press, Baltimore, Maryland, 1989.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1297 ## G.W. Stewart, Introduction to Matrix Computations, Academic Press,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1298 ## New York, 1973,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1299 ## J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1300 ## Press, 1965.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1301
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1302 if (nargin != 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1303 error ("gallery: 1 argument is required for house matrix.");
20295
557979395ca9 gallery: allow single element vectors for 'house'.
Carnë Draug <carandraug@octave.org>
parents: 20294
diff changeset
1304 elseif (! isnumeric (x) || ! isvector (x))
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1305 error ("gallery: X must be a vector for house matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1306 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1307
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1308 ## must be a column vector
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1309 x = x(:);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1310
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1311 s = norm (x) * (sign (x(1)) + (x(1) == 0)); # Modification for sign (0) == 1.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1312 v = x;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1313 if (s == 0)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1314 ## Quit if x is the zero vector.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1315 beta = 1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1316 else
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1317 v(1) = v(1) + s;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1318 beta = 1/(s'*v(1)); # NB the conjugated s.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1319 ## beta = 1/(abs (s) * (abs (s) +abs(x(1)) would guarantee beta real.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1320 ## But beta as above can be non-real (due to rounding) only when x is complex.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1321 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1322 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1323
16980
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1324 function A = integerdata (varargin)
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1325
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1326 if (nargin < 3)
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1327 error ("gallery: At least 3 arguments required for integerdata matrix.");
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1328 endif
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1329
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1330 if (isnumeric (varargin{end}))
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1331 jidx = varargin{end};
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1332 svec = [varargin{:}];
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1333 varargin(end) = [];
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1334 elseif (ischar (varargin{end}))
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1335 if (nargin < 4)
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1336 error (["gallery: CLASS argument requires 4 inputs " ...
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1337 "for integerdata matrix."]);
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1338 endif
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1339 jidx = varargin{end-1};
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1340 svec = [varargin{1:end-1}];
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1341 varargin(end-1) = [];
19593
446c46af4b42 strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents: 17386
diff changeset
1342 else
16980
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1343 error (["gallery: J must be an integer in the range [0, 2^32-1] " ...
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1344 "for integerdata matrix"]);
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1345 endif
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1346
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1347 if (! (isnumeric (jidx) && isscalar (jidx)
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1348 && jidx == fix (jidx)
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1349 && jidx >= 0 && jidx <= 0xFFFFFFFF))
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1350 error (["gallery: J must be an integer in the range [0, 2^32-1] " ...
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1351 "for integerdata matrix"]);
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1352 endif
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1353
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1354 ## Save and restore random state. Initialization done so that reproducible
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1355 ## data is available from gallery depending on the jidx and size vector.
19593
446c46af4b42 strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents: 17386
diff changeset
1356 randstate = rand ("state");
16980
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1357 unwind_protect
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1358 rand ("state", svec);
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1359 A = randi (varargin{:});
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1360 unwind_protect_cleanup
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1361 rand ("state", randstate);
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1362 end_unwind_protect
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1363
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1364 endfunction
1909e1ed63e6 gallery.m: Add 'integerdata' matrix to function.
Rik <rik@octave.org>
parents: 16979
diff changeset
1365
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1366 function A = invhess (x, y)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1367 ## INVHESS Inverse of an upper Hessenberg matrix.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1368 ## INVHESS(X, Y), where X is an N-vector and Y an N-1 vector,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1369 ## is the matrix whose lower triangle agrees with that of
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1370 ## ONES(N,1)*X' and whose strict upper triangle agrees with
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1371 ## that of [1 Y]*ONES(1,N).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1372 ## The matrix is nonsingular if X(1) ~= 0 and X(i+1) ~= Y(i)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1373 ## for all i, and its inverse is an upper Hessenberg matrix.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1374 ## If Y is omitted it defaults to -X(1:N-1).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1375 ## Special case: if X is a scalar INVHESS(X) is the same as
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1376 ## INVHESS(1:X).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1377 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1378 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1379 ## F.N. Valvi and V.S. Geroyannis, Analytic inverses and
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1380 ## determinants for a class of matrices, IMA Journal of Numerical
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1381 ## Analysis, 7 (1987), pp. 123-128.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1382 ## W.-L. Cao and W.J. Stewart, A note on inverses of Hessenberg-like
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1383 ## matrices, Linear Algebra and Appl., 76 (1986), pp. 233-240.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1384 ## Y. Ikebe, On inverses of Hessenberg matrices, Linear Algebra and
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1385 ## Appl., 24 (1979), pp. 93-97.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1386 ## P. Rozsa, On the inverse of band matrices, Integral Equations and
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1387 ## Operator Theory, 10 (1987), pp. 82-95.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1388
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1389 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1390 error ("gallery: 1 to 2 arguments are required for invhess matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1391 elseif (! isnumeric (x))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1392 error ("gallery: X must be numeric for invhess matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1393 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1394
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1395 if (isscalar (x) && fix (x) == x)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1396 n = x;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1397 x = 1:n;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1398 elseif (! isscalar (x) && isvector (x))
20297
26fc9bbb8762 gallery: fix typo on variable name for invhess matrix.
Carnë Draug <carandraug@octave.org>
parents: 20296
diff changeset
1399 n = numel (x);
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1400 else
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1401 error ("gallery: X must be an integer scalar, or a vector for invhess matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1402 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1403
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1404 if (nargin < 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1405 y = -x(1:end-1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1406 elseif (! isvector (y) || numel (y) != numel (x) -1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1407 error ("gallery: Y must be a vector of length -1 than X for invhess matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1408 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1409
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1410 x = x(:);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1411 y = y(:);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1412
16933
e39f00a32dc7 maint: Use parentheses around condition for switch(),while(),if() statements.
Rik <rik@octave.org>
parents: 16816
diff changeset
1413 ## FIXME: On next line, z = x'; A = z(ones(n,1),:) would be more efficient.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1414 A = ones (n, 1) * x';
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1415 for j = 2:n
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1416 A(1:j-1,j) = y(1:j-1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1417 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1418 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1419
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1420 function A = invol (n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1421 ## INVOL An involutory matrix.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1422 ## A = INVOL(N) is an N-by-N involutory (A*A = EYE(N)) and
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1423 ## ill-conditioned matrix.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1424 ## It is a diagonally scaled version of HILB(N).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1425 ## NB: B = (EYE(N)-A)/2 and B = (EYE(N)+A)/2 are idempotent (B*B = B).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1426 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1427 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1428 ## A.S. Householder and J.A. Carpenter, The singular values
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1429 ## of involutory and of idempotent matrices, Numer. Math. 5 (1963),
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1430 ## pp. 234-237.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1431
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1432 if (nargin != 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1433 error ("gallery: 1 argument is required for invol matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1434 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1435 error ("gallery: N must be an integer for invol matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1436 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1437
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1438 A = hilb (n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1439
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1440 d = -n;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1441 A(:, 1) = d * A(:, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1442
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1443 for i = 1:n-1
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1444 d = -(n+i)*(n-i)*d/(i*i);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1445 A(i+1,:) = d * A(i+1,:);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1446 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1447 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1448
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1449 function [A, detA] = ipjfact (n, k = 0)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1450 ## IPJFACT A Hankel matrix with factorial elements.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1451 ## A = IPJFACT(N, K) is the matrix with
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1452 ## A(i,j) = (i+j)! (K = 0, default)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1453 ## A(i,j) = 1/(i+j)! (K = 1)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1454 ## Both are Hankel matrices.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1455 ## The determinant and inverse are known explicitly.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1456 ## If a second output argument is present, d = DET(A) is returned:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1457 ## [A, d] = IPJFACT(N, K);
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1458 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1459 ## Suggested by P. R. Graves-Morris.
19593
446c46af4b42 strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents: 17386
diff changeset
1460 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1461 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1462 ## M.J.C. Gover, The explicit inverse of factorial Hankel matrices,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1463 ## Dept. of Mathematics, University of Bradford, 1993.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1464
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1465 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1466 error ("gallery: 1 to 2 arguments are required for ipjfact matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1467 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1468 error ("gallery: N must be an integer for ipjfact matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1469 elseif (! isnumeric (k) || ! isscalar (k))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1470 error ("gallery: K must be a numeric scalar for ipjfact matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1471 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1472
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1473 c = cumprod (2:n+1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1474 d = cumprod (n+1:2*n) * c(n-1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1475
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1476 A = hankel (c, d);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1477
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1478 switch (k)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1479 case (0), # do nothing
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1480 case (1), A = ones (n) ./ A;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1481 otherwise
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1482 error ("gallery: K must have a value of 0 or 1 for ipjfact matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1483 endswitch
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1484
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1485 if (nargout == 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1486 d = 1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1487
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1488 if (k == 0)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1489 for i = 1:n-1
20231
83792dd9bcc1 Use in-place operators in m-files where possible.
Rik <rik@octave.org>
parents: 20162
diff changeset
1490 d *= prod (1:i+1) * prod (1:n-i);
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1491 endfor
20231
83792dd9bcc1 Use in-place operators in m-files where possible.
Rik <rik@octave.org>
parents: 20162
diff changeset
1492 d *= prod (1:n+1);
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1493
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1494 elseif (k == 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1495 for i = 0:n-1
20231
83792dd9bcc1 Use in-place operators in m-files where possible.
Rik <rik@octave.org>
parents: 20162
diff changeset
1496 d *= prod (1:i) / prod (1:n+1+i);
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1497 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1498 if (rem (n*(n-1)/2, 2))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1499 d = -d;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1500 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1501
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1502 else
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1503 error ("gallery: K must have a value of 0 or 1 for ipjfact matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1504 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1505
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1506 detA = d;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1507 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1508 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1509
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1510 function J = jordbloc (n, lambda = 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1511 ## JORDBLOC Jordan block.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1512 ## JORDBLOC(N, LAMBDA) is the N-by-N Jordan block with eigenvalue
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1513 ## LAMBDA. LAMBDA = 1 is the default.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1514
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1515 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1516 error ("gallery: 1 to 2 arguments are required for jordbloc matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1517 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1518 error ("gallery: N must be an integer for jordbloc matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1519 elseif (! isnumeric (lambda) || ! isscalar (lambda))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1520 error ("gallery: LAMBDA must be a numeric scalar for jordbloc matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1521 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1522
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1523 J = lambda * eye (n) + diag (ones (n-1, 1), 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1524 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1525
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1526 function U = kahan (n, theta = 1.2, pert = 25)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1527 ## KAHAN Kahan matrix - upper trapezoidal.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1528 ## KAHAN(N, THETA) is an upper trapezoidal matrix
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1529 ## that has some interesting properties regarding estimation of
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1530 ## condition and rank.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1531 ## The matrix is N-by-N unless N is a 2-vector, in which case it
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1532 ## is N(1)-by-N(2).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1533 ## The parameter THETA defaults to 1.2.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1534 ## The useful range of THETA is 0 < THETA < PI.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1535 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1536 ## To ensure that the QR factorization with column pivoting does not
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1537 ## interchange columns in the presence of rounding errors, the diagonal
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1538 ## is perturbed by PERT*EPS*diag( [N:-1:1] ).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1539 ## The default is PERT = 25, which ensures no interchanges for KAHAN(N)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1540 ## up to at least N = 90 in IEEE arithmetic.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1541 ## KAHAN(N, THETA, PERT) uses the given value of PERT.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1542 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1543 ## The inverse of KAHAN(N, THETA) is known explicitly: see
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1544 ## Higham (1987, p. 588), for example.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1545 ## The diagonal perturbation was suggested by Christian Bischof.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1546 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1547 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1548 ## W. Kahan, Numerical linear algebra, Canadian Math. Bulletin,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1549 ## 9 (1966), pp. 757-801.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1550 ## N.J. Higham, A survey of condition number estimation for
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1551 ## triangular matrices, SIAM Review, 29 (1987), pp. 575-596.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1552
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1553 if (nargin < 1 || nargin > 3)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1554 error ("gallery: 1 to 3 arguments are required for kahan matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1555 elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1556 error ("gallery: N must be a 1 or 2 element integer for kahan matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1557 elseif (! isnumeric (theta) || ! isscalar (theta))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1558 error ("gallery: THETA must be a numeric scalar for kahan matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1559 elseif (! isnumeric (pert) || ! isscalar (pert))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1560 error ("gallery: PERT must be a numeric scalar for kahan matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1561 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1562
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1563 ## Parameter n specifies dimension: r-by-n
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1564 r = n(1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1565 n = n(end);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1566
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1567 s = sin (theta);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1568 c = cos (theta);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1569
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1570 U = eye (n) - c * triu (ones (n), 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1571 U = diag (s.^[0:n-1]) * U + pert*eps* diag ([n:-1:1]);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1572 if (r > n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1573 U(r,n) = 0; # Extend to an r-by-n matrix
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1574 else
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1575 U = U(1:r,:); # Reduce to an r-by-n matrix
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1576 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1577 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1578
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1579 function A = kms (n, rho = 0.5)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1580 ## KMS Kac-Murdock-Szego Toeplitz matrix.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1581 ## A = KMS(N, RHO) is the N-by-N Kac-Murdock-Szego Toeplitz matrix with
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1582 ## A(i,j) = RHO^(ABS((i-j))) (for real RHO).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1583 ## If RHO is complex, then the same formula holds except that elements
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1584 ## below the diagonal are conjugated.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1585 ## RHO defaults to 0.5.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1586 ## Properties:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1587 ## A has an LDL' factorization with
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1588 ## L = INV(TRIW(N,-RHO,1)'),
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1589 ## D(i,i) = (1-ABS(RHO)^2)*EYE(N) except D(1,1) = 1.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1590 ## A is positive definite if and only if 0 < ABS(RHO) < 1.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1591 ## INV(A) is tridiagonal.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1592 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1593 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1594 ## W.F. Trench, Numerical solution of the eigenvalue problem
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1595 ## for Hermitian Toeplitz matrices, SIAM J. Matrix Analysis and Appl.,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1596 ## 10 (1989), pp. 135-146 (and see the references therein).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1597
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1598 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1599 error ("gallery: 1 to 2 arguments are required for lauchli matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1600 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
21385
89fa0694aa2e Fix check-missing-semicolon QA target (bug #47277).
Rik <rik@octave.org>
parents: 20929
diff changeset
1601 error ("gallery: N must be an integer for lauchli matrix.");
20294
2102c4582e32 gallery: fix chebspec, cycol, gearmat, hanowa, lauchli, and pei (bug #45466)
Massimiliano Fasi <massimiliano.fasi@gmail.com>
parents: 20231
diff changeset
1602 elseif (! isscalar (rho))
21385
89fa0694aa2e Fix check-missing-semicolon QA target (bug #47277).
Rik <rik@octave.org>
parents: 20929
diff changeset
1603 error ("gallery: RHO must be a scalar for lauchli matrix.");
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1604 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1605
16933
e39f00a32dc7 maint: Use parentheses around condition for switch(),while(),if() statements.
Rik <rik@octave.org>
parents: 16816
diff changeset
1606 A = (1:n)'*ones (1,n);
e39f00a32dc7 maint: Use parentheses around condition for switch(),while(),if() statements.
Rik <rik@octave.org>
parents: 16816
diff changeset
1607 A = abs (A - A');
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1608 A = rho .^ A;
16933
e39f00a32dc7 maint: Use parentheses around condition for switch(),while(),if() statements.
Rik <rik@octave.org>
parents: 16816
diff changeset
1609 if (imag (rho))
e39f00a32dc7 maint: Use parentheses around condition for switch(),while(),if() statements.
Rik <rik@octave.org>
parents: 16816
diff changeset
1610 A = conj (tril (A,-1)) + triu (A);
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1611 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1612 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1613
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1614 function B = krylov (A, x, j)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1615 ## KRYLOV Krylov matrix.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1616 ## KRYLOV(A, x, j) is the Krylov matrix
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1617 ## [x, Ax, A^2x, ..., A^(j-1)x],
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1618 ## where A is an n-by-n matrix and x is an n-vector.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1619 ## Defaults: x = ONES(n,1), j = n.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1620 ## KRYLOV(n) is the same as KRYLOV(RANDN(n)).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1621 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1622 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1623 ## G.H. Golub and C.F. Van Loan, Matrix Computations, second edition,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1624 ## Johns Hopkins University Press, Baltimore, Maryland, 1989, p. 369.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1625
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1626 if (nargin < 1 || nargin > 3)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1627 error ("gallery: 1 to 3 arguments are required for krylov matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1628 elseif (! isnumeric (A) || ! issquare (A) || ndims (A) != 2)
17199
9deb214ae9d5 Use 2-D, not 2D, in error messages.
Rik <rik@octave.org>
parents: 16982
diff changeset
1629 error ("gallery: A must be a square 2-D matrix for krylov matrix.");
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1630 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1631
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1632 n = length (A);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1633 if (isscalar (A))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1634 n = A;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1635 A = randn (n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1636 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1637
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1638 if (nargin < 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1639 x = ones (n, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1640 elseif (! isvector (x) || numel (x) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1641 error ("gallery: X must be a vector of length equal to A for krylov matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1642 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1643
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1644 if (nargin < 3)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1645 j = n;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1646 elseif (! isnumeric (j) || ! isscalar (j) || fix (j) != j)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1647 error ("gallery: J must be an integer for krylov matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1648 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1649
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1650 B = ones (n, j);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1651 B(:,1) = x(:);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1652 for i = 2:j
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1653 B(:,i) = A*B(:,i-1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1654 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1655 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1656
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1657 function A = lauchli (n, mu = sqrt (eps))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1658 ## LAUCHLI Lauchli matrix - rectangular.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1659 ## LAUCHLI(N, MU) is the (N+1)-by-N matrix [ONES(1,N); MU*EYE(N))].
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1660 ## It is a well-known example in least squares and other problems
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1661 ## that indicates the dangers of forming A'*A.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1662 ## MU defaults to SQRT(EPS).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1663 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1664 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1665 ## P. Lauchli, Jordan-Elimination und Ausgleichung nach
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1666 ## kleinsten Quadraten, Numer. Math, 3 (1961), pp. 226-240.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1667
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1668 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1669 error ("gallery: 1 to 2 arguments are required for lauchli matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1670 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1671 error ("gallery: N must be an integer for lauchli matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1672 elseif (! isscalar (mu))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1673 error ("gallery: MU must be a scalar for lauchli matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1674 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1675
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1676 A = [ones(1, n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1677 mu*eye(n) ];
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1678 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1679
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1680 function A = lehmer (n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1681 ## LEHMER Lehmer matrix - symmetric positive definite.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1682 ## A = LEHMER(N) is the symmetric positive definite N-by-N matrix with
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1683 ## A(i,j) = i/j for j >= i.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1684 ## A is totally nonnegative. INV(A) is tridiagonal, and explicit
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1685 ## formulas are known for its entries.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1686 ## N <= COND(A) <= 4*N*N.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1687 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1688 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1689 ## M. Newman and J. Todd, The evaluation of matrix inversion
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1690 ## programs, J. Soc. Indust. Appl. Math., 6 (1958), pp. 466-476.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1691 ## Solutions to problem E710 (proposed by D.H. Lehmer): The inverse
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1692 ## of a matrix, Amer. Math. Monthly, 53 (1946), pp. 534-535.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1693 ## J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1694 ## Birkhauser, Basel, and Academic Press, New York, 1977, p. 154.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1695
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1696 if (nargin != 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1697 error ("gallery: 1 argument is required for lehmer matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1698 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1699 error ("gallery: N must be an integer for lehmer matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1700 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1701
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1702 A = ones (n, 1) * (1:n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1703 A = A./A';
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1704 A = tril (A) + tril (A, -1)';
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1705 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1706
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1707 function T = lesp (n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1708 ## LESP A tridiagonal matrix with real, sensitive eigenvalues.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1709 ## LESP(N) is an N-by-N matrix whose eigenvalues are real and smoothly
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1710 ## distributed in the interval approximately [-2*N-3.5, -4.5].
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1711 ## The sensitivities of the eigenvalues increase exponentially as
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1712 ## the eigenvalues grow more negative.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1713 ## The matrix is similar to the symmetric tridiagonal matrix with
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1714 ## the same diagonal entries and with off-diagonal entries 1,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1715 ## via a similarity transformation with D = diag(1!,2!,...,N!).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1716 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1717 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1718 ## H.W.J. Lenferink and M.N. Spijker, On the use of stability regions in
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1719 ## the numerical analysis of initial value problems,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1720 ## Math. Comp., 57 (1991), pp. 221-237.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1721 ## L.N. Trefethen, Pseudospectra of matrices, in Numerical Analysis 1991,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1722 ## Proceedings of the 14th Dundee Conference,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1723 ## D.F. Griffiths and G.A. Watson, eds, Pitman Research Notes in
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1724 ## Mathematics, volume 260, Longman Scientific and Technical, Essex,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1725 ## UK, 1992, pp. 234-266.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1726
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1727 if (nargin != 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1728 error ("gallery: 1 argument is required for lesp matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1729 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1730 error ("gallery: N must be an integer for lesp matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1731 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1732
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1733 x = 2:n;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1734 T = full (tridiag (ones (size (x)) ./x, -(2*[x n+1]+1), x));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1735 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1736
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1737 function A = lotkin (n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1738 ## LOTKIN Lotkin matrix.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1739 ## A = LOTKIN(N) is the Hilbert matrix with its first row altered to
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1740 ## all ones. A is unsymmetric, ill-conditioned, and has many negative
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1741 ## eigenvalues of small magnitude.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1742 ## The inverse has integer entries and is known explicitly.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1743 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1744 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1745 ## M. Lotkin, A set of test matrices, MTAC, 9 (1955), pp. 153-161.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1746
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1747 if (nargin != 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1748 error ("gallery: 1 argument is required for lotkin matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1749 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1750 error ("gallery: N must be an integer for lotkin matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1751 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1752
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1753 A = hilb (n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1754 A(1,:) = ones (1, n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1755 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1756
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1757 function A = minij (n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1758 ## MINIJ Symmetric positive definite matrix MIN(i,j).
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1759 ## A = MINIJ(N) is the N-by-N symmetric positive definite matrix with
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1760 ## A(i,j) = MIN(i,j).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1761 ## Properties, variations:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1762 ## INV(A) is tridiagonal: it is minus the second difference matrix
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1763 ## except its (N,N) element is 1.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1764 ## 2*A-ONES(N) (Givens' matrix) has tridiagonal inverse and
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1765 ## eigenvalues .5*sec^2([2r-1)PI/4N], r=1:N.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1766 ## (N+1)*ONES(N)-A also has a tridiagonal inverse.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1767 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1768 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1769 ## J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1770 ## Birkhauser, Basel, and Academic Press, New York, 1977, p. 158.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1771 ## D.E. Rutherford, Some continuant determinants arising in physics and
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1772 ## chemistry---II, Proc. Royal Soc. Edin., 63, A (1952), pp. 232-241.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1773 ## (For the eigenvalues of Givens' matrix.)
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1774
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1775 if (nargin != 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1776 error ("gallery: 1 argument is required for minij matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1777 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1778 error ("gallery: N must be an integer for minij matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1779 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1780
19944
941e782d0429 gallery: performance increase for minij matrix.
Carnë Draug <carandraug@octave.org>
parents: 19833
diff changeset
1781 A = bsxfun (@min, 1:n, (1:n)');
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1782 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1783
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1784 function A = moler (n, alpha = -1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1785 ## MOLER Moler matrix - symmetric positive definite.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1786 ## A = MOLER(N, ALPHA) is the symmetric positive definite N-by-N matrix
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1787 ## U'*U where U = TRIW(N, ALPHA).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1788 ## For ALPHA = -1 (the default) A(i,j) = MIN(i,j)-2, A(i,i) = i.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1789 ## A has one small eigenvalue.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1790 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1791 ## Nash (1990) attributes the ALPHA = -1 matrix to Moler.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1792 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1793 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1794 ## J.C. Nash, Compact Numerical Methods for Computers: Linear
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1795 ## Algebra and Function Minimisation, second edition, Adam Hilger,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1796 ## Bristol, 1990 (Appendix 1).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1797
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1798 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1799 error ("gallery: 1 to 2 arguments are required for moler matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1800 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1801 error ("gallery: N must be an integer for moler matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1802 elseif (! isscalar (alpha))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1803 error ("gallery: ALPHA must be a scalar for moler matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1804 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1805
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1806 A = triw (n, alpha)' * triw (n, alpha);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1807 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1808
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1809 function [A, T] = neumann (n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1810 ## NEUMANN Singular matrix from the discrete Neumann problem (sparse).
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1811 ## NEUMANN(N) is the singular, row diagonally dominant matrix resulting
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1812 ## from discretizing the Neumann problem with the usual five point
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1813 ## operator on a regular mesh.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1814 ## It has a one-dimensional null space with null vector ONES(N,1).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1815 ## The dimension N should be a perfect square, or else a 2-vector,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1816 ## in which case the dimension of the matrix is N(1)*N(2).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1817 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1818 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1819 ## R.J. Plemmons, Regular splittings and the discrete Neumann
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1820 ## problem, Numer. Math., 25 (1976), pp. 153-161.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1821
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1822 if (nargin != 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1823 error ("gallery: 1 argument is required for neumann matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1824 elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1825 error ("gallery: N must be a 1 or 2 element integer for neumann matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1826 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1827
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1828 if (isscalar (n))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1829 m = sqrt (n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1830 if (m^2 != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1831 error ("gallery: N must be a perfect square for neumann matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1832 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1833 n(1) = m;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1834 n(2) = m;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1835 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1836
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1837 T = tridiag (n(1), -1, 2, -1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1838 T(1,2) = -2;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1839 T(n(1),n(1)-1) = -2;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1840
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1841 A = kron (T, eye (n(2))) + kron (eye (n(2)), T);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1842 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1843
16979
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1844 function A = normaldata (varargin)
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1845
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1846 if (nargin < 2)
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1847 error ("gallery: At least 2 arguments required for normaldata matrix.");
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1848 endif
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1849 if (isnumeric (varargin{end}))
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1850 jidx = varargin{end};
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1851 svec = [varargin{:}];
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1852 varargin(end) = [];
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1853 elseif (ischar (varargin{end}))
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1854 if (nargin < 3)
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1855 error (["gallery: CLASS argument requires 3 inputs " ...
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1856 "for normaldata matrix."]);
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1857 endif
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1858 jidx = varargin{end-1};
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1859 svec = [varargin{1:end-1}];
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1860 varargin(end-1) = [];
19593
446c46af4b42 strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents: 17386
diff changeset
1861 else
16979
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1862 error (["gallery: J must be an integer in the range [0, 2^32-1] " ...
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1863 "for normaldata matrix"]);
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1864 endif
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1865
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1866 if (! (isnumeric (jidx) && isscalar (jidx)
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1867 && jidx == fix (jidx)
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1868 && jidx >= 0 && jidx <= 0xFFFFFFFF))
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1869 error (["gallery: J must be an integer in the range [0, 2^32-1] " ...
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1870 "for normaldata matrix"]);
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1871 endif
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1872
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1873 ## Save and restore random state. Initialization done so that reproducible
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1874 ## data is available from gallery depending on the jidx and size vector.
19593
446c46af4b42 strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents: 17386
diff changeset
1875 randstate = randn ("state");
16979
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1876 unwind_protect
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1877 randn ("state", svec);
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1878 A = randn (varargin{:});
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1879 unwind_protect_cleanup
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1880 randn ("state", randstate);
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1881 end_unwind_protect
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1882
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1883 endfunction
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
1884
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1885 function Q = orthog (n, k = 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1886 ## ORTHOG Orthogonal and nearly orthogonal matrices.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1887 ## Q = ORTHOG(N, K) selects the K'th type of matrix of order N.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1888 ## K > 0 for exactly orthogonal matrices, K < 0 for diagonal scalings of
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1889 ## orthogonal matrices.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1890 ## Available types: (K = 1 is the default)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1891 ## K = 1: Q(i,j) = SQRT(2/(n+1)) * SIN( i*j*PI/(n+1) )
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1892 ## Symmetric eigenvector matrix for second difference matrix.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1893 ## K = 2: Q(i,j) = 2/SQRT(2*n+1)) * SIN( 2*i*j*PI/(2*n+1) )
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1894 ## Symmetric.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1895 ## K = 3: Q(r,s) = EXP(2*PI*i*(r-1)*(s-1)/n) / SQRT(n) (i=SQRT(-1))
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1896 ## Unitary, the Fourier matrix. Q^4 is the identity.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1897 ## This is essentially the same matrix as FFT(EYE(N))/SQRT(N)!
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1898 ## K = 4: Helmert matrix: a permutation of a lower Hessenberg matrix,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1899 ## whose first row is ONES(1:N)/SQRT(N).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1900 ## K = 5: Q(i,j) = SIN( 2*PI*(i-1)*(j-1)/n ) + COS( 2*PI*(i-1)*(j-1)/n ).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1901 ## Symmetric matrix arising in the Hartley transform.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1902 ## K = -1: Q(i,j) = COS( (i-1)*(j-1)*PI/(n-1) )
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1903 ## Chebyshev Vandermonde-like matrix, based on extrema of T(n-1).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1904 ## K = -2: Q(i,j) = COS( (i-1)*(j-1/2)*PI/n) )
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1905 ## Chebyshev Vandermonde-like matrix, based on zeros of T(n).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1906 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1907 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1908 ## N.J. Higham and D.J. Higham, Large growth factors in Gaussian
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1909 ## elimination with pivoting, SIAM J. Matrix Analysis and Appl.,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1910 ## 10 (1989), pp. 155-164.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1911 ## P. Morton, On the eigenvectors of Schur's matrix, J. Number Theory,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1912 ## 12 (1980), pp. 122-127. (Re. ORTHOG(N, 3))
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1913 ## H.O. Lancaster, The Helmert Matrices, Amer. Math. Monthly, 72 (1965),
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1914 ## pp. 4-12.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1915 ## D. Bini and P. Favati, On a matrix algebra related to the discrete
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1916 ## Hartley transform, SIAM J. Matrix Anal. Appl., 14 (1993),
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1917 ## pp. 500-507.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1918
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1919 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1920 error ("gallery: 1 to 2 arguments are required for orthog matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1921 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1922 error ("gallery: N must be an integer for orthog matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1923 elseif (! isnumeric (k) || ! isscalar (k))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1924 error ("gallery: K must be a numeric scalar for orthog matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1925 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1926
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1927 switch (k)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1928 case (1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1929 ## E'vectors second difference matrix
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1930 m = (1:n)'*(1:n) * (pi/(n+1));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1931 Q = sin (m) * sqrt (2/(n+1));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1932
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1933 case (2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1934 m = (1:n)'*(1:n) * (2*pi/(2*n+1));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1935 Q = sin (m) * (2/ sqrt (2*n+1));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1936
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1937 case (3)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1938 ## Vandermonde based on roots of unity
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1939 m = 0:n-1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1940 Q = exp (m'*m*2*pi* sqrt (-1) / n) / sqrt (n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1941
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1942 case (4)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1943 ## Helmert matrix
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1944 Q = tril (ones (n));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1945 Q(1,2:n) = ones (1, n-1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1946 for i = 2:n
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1947 Q(i,i) = -(i-1);
17306
09543e9c8f40 Use explicit form of end (endif, endfor, etc.) in core m-files.
Rik <rik@octave.org>
parents: 17281
diff changeset
1948 endfor
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1949 Q = diag (sqrt ([n 1:n-1] .* [1:n])) \ Q;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1950
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1951 case (5)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1952 ## Hartley matrix
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1953 m = (0:n-1)'*(0:n-1) * (2*pi/n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1954 Q = (cos (m) + sin (m)) / sqrt (n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1955
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1956 case (-1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1957 ## extrema of T(n-1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1958 m = (0:n-1)'*(0:n-1) * (pi/(n-1));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1959 Q = cos (m);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1960
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1961 case (-2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1962 ## zeros of T(n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1963 m = (0:n-1)'*(.5:n-.5) * (pi/n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1964 Q = cos (m);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1965
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1966 otherwise
16766
7268845c0a1e avoid backquote in error messages, some uses in doc strings
John W. Eaton <jwe@octave.org>
parents: 16734
diff changeset
1967 error ("gallery: unknown K '%d' for orthog matrix.", k);
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1968 endswitch
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1969 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1970
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1971 function A = parter (n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1972 ## PARTER Parter matrix - a Toeplitz matrix with singular values near PI.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1973 ## PARTER(N) is the matrix with (i,j) element 1/(i-j+0.5).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1974 ## It is a Cauchy matrix and a Toeplitz matrix.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1975 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1976 ## At the Second SIAM Conference on Linear Algebra, Raleigh, N.C.,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1977 ## 1985, Cleve Moler noted that most of the singular values of
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1978 ## PARTER(N) are very close to PI. An explanation of the phenomenon
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1979 ## was given by Parter; see also the paper by Tyrtyshnikov.
19593
446c46af4b42 strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents: 17386
diff changeset
1980 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1981 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1982 ## The MathWorks Newsletter, Volume 1, Issue 1, March 1986, page 2.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1983 ## S.V. Parter, On the distribution of the singular values of Toeplitz
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1984 ## matrices, Linear Algebra and Appl., 80 (1986), pp. 115-130.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1985 ## E.E. Tyrtyshnikov, Cauchy-Toeplitz matrices and some applications,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1986 ## Linear Algebra and Appl., 149 (1991), pp. 1-18.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1987
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1988 if (nargin != 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1989 error ("gallery: 1 argument is required for parter matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1990 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1991 error ("gallery: N must be an integer for parter matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1992 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1993
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1994 A = cauchy ((1:n) + 0.5, -(1:n));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1995 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1996
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1997 function P = pei (n, alpha = 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
1998 ## PEI Pei matrix.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
1999 ## PEI(N, ALPHA), where ALPHA is a scalar, is the symmetric matrix
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2000 ## ALPHA*EYE(N) + ONES(N).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2001 ## If ALPHA is omitted then ALPHA = 1 is used.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2002 ## The matrix is singular for ALPHA = 0, -N.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2003 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2004 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2005 ## M.L. Pei, A test matrix for inversion procedures,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2006 ## Comm. ACM, 5 (1962), p. 508.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2007
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2008 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2009 error ("gallery: 1 to 2 arguments are required for pei matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2010 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2011 error ("gallery: N must be an integer for pei matrix.");
20294
2102c4582e32 gallery: fix chebspec, cycol, gearmat, hanowa, lauchli, and pei (bug #45466)
Massimiliano Fasi <massimiliano.fasi@gmail.com>
parents: 20231
diff changeset
2012 elseif (! isnumeric (alpha) || ! isscalar (alpha))
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2013 error ("gallery: ALPHA must be a scalar for pei matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2014 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2015
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2016 P = alpha * eye (n) + ones (n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2017 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2018
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2019 function A = poisson (n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2020 ## POISSON Block tridiagonal matrix from Poisson's equation (sparse).
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2021 ## POISSON(N) is the block tridiagonal matrix of order N^2
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2022 ## resulting from discretizing Poisson's equation with the
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2023 ## 5-point operator on an N-by-N mesh.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2024 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2025 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2026 ## G.H. Golub and C.F. Van Loan, Matrix Computations, second edition,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2027 ## Johns Hopkins University Press, Baltimore, Maryland, 1989
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2028 ## (Section 4.5.4).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2029
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2030 if (nargin != 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2031 error ("gallery: 1 argument is required for poisson matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2032 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2033 error ("gallery: N must be an integer for poisson matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2034 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2035
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2036 S = tridiag (n, -1, 2, -1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2037 I = speye (n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2038 A = kron (I, S) + kron (S, I);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2039 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2040
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2041 function A = prolate (n, w = 0.25)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2042 ## PROLATE Prolate matrix - symmetric, ill-conditioned Toeplitz matrix.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2043 ## A = PROLATE(N, W) is the N-by-N prolate matrix with parameter W.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2044 ## It is a symmetric Toeplitz matrix.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2045 ## If 0 < W < 0.5 then
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2046 ## - A is positive definite
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2047 ## - the eigenvalues of A are distinct, lie in (0, 1), and
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2048 ## tend to cluster around 0 and 1.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2049 ## W defaults to 0.25.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2050 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2051 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2052 ## J.M. Varah. The Prolate matrix. Linear Algebra and Appl.,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2053 ## 187:269--278, 1993.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2054
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2055 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2056 error ("gallery: 1 to 2 arguments are required for prolate matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2057 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2058 error ("gallery: N must be an integer for prolate matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2059 elseif (! isnumeric (w) || ! isscalar (w))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2060 error ("gallery: W must be a scalar for prolate matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2061 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2062
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2063 a = zeros (n, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2064 a(1) = 2*w;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2065 a(2:n) = sin (2*pi*w*(1:n-1)) ./ (pi*(1:n-1));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2066
16933
e39f00a32dc7 maint: Use parentheses around condition for switch(),while(),if() statements.
Rik <rik@octave.org>
parents: 16816
diff changeset
2067 A = toeplitz (a);
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2068 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2069
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2070 function H = randhess (x)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2071 ## NOTE: this function was named ohess in the original Test Matrix Toolbox
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2072 ## RANDHESS Random, orthogonal upper Hessenberg matrix.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2073 ## H = RANDHESS(N) is an N-by-N real, random, orthogonal
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2074 ## upper Hessenberg matrix.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2075 ## Alternatively, H = RANDHESS(X), where X is an arbitrary real
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2076 ## N-vector (N > 1) constructs H non-randomly using the elements
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2077 ## of X as parameters.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2078 ## In both cases H is constructed via a product of N-1 Givens rotations.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2079 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2080 ## Note: See Gragg (1986) for how to represent an N-by-N (complex)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2081 ## unitary Hessenberg matrix with positive subdiagonal elements in terms
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2082 ## of 2N-1 real parameters (the Schur parametrization).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2083 ## This M-file handles the real case only and is intended simply as a
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2084 ## convenient way to generate random or non-random orthogonal Hessenberg
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2085 ## matrices.
19593
446c46af4b42 strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents: 17386
diff changeset
2086 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2087 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2088 ## W.B. Gragg, The QR algorithm for unitary Hessenberg matrices,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2089 ## J. Comp. Appl. Math., 16 (1986), pp. 1-8.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2090
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2091 if (nargin != 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2092 error ("gallery: 1 argument is required for randhess matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2093 elseif (! isnumeric (x) || ! isreal (x))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2094 error ("gallery: N or X must be numeric real values for randhess matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2095 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2096
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2097 if (isscalar (x))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2098 n = x;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2099 x = rand (n-1, 1) * 2*pi;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2100 H = eye (n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2101 H(n,n) = sign (randn);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2102 elseif (isvector (x))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2103 n = numel (x);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2104 H = eye (n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2105 H(n,n) = sign (x(n)) + (x(n) == 0); # Second term ensures H(n,n) nonzero.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2106 else
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2107 error ("gallery: N or X must be a scalar or a vector for randhess matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2108 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2109
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2110 for i = n:-1:2
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2111 ## Apply Givens rotation through angle x(i-1).
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2112 theta = x(i-1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2113 c = cos (theta);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2114 s = sin (theta);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2115 H([i-1 i], :) = [ c*H(i-1,:)+s*H(i,:)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2116 -s*H(i-1,:)+c*H(i,:) ];
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2117 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2118 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2119
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2120 function A = rando (n, k = 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2121 ## RANDO Random matrix with elements -1, 0 or 1.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2122 ## A = RANDO(N, K) is a random N-by-N matrix with elements from
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2123 ## one of the following discrete distributions (default K = 1):
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2124 ## K = 1: A(i,j) = 0 or 1 with equal probability,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2125 ## K = 2: A(i,j) = -1 or 1 with equal probability,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2126 ## K = 3: A(i,j) = -1, 0 or 1 with equal probability.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2127 ## N may be a 2-vector, in which case the matrix is N(1)-by-N(2).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2128
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2129 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2130 error ("gallery: 1 to 2 arguments are required for rando matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2131 elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2132 error ("gallery: N must be an integer for rando matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2133 elseif (! isnumeric (k) || ! isscalar (k))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2134 error ("gallery: K must be a numeric scalar for smoke matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2135 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2136
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2137 ## Parameter n specifies dimension: m-by-n.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2138 m = n(1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2139 n = n(end);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2140
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2141 switch (k)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2142 case (1), A = floor ( rand(m, n) + 0.5); # {0, 1}
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2143 case (2), A = 2*floor ( rand(m, n) + 0.5) -1; # {-1, 1}
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2144 case (3), A = round (3*rand(m, n) - 1.5); # {-1, 0, 1}
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2145 otherwise
16766
7268845c0a1e avoid backquote in error messages, some uses in doc strings
John W. Eaton <jwe@octave.org>
parents: 16734
diff changeset
2146 error ("gallery: unknown K '%d' for smoke matrix.", k);
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2147 endswitch
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2148
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2149 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2150
20296
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2151 function A = randsvd (n, kappa = sqrt (1/eps), mode = 3, kl = max (n) -1,
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2152 ku = kl)
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2153 ## RANDSVD Random matrix with pre-assigned singular values.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2154 ## RANDSVD(N, KAPPA, MODE, KL, KU) is a (banded) random matrix of order N
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2155 ## with COND(A) = KAPPA and singular values from the distribution MODE.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2156 ## N may be a 2-vector, in which case the matrix is N(1)-by-N(2).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2157 ## Available types:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2158 ## MODE = 1: one large singular value,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2159 ## MODE = 2: one small singular value,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2160 ## MODE = 3: geometrically distributed singular values,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2161 ## MODE = 4: arithmetically distributed singular values,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2162 ## MODE = 5: random singular values with unif. dist. logarithm.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2163 ## If omitted, MODE defaults to 3, and KAPPA defaults to SQRT(1/EPS).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2164 ## If MODE < 0 then the effect is as for ABS(MODE) except that in the
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2165 ## original matrix of singular values the order of the diagonal entries
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2166 ## is reversed: small to large instead of large to small.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2167 ## KL and KU are the lower and upper bandwidths respectively; if they
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2168 ## are omitted a full matrix is produced.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2169 ## If only KL is present, KU defaults to KL.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2170 ## Special case: if KAPPA < 0 then a random full symmetric positive
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2171 ## definite matrix is produced with COND(A) = -KAPPA and
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2172 ## eigenvalues distributed according to MODE.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2173 ## KL and KU, if present, are ignored.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2174 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2175 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2176 ## N.J. Higham, Accuracy and Stability of Numerical Algorithms,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2177 ## Society for Industrial and Applied Mathematics, Philadelphia, PA,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2178 ## USA, 1996; sec. 26.3.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2179 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2180 ## This routine is similar to the more comprehensive Fortran routine xLATMS
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2181 ## in the following reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2182 ## J.W. Demmel and A. McKenney, A test matrix generation suite,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2183 ## LAPACK Working Note #9, Courant Institute of Mathematical Sciences,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2184 ## New York, 1989.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2185
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2186 if (nargin < 1 || nargin > 5)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2187 error ("gallery: 1 to 5 arguments are required for randsvd matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2188 elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2189 error ("gallery: N must be a 1 or 2 element integer vector for randsvd matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2190 elseif (! isnumeric (kappa) || ! isscalar (kappa))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2191 error ("gallery: KAPPA must be a numeric scalar for randsvd matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2192 elseif (abs (kappa) < 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2193 error ("gallery: KAPPA must larger than or equal to 1 for randsvd matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2194 elseif (! isnumeric (mode) || ! isscalar (mode))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2195 error ("gallery: MODE must be a numeric scalar for randsvd matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2196 elseif (! isnumeric (kl) || ! isscalar (kl))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2197 error ("gallery: KL must be a numeric scalar for randsvd matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2198 elseif (! isnumeric (ku) || ! isscalar (ku))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2199 error ("gallery: KU must be a numeric scalar for randsvd matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2200 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2201
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2202 posdef = 0;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2203 if (kappa < 0)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2204 posdef = 1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2205 kappa = -kappa;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2206 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2207
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2208 ## Parameter n specifies dimension: m-by-n.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2209 m = n(1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2210 n = n(end);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2211 p = min ([m n]);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2212
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2213 ## If A will be a vector
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2214 if (p == 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2215 A = randn (m, n);
20231
83792dd9bcc1 Use in-place operators in m-files where possible.
Rik <rik@octave.org>
parents: 20162
diff changeset
2216 A /= norm (A);
17306
09543e9c8f40 Use explicit form of end (endif, endfor, etc.) in core m-files.
Rik <rik@octave.org>
parents: 17281
diff changeset
2217 return;
09543e9c8f40 Use explicit form of end (endif, endfor, etc.) in core m-files.
Rik <rik@octave.org>
parents: 17281
diff changeset
2218 endif
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2219
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2220 ## Set up vector sigma of singular values.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2221 switch (abs (mode))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2222 case (1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2223 sigma = ones (p, 1) ./ kappa;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2224 sigma(1) = 1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2225 case (2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2226 sigma = ones (p, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2227 sigma(p) = 1 / kappa;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2228 case (3)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2229 factor = kappa^(-1/(p-1));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2230 sigma = factor.^[0:p-1];
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2231 case (4)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2232 sigma = ones (p, 1) - (0:p-1)'/(p-1)*(1-1/kappa);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2233 case (5)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2234 ## In this case cond (A) <= kappa.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2235 rand ("uniform");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2236 sigma = exp (-rand (p, 1) * log (kappa));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2237 otherwise
16766
7268845c0a1e avoid backquote in error messages, some uses in doc strings
John W. Eaton <jwe@octave.org>
parents: 16734
diff changeset
2238 error ("gallery: unknown MODE '%d' for randsvd matrix.", mode);
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2239 endswitch
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2240
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2241 ## Convert to diagonal matrix of singular values.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2242 if (mode < 0)
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2243 sigma = sigma(p:-1:1);
17306
09543e9c8f40 Use explicit form of end (endif, endfor, etc.) in core m-files.
Rik <rik@octave.org>
parents: 17281
diff changeset
2244 endif
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2245 sigma = diag (sigma);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2246
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2247 if (posdef)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2248 ## handle case where KAPPA was negative
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2249 Q = qmult (p);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2250 A = Q' * sigma * Q;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2251 A = (A + A') / 2; # Ensure matrix is symmetric.
17312
088d014a7fe2 Use semicolon after "return" statement in core m-files.
Rik <rik@octave.org>
parents: 17306
diff changeset
2252 return;
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2253 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2254
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2255 if (m != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2256 ## Expand to m-by-n diagonal matrix
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2257 sigma(m, n) = 0;
17306
09543e9c8f40 Use explicit form of end (endif, endfor, etc.) in core m-files.
Rik <rik@octave.org>
parents: 17281
diff changeset
2258 endif
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2259
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2260 if (kl == 0 && ku == 0)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2261 ## Diagonal matrix requested - nothing more to do.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2262 A = sigma;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2263 else
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2264 ## A = U*sigma*V, where U, V are random orthogonal matrices from the
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2265 ## Haar distribution.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2266 A = qmult (sigma');
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2267 A = qmult (A');
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2268
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2269 if (kl < n-1 || ku < n-1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2270 ## Bandwidth reduction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2271 A = bandred (A, kl, ku);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2272 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2273 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2274 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2275
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2276 function A = redheff (n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2277 ## REDHEFF A (0,1) matrix of Redheffer associated with the Riemann hypothesis.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2278 ## A = REDHEFF(N) is an N-by-N matrix of 0s and 1s defined by
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2279 ## A(i,j) = 1 if j = 1 or if i divides j,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2280 ## A(i,j) = 0 otherwise.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2281 ## It has N - FLOOR(LOG2(N)) - 1 eigenvalues equal to 1,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2282 ## a real eigenvalue (the spectral radius) approximately SQRT(N),
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2283 ## a negative eigenvalue approximately -SQRT(N),
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2284 ## and the remaining eigenvalues are provably ``small''.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2285 ## Barrett and Jarvis (1992) conjecture that
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2286 ## ``the small eigenvalues all lie inside the unit circle
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2287 ## ABS(Z) = 1'',
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2288 ## and a proof of this conjecture, together with a proof that some
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2289 ## eigenvalue tends to zero as N tends to infinity, would yield
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2290 ## a new proof of the prime number theorem.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2291 ## The Riemann hypothesis is true if and only if
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2292 ## DET(A) = O( N^(1/2+epsilon) ) for every epsilon > 0
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2293 ## (`!' denotes factorial).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2294 ## See also RIEMANN.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2295 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2296 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2297 ## W.W. Barrett and T.J. Jarvis,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2298 ## Spectral Properties of a Matrix of Redheffer,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2299 ## Linear Algebra and Appl., 162 (1992), pp. 673-683.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2300
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2301 if (nargin != 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2302 error ("gallery: 1 argument is required for redheff matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2303 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2304 error ("gallery: N must be an integer for redheff matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2305 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2306
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2307 i = (1:n)' * ones (1, n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2308 A = ! rem (i', i);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2309 A(:,1) = ones (n, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2310 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2311
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2312 function A = riemann (n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2313 ## RIEMANN A matrix associated with the Riemann hypothesis.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2314 ## A = RIEMANN(N) is an N-by-N matrix for which the
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2315 ## Riemann hypothesis is true if and only if
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2316 ## DET(A) = O( N! N^(-1/2+epsilon) ) for every epsilon > 0
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2317 ## (`!' denotes factorial).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2318 ## A = B(2:N+1, 2:N+1), where
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2319 ## B(i,j) = i-1 if i divides j and -1 otherwise.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2320 ## Properties include, with M = N+1:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2321 ## Each eigenvalue E(i) satisfies ABS(E(i)) <= M - 1/M.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2322 ## i <= E(i) <= i+1 with at most M-SQRT(M) exceptions.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2323 ## All integers in the interval (M/3, M/2] are eigenvalues.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2324 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2325 ## See also REDHEFF.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2326 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2327 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2328 ## F. Roesler, Riemann's hypothesis as an eigenvalue problem,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2329 ## Linear Algebra and Appl., 81 (1986), pp. 153-198.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2330
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2331 if (nargin != 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2332 error ("gallery: 1 argument is required for riemann matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2333 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2334 error ("gallery: N must be an integer for riemann matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2335 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2336
20231
83792dd9bcc1 Use in-place operators in m-files where possible.
Rik <rik@octave.org>
parents: 20162
diff changeset
2337 n += 1;
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2338 i = (2:n)' * ones (1, n-1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2339 j = i';
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2340 A = i .* (! rem (j, i)) - ones (n-1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2341 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2342
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2343 function A = ris (n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2344 ## NOTE: this function was named dingdong in the original Test Matrix Toolbox
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2345 ## RIS Dingdong matrix - a symmetric Hankel matrix.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2346 ## A = RIS(N) is the symmetric N-by-N Hankel matrix with
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2347 ## A(i,j) = 0.5/(N-i-j+1.5).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2348 ## The eigenvalues of A cluster around PI/2 and -PI/2.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2349 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2350 ## Invented by F.N. Ris.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2351 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2352 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2353 ## J.C. Nash, Compact Numerical Methods for Computers: Linear
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2354 ## Algebra and Function Minimisation, second edition, Adam Hilger,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2355 ## Bristol, 1990 (Appendix 1).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2356
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2357 if (nargin != 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2358 error ("gallery: 1 argument is required for ris matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2359 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2360 error ("gallery: N must be an integer for ris matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2361 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2362
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2363 p = -2*(1:n) + (n+1.5);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2364 A = cauchy (p);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2365 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2366
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2367 function A = smoke (n, k = 0)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2368 ## SMOKE Smoke matrix - complex, with a `smoke ring' pseudospectrum.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2369 ## SMOKE(N) is an N-by-N matrix with 1s on the
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2370 ## superdiagonal, 1 in the (N,1) position, and powers of
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2371 ## roots of unity along the diagonal.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2372 ## SMOKE(N, 1) is the same except for a zero (N,1) element.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2373 ## The eigenvalues of SMOKE(N, 1) are the N'th roots of unity;
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2374 ## those of SMOKE(N) are the N'th roots of unity times 2^(1/N).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2375 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2376 ## Try PS(SMOKE(32)). For SMOKE(N, 1) the pseudospectrum looks
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2377 ## like a sausage folded back on itself.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2378 ## GERSH(SMOKE(N, 1)) is interesting.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2379 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2380 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2381 ## L. Reichel and L.N. Trefethen, Eigenvalues and pseudo-eigenvalues of
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2382 ## Toeplitz matrices, Linear Algebra and Appl., 162-164:153-185, 1992.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2383
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2384 if (nargin < 1 || nargin > 2)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2385 error ("gallery: 1 to 2 arguments are required for smoke matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2386 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2387 error ("gallery: N must be an integer for smoke matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2388 elseif (! isnumeric (n) || ! isscalar (n))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2389 error ("gallery: K must be a numeric scalar for smoke matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2390 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2391
16933
e39f00a32dc7 maint: Use parentheses around condition for switch(),while(),if() statements.
Rik <rik@octave.org>
parents: 16816
diff changeset
2392 w = exp (2*pi*i/n);
e39f00a32dc7 maint: Use parentheses around condition for switch(),while(),if() statements.
Rik <rik@octave.org>
parents: 16816
diff changeset
2393 A = diag ( [w.^(1:n-1) 1] ) + diag (ones (n-1,1), 1);
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2394
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2395 switch (k)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2396 case (0), A(n,1) = 1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2397 case (1), # do nothing
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2398 otherwise,
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2399 error ("gallery: K must have a value of 0 or 1 for smoke matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2400 endswitch
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2401 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2402
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2403 function T = toeppd (n, m = n, w = rand (m,1), theta = rand (m,1))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2404 ## NOTE: this function was named pdtoep in the original Test Matrix Toolbox
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2405 ## TOEPPD Symmetric positive definite Toeplitz matrix.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2406 ## TOEPPD(N, M, W, THETA) is an N-by-N symmetric positive (semi-)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2407 ## definite (SPD) Toeplitz matrix, comprised of the sum of M rank 2
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2408 ## (or, for certain THETA, rank 1) SPD Toeplitz matrices.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2409 ## Specifically,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2410 ## T = W(1)*T(THETA(1)) + ... + W(M)*T(THETA(M)),
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2411 ## where T(THETA(k)) has (i,j) element COS(2*PI*THETA(k)*(i-j)).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2412 ## Defaults: M = N, W = RAND(M,1), THETA = RAND(M,1).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2413 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2414 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2415 ## G. Cybenko and C.F. Van Loan, Computing the minimum eigenvalue of
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2416 ## a symmetric positive definite Toeplitz matrix, SIAM J. Sci. Stat.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2417 ## Comput., 7 (1986), pp. 123-131.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2418
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2419 if (nargin < 1 || nargin > 4)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2420 error ("gallery: 1 to 4 arguments are required for toeppd matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2421 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2422 error ("gallery: N must be a numeric integer for toeppd matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2423 elseif (! isnumeric (m) || ! isscalar (m) || fix (m) != m)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2424 error ("gallery: M must be a numeric integer for toeppd matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2425 elseif (numel (w) != m || numel (theta) != m)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2426 error ("gallery: W and THETA must be vectors of length M for toeppd matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2427 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2428
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2429 T = zeros (n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2430 E = 2*pi * ((1:n)' * ones (1, n) - ones (n, 1) * (1:n));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2431
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2432 for i = 1:m
20231
83792dd9bcc1 Use in-place operators in m-files where possible.
Rik <rik@octave.org>
parents: 20162
diff changeset
2433 T += w(i) * cos (theta(i)*E);
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2434 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2435 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2436
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2437 function P = toeppen (n, a = 1, b = -10, c = 0, d = 10, e = 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2438 ## NOTE: this function was named pentoep in the original Test Matrix Toolbox
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2439 ## TOEPPEN Pentadiagonal Toeplitz matrix (sparse).
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2440 ## P = TOEPPEN(N, A, B, C, D, E) is the N-by-N pentadiagonal
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2441 ## Toeplitz matrix with diagonals composed of the numbers
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2442 ## A =: P(3,1), B =: P(2,1), C =: P(1,1), D =: P(1,2), E =: P(1,3).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2443 ## Default: (A,B,C,D,E) = (1,-10,0,10,1) (a matrix of Rutishauser).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2444 ## This matrix has eigenvalues lying approximately on
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2445 ## the line segment 2*cos(2*t) + 20*i*sin(t).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2446 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2447 ## Interesting plots are
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2448 ## PS(FULL(TOEPPEN(32,0,1,0,0,1/4))) - `triangle'
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2449 ## PS(FULL(TOEPPEN(32,0,1/2,0,0,1))) - `propeller'
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2450 ## PS(FULL(TOEPPEN(32,0,1/2,1,1,1))) - `fish'
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2451 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2452 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2453 ## R.M. Beam and R.F. Warming, The asymptotic spectra of
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2454 ## banded Toeplitz and quasi-Toeplitz matrices, SIAM J. Sci.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2455 ## Comput. 14 (4), 1993, pp. 971-1006.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2456 ## H. Rutishauser, On test matrices, Programmation en Mathematiques
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2457 ## Numeriques, Editions Centre Nat. Recherche Sci., Paris, 165,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2458 ## 1966, pp. 349-365.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2459
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2460 if (nargin < 1 || nargin > 6)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2461 error ("gallery: 1 to 6 arguments are required for toeppen matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2462 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2463 error ("gallery: N must be a numeric integer for toeppen matrix.");
17386
6dbc866379e2 Replace cellfun() occurrences with faster code where possible.
Rik <rik@octave.org>
parents: 17338
diff changeset
2464 elseif (any (! cellfun ("isnumeric", {a b c d e})) || any (cellfun ("numel", {a b c d e}) != 1))
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2465 error ("gallery: A, B, C, D and E must be numeric scalars for toeppen matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2466 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2467
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2468 P = spdiags ([a*ones(n,1) b*ones(n,1) c*ones(n,1) d*ones(n,1) e*ones(n,1)],
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2469 -2:2, n, n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2470 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2471
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2472 function T = tridiag (n, x = -1, y = 2, z = -1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2473 ## TRIDIAG Tridiagonal matrix (sparse).
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2474 ## TRIDIAG(X, Y, Z) is the tridiagonal matrix with subdiagonal X,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2475 ## diagonal Y, and superdiagonal Z.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2476 ## X and Z must be vectors of dimension one less than Y.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2477 ## Alternatively TRIDIAG(N, C, D, E), where C, D, and E are all
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2478 ## scalars, yields the Toeplitz tridiagonal matrix of order N
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2479 ## with subdiagonal elements C, diagonal elements D, and superdiagonal
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2480 ## elements E. This matrix has eigenvalues (Todd 1977)
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2481 ## D + 2*SQRT(C*E)*COS(k*PI/(N+1)), k=1:N.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2482 ## TRIDIAG(N) is the same as TRIDIAG(N,-1,2,-1), which is
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2483 ## a symmetric positive definite M-matrix (the negative of the
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2484 ## second difference matrix).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2485 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2486 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2487 ## J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2488 ## Birkhauser, Basel, and Academic Press, New York, 1977, p. 155.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2489 ## D.E. Rutherford, Some continuant determinants arising in physics and
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2490 ## chemistry---II, Proc. Royal Soc. Edin., 63, A (1952), pp. 232-241.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2491
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2492 if (nargin != 1 && nargin != 3 && nargin != 4)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2493 error ("gallery: 1, 3, or 4 arguments are required for tridiag matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2494 elseif (nargin == 3)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2495 z = y;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2496 y = x;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2497 x = n;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2498 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2499
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2500 ## Force column vectors
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2501 x = x(:);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2502 y = y(:);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2503 z = z(:);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2504
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2505 if (isscalar (x) && isscalar (y) && isscalar (z))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2506 x *= ones (n-1, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2507 z *= ones (n-1, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2508 y *= ones (n, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2509 elseif (numel (y) != numel (x) + 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2510 error ("gallery: X must have one element less than Y for tridiag matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2511 elseif (numel (y) != numel (z) + 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2512 error ("gallery: Z must have one element less than Y for tridiag matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2513 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2514
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2515 ## T = diag (x, -1) + diag (y) + diag (z, 1); # For non-sparse matrix.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2516 n = numel (y);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2517 T = spdiags ([[x;0] y [0;z]], -1:1, n, n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2518 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2519
18846
3d33fe79816c gallery.m: Return correct matrix if 3rd argument is not specified.
Rik <rik@octave.org>
parents: 18812
diff changeset
2520 function t = triw (n, alpha = -1, k = n(end) - 1)
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2521 ## TRIW Upper triangular matrix discussed by Wilkinson and others.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2522 ## TRIW(N, ALPHA, K) is the upper triangular matrix with ones on
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2523 ## the diagonal and ALPHAs on the first K >= 0 superdiagonals.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2524 ## N may be a 2-vector, in which case the matrix is N(1)-by-N(2) and
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2525 ## upper trapezoidal.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2526 ## Defaults: ALPHA = -1,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2527 ## K = N - 1 (full upper triangle).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2528 ## TRIW(N) is a matrix discussed by Kahan, Golub and Wilkinson.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2529 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2530 ## Ostrowski (1954) shows that
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2531 ## COND(TRIW(N,2)) = COT(PI/(4*N))^2,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2532 ## and for large ABS(ALPHA),
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2533 ## COND(TRIW(N,ALPHA)) is approximately ABS(ALPHA)^N*SIN(PI/(4*N-2)).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2534 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2535 ## Adding -2^(2-N) to the (N,1) element makes TRIW(N) singular,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2536 ## as does adding -2^(1-N) to all elements in the first column.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2537 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2538 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2539 ## G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2540 ## computation of the Jordan canonical form, SIAM Review,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2541 ## 18(4), 1976, pp. 578-619.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2542 ## W. Kahan, Numerical linear algebra, Canadian Math. Bulletin,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2543 ## 9 (1966), pp. 757-801.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2544 ## A.M. Ostrowski, On the spectrum of a one-parametric family of
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2545 ## matrices, J. Reine Angew. Math., 193 (3/4), 1954, pp. 143-160.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2546 ## J.H. Wilkinson, Singular-value decomposition---basic aspects,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2547 ## in D.A.H. Jacobs, ed., Numerical Software---Needs and Availability,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2548 ## Academic Press, London, 1978, pp. 109-135.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2549
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2550 if (nargin < 1 || nargin > 3)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2551 error ("gallery: 1 to 3 arguments are required for triw matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2552 elseif (! isnumeric (n) || all (numel (n) != [1 2]))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2553 error ("gallery: N must be a 1 or 2 elements vector for triw matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2554 elseif (! isscalar (alpha))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2555 error ("gallery: ALPHA must be a scalar for triw matrix.");
18846
3d33fe79816c gallery.m: Return correct matrix if 3rd argument is not specified.
Rik <rik@octave.org>
parents: 18812
diff changeset
2556 elseif (! isscalar (k) || ! isnumeric (k) || fix (k) != k || k < 0)
3d33fe79816c gallery.m: Return correct matrix if 3rd argument is not specified.
Rik <rik@octave.org>
parents: 18812
diff changeset
2557 error ("gallery: K must be a numeric integer >= 0 for triw matrix.");
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2558 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2559
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2560 m = n(1); # Parameter n specifies dimension: m-by-n.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2561 n = n(end);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2562
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2563 t = tril (eye (m, n) + alpha * triu (ones (m, n), 1), k);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2564 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2565
16978
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2566 function A = uniformdata (varargin)
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2567
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2568 if (nargin < 2)
16979
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
2569 error ("gallery: At least 2 arguments required for uniformdata matrix.");
16978
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2570 endif
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2571 if (isnumeric (varargin{end}))
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2572 jidx = varargin{end};
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2573 svec = [varargin{:}];
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2574 varargin(end) = [];
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2575 elseif (ischar (varargin{end}))
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2576 if (nargin < 3)
16979
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
2577 error (["gallery: CLASS argument requires 3 inputs " ...
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
2578 "for uniformdata matrix."]);
16978
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2579 endif
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2580 jidx = varargin{end-1};
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2581 svec = [varargin{1:end-1}];
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2582 varargin(end-1) = [];
19593
446c46af4b42 strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents: 17386
diff changeset
2583 else
16979
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
2584 error (["gallery: J must be an integer in the range [0, 2^32-1] " ...
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
2585 "for uniformdata matrix"]);
16978
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2586 endif
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2587
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2588 if (! (isnumeric (jidx) && isscalar (jidx)
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2589 && jidx == fix (jidx)
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2590 && jidx >= 0 && jidx <= 0xFFFFFFFF))
16979
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
2591 error (["gallery: J must be an integer in the range [0, 2^32-1] " ...
9aa293e00475 gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents: 16978
diff changeset
2592 "for uniformdata matrix"]);
16978
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2593 endif
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2594
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2595 ## Save and restore random state. Initialization done so that reproducible
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2596 ## data is available from gallery depending on the jidx and size vector.
19593
446c46af4b42 strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents: 17386
diff changeset
2597 randstate = rand ("state");
16978
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2598 unwind_protect
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2599 rand ("state", svec);
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2600 A = rand (varargin{:});
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2601 unwind_protect_cleanup
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2602 rand ("state", randstate);
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2603 end_unwind_protect
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2604
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2605 endfunction
00379f9f8773 gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents: 16933
diff changeset
2606
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2607 function A = wathen (nx, ny, k = 0)
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2608 ## WATHEN returns the Wathen matrix.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2609 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2610 ## Discussion:
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2611 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2612 ## The Wathen matrix is a finite element matrix which is sparse.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2613 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2614 ## The entries of the matrix depend in part on a physical quantity
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2615 ## related to density. That density is here assigned random values between
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2616 ## 0 and 100.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2617 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2618 ## A = WATHEN ( NX, NY ) is a sparse random N-by-N finite element matrix
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2619 ## where N = 3*NX*NY + 2*NX + 2*NY + 1.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2620 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2621 ## A is the consistent mass matrix for a regular NX-by-NY
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2622 ## grid of 8-node (serendipity) elements in 2 space dimensions.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2623 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2624 ## Here is an illustration for NX = 3, NX = 2:
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2625 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2626 ## 23-24-25-26-27-28-29
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2627 ## | | | |
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2628 ## 19 20 21 22
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2629 ## | | | |
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2630 ## 12-13-14-15-16-17-18
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2631 ## | | | |
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2632 ## 8 9 10 11
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2633 ## | | | |
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2634 ## 1--2--3--4--5--6--7
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2635 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2636 ## For this example, the total number of nodes is, as expected,
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2637 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2638 ## N = 3 * 3 * 2 + 2 * 2 + 2 * 3 + 1 = 29.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2639 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2640 ## A is symmetric positive definite for any (positive) values of
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2641 ## the density, RHO(NX,NY), which is chosen randomly in this routine.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2642 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2643 ## In particular, if D = DIAG(DIAG(A)), then
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2644 ## 0.25 <= EIG(INV(D)*A) <= 4.5
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2645 ## for any positive integers NX and NY and any densities RHO(NX,NY).
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2646 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2647 ## A = WATHEN ( NX, NY, 1 ) returns the diagonally scaled matrix.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2648 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2649 ## Modified:
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2650 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2651 ## 17 September 2007
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2652 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2653 ## Author:
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2654 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2655 ## Nicholas Higham
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2656 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2657 ## Reference:
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2658 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2659 ## Nicholas Higham,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2660 ## Algorithm 694: A Collection of Test Matrices in MATLAB,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2661 ## ACM Transactions on Mathematical Software,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2662 ## Volume 17, Number 3, September 1991, pages 289-305.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2663 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2664 ## Andrew Wathen,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2665 ## Realistic eigenvalue bounds for the Galerkin mass matrix,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2666 ## IMA Journal of Numerical Analysis,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2667 ## Volume 7, 1987, pages 449-457.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2668 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2669 ## Parameters:
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2670 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2671 ## Input, integer NX, NY, the number of elements in the X and Y directions
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2672 ## of the finite element grid. NX and NY must each be at least 1.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2673 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2674 ## Optional input, integer K, is used to request that the diagonally scaled
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2675 ## version of the matrix be returned. This happens if K is specified with
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2676 ## the value 1.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2677 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2678 ## Output, sparse real A(N,N), the matrix. The dimension N is determined by
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2679 ## NX and NY, as described above. A is stored in the MATLAB sparse matrix
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2680 ## format.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2681
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2682 if (nargin < 2 || nargin > 3)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2683 error ("gallery: 2 or 3 arguments are required for wathen matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2684 elseif (! isnumeric (nx) || ! isscalar (nx) || nx < 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2685 error ("gallery: NX must be a positive scalar for wathen matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2686 elseif (! isnumeric (ny) || ! isscalar (ny) || ny < 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2687 error ("gallery: NY must be a positive scalar for wathen matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2688 elseif (! isscalar (k))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2689 error ("gallery: K must be a scalar for wathen matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2690 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2691
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2692 e1 = [ 6 -6 2 -8
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2693 -6 32 -6 20
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2694 2 -6 6 -6
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2695 -8 20 -6 32 ];
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2696
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2697 e2 = [ 3 -8 2 -6
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2698 -8 16 -8 20
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2699 2 -8 3 -8
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2700 -6 20 -8 16 ];
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2701
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2702 e = [ e1 e2
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2703 e2' e1] / 45;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2704
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2705 n = 3*nx*ny + 2*nx + 2*ny + 1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2706
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2707 A = sparse (n, n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2708
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2709 rho = 100 * rand (nx, ny);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2710
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2711 for j = 1:ny
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2712 for i = 1:nx
19593
446c46af4b42 strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents: 17386
diff changeset
2713 ##
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2714 ## For the element (I,J), determine the indices of the 8 nodes.
19593
446c46af4b42 strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents: 17386
diff changeset
2715 ##
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2716 nn(1) = 3*j*nx + 2*i + 2*j + 1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2717 nn(2) = nn(1) - 1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2718 nn(3) = nn(2) - 1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2719 nn(4) = (3*j - 1) * nx + 2*j + i - 1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2720 nn(5) = 3 * (j-1) * nx + 2*i + 2*j - 3;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2721 nn(6) = nn(5) + 1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2722 nn(7) = nn(6) + 1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2723 nn(8) = nn(4) + 1;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2724
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2725 em = e * rho(i,j);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2726
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2727 for krow = 1:8
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2728 for kcol = 1:8
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2729 A(nn(krow),nn(kcol)) = A(nn(krow),nn(kcol)) + em(krow,kcol);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2730 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2731 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2732
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2733 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2734 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2735
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2736 ## If requested, return A with diagonal scaling.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2737 if (k)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2738 A = diag (diag (A)) \ A;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2739 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2740 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2741
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2742 function [A, b] = wilk (n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2743 ## WILK Various specific matrices devised/discussed by Wilkinson.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2744 ## [A, b] = WILK(N) is the matrix or system of order N.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2745 ## N = 3: upper triangular system Ux=b illustrating inaccurate solution.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2746 ## N = 4: lower triangular system Lx=b, ill-conditioned.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2747 ## N = 5: HILB(6)(1:5,2:6)*1.8144. Symmetric positive definite.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2748 ## N = 21: W21+, tridiagonal. Eigenvalue problem.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2749 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2750 ## References:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2751 ## J.H. Wilkinson, Error analysis of direct methods of matrix inversion,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2752 ## J. Assoc. Comput. Mach., 8 (1961), pp. 281-330.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2753 ## J.H. Wilkinson, Rounding Errors in Algebraic Processes, Notes on Applied
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2754 ## Science No. 32, Her Majesty's Stationery Office, London, 1963.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2755 ## J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2756 ## Press, 1965.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2757
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2758 if (nargin != 1)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2759 error ("gallery: 1 argument is required for wilk matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2760 elseif (! isnumeric (n) || ! isscalar (n))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2761 error ("gallery: N must be a numeric scalar for wilk matrix.");
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2762 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2763
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2764 if (n == 3)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2765 ## Wilkinson (1961) p.323.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2766 A = [ 1e-10 0.9 -0.4
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2767 0 0.9 -0.4
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2768 0 0 1e-10 ];
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2769
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2770 b = [ 0
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2771 0
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2772 1];
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2773
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2774 elseif (n == 4)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2775 ## Wilkinson (1963) p.105.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2776 A = [0.9143e-4 0 0 0
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2777 0.8762 0.7156e-4 0 0
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2778 0.7943 0.8143 0.9504e-4 0
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2779 0.8017 0.6123 0.7165 0.7123e-4];
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2780
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2781 b = [0.6524
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2782 0.3127
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2783 0.4186
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2784 0.7853];
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2785
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2786 elseif (n == 5)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2787 ## Wilkinson (1965), p.234.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2788 A = hilb (6);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2789 A = A(1:5, 2:6) * 1.8144;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2790
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2791 elseif (n == 21)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2792 ## Wilkinson (1965), p.308.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2793 E = diag (ones (n-1, 1), 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2794 m = (n-1)/2;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2795 A = diag (abs (-m:m)) + E + E';
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2796
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2797 else
16766
7268845c0a1e avoid backquote in error messages, some uses in doc strings
John W. Eaton <jwe@octave.org>
parents: 16734
diff changeset
2798 error ("gallery: unknown N '%d' for wilk matrix.", n);
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2799 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2800 endfunction
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2801
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2802 ## NOTE: bandred is part of the Test Matrix Toolbox and is used by randsvd()
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2803 function A = bandred (A, kl, ku)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2804 ## BANDRED Band reduction by two-sided unitary transformations.
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2805 ## B = BANDRED(A, KL, KU) is a matrix unitarily equivalent to A
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2806 ## with lower bandwidth KL and upper bandwidth KU
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2807 ## (i.e. B(i,j) = 0 if i > j+KL or j > i+KU).
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2808 ## The reduction is performed using Householder transformations.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2809 ## If KU is omitted it defaults to KL.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2810 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2811 ## Called by RANDSVD.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2812 ## This is a `standard' reduction. Cf. reduction to bidiagonal form
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2813 ## prior to computing the SVD. This code is a little wasteful in that
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2814 ## it computes certain elements which are immediately set to zero!
19593
446c46af4b42 strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents: 17386
diff changeset
2815 ##
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2816 ## Reference:
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2817 ## G.H. Golub and C.F. Van Loan, Matrix Computations, second edition,
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2818 ## Johns Hopkins University Press, Baltimore, Maryland, 1989.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
2819 ## Section 5.4.3.
16634
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2820
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2821 ## Check for special case where order of left/right transformations matters.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2822 ## Easiest approach is to work on the transpose, flipping back at the end.
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2823 flip = false;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2824 if (ku == 0)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2825 flip = true;
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2826 A = A';
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2827 [ku, kl] = deal (kl, ku);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2828 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2829
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2830 [m, n] = size (A);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2831
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2832 for j = 1:min (min (m, n), max (m-kl-1, n-ku-1))
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2833 if (j+kl+1 <= m)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2834 [v, beta] = house (A(j+kl:m,j));
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2835 temp = A(j+kl:m,j:n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2836 A(j+kl:m,j:n) = temp - beta*v*(v'*temp);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2837 A(j+kl+1:m,j) = zeros (m-j-kl, 1);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2838 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2839
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2840 if (j+ku+1 <= n)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2841 [v, beta] = house (A(j,j+ku:n)');
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2842 temp = A(j:m,j+ku:n);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2843 A(j:m,j+ku:n) = temp - beta*(temp*v)*v';
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2844 A(j,j+ku+1:n) = zeros (1, n-j-ku);
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2845 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2846 endfor
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2847
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2848 if (flip)
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2849 A = A';
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2850 endif
2510fffc05e1 gallery: new function
Carnë Draug <carandraug@octave.org>
parents:
diff changeset
2851 endfunction
17338
1c89599167a6 maint: End m-files with 1 blank line.
Rik <rik@octave.org>
parents: 17312
diff changeset
2852
20296
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2853 ## NOTE: qmult is part of the Test Matrix Toolbox and is used by randsvd()
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2854 function B = qmult (A)
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2855 ## QMULT Pre-multiply by random orthogonal matrix.
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2856 ## QMULT(A) is Q*A where Q is a random real orthogonal matrix from
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2857 ## the Haar distribution, of dimension the number of rows in A.
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2858 ## Special case: if A is a scalar then QMULT(A) is the same as
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2859 ## QMULT(EYE(A)).
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2860 ##
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2861 ## Called by RANDSVD.
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2862 ##
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2863 ## Reference:
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2864 ## G.W. Stewart, The efficient generation of random
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2865 ## orthogonal matrices with an application to condition estimators,
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2866 ## SIAM J. Numer. Anal., 17 (1980), 403-409.
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2867
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2868 [n, m] = size (A);
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2869
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2870 ## Handle scalar A
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2871 if (isscalar (A))
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2872 n = A;
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2873 A = eye (n);
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2874 endif
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2875
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2876 d = zeros (n);
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2877
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2878 for k = n-1:-1:1
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2879 ## Generate random Householder transformation.
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2880 x = randn (n-k+1, 1);
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2881 s = norm (x);
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2882 sgn = sign (x(1)) + (x(1) == 0); # Modification for sign(1)=1.
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2883 s = sgn*s;
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2884 d(k) = -sgn;
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2885 x(1) = x(1) + s;
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2886 beta = s*x(1);
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2887
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2888 ## Apply the transformation to A.
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2889 y = x'*A(k:n,:);
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2890 A(k:n,:) = A(k:n,:) - x*(y/beta);
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2891 endfor
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2892
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2893 ## Tidy up signs
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2894 for i = 1:n-1
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2895 A(i,:) = d(i)*A(i,:);
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2896 endfor
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2897 A(n,:) = A(n,:) * sign (randn);
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2898 B = A;
0b9d23557506 gallery: fix randsvd by adding missing dependency qmult().
Carnë Draug <carandraug@octave.org>
parents: 20295
diff changeset
2899 endfunction
19226
cdfc8bc9ab62 gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents: 19040
diff changeset
2900
21759
b002b4331a12 maint: Use two newlines after endfunction and start of BIST tests.
Rik <rik@octave.org>
parents: 21751
diff changeset
2901
19226
cdfc8bc9ab62 gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents: 19040
diff changeset
2902 ## BIST testing for just a few functions to verify that the main gallery
cdfc8bc9ab62 gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents: 19040
diff changeset
2903 ## dispatch function works.
20929
45a64a6c7273 fix comment character style in most .m files to be consistent
John W. Eaton <jwe@octave.org>
parents: 20852
diff changeset
2904 %!assert (gallery ("clement", 3), [0 1 0; 2 0 2; 0 1 0])
45a64a6c7273 fix comment character style in most .m files to be consistent
John W. Eaton <jwe@octave.org>
parents: 20852
diff changeset
2905 %!assert (gallery ("invhess", 2), [1 -1; 1 2])
19226
cdfc8bc9ab62 gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents: 19040
diff changeset
2906
cdfc8bc9ab62 gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents: 19040
diff changeset
2907 ## Test input validation of main dispatch function only
cdfc8bc9ab62 gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents: 19040
diff changeset
2908 %!error gallery ()
cdfc8bc9ab62 gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents: 19040
diff changeset
2909 %!error <NAME must be a string> gallery (123)
cdfc8bc9ab62 gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents: 19040
diff changeset
2910 %!error <matrix binomial not implemented> gallery ("binomial")
cdfc8bc9ab62 gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents: 19040
diff changeset
2911 %!error <unknown matrix with NAME foobar> gallery ("foobar")
cdfc8bc9ab62 gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents: 19040
diff changeset
2912
21759
b002b4331a12 maint: Use two newlines after endfunction and start of BIST tests.
Rik <rik@octave.org>
parents: 21751
diff changeset
2913 ## BIST testing for individual gallery functions
19944
941e782d0429 gallery: performance increase for minij matrix.
Carnë Draug <carandraug@octave.org>
parents: 19833
diff changeset
2914 %!assert (gallery ("minij", 4), [1 1 1 1; 1 2 2 2; 1 2 3 3; 1 2 3 4])
941e782d0429 gallery: performance increase for minij matrix.
Carnë Draug <carandraug@octave.org>
parents: 19833
diff changeset
2915 %!assert (gallery ("minij", 1), 1)
941e782d0429 gallery: performance increase for minij matrix.
Carnë Draug <carandraug@octave.org>
parents: 19833
diff changeset
2916 %!assert (gallery ("minij", 0), [])
941e782d0429 gallery: performance increase for minij matrix.
Carnë Draug <carandraug@octave.org>
parents: 19833
diff changeset
2917 %!assert (gallery ("minij", -1), [])
20299
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2918
23381
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2919 %!test
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2920 %! exp = 1 ./ [
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2921 %! 2 3 4 5 6
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2922 %! 3 4 5 6 7
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2923 %! 4 5 6 7 8
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2924 %! 5 6 7 8 9
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2925 %! 6 7 8 9 10];
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2926 %! assert (gallery ("cauchy", 5), exp)
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2927 %! assert (gallery ("cauchy", 1:5), exp)
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2928 %! assert (gallery ("cauchy", 1:5, 1:5), exp)
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2929 %!
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2930 %! exp = 1 ./ [
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2931 %! 1 2 3 4 5
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2932 %! 2 3 4 5 6
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2933 %! 3 4 5 6 7
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2934 %! 4 5 6 7 8
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2935 %! 5 6 7 8 9];
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2936 %! assert (gallery ("cauchy", 0:4, 1:5), exp)
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2937 %! assert (gallery ("cauchy", 1:5, 0:4), exp)
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2938 %! assert (gallery ("cauchy", 1:5, 4:-1:0), fliplr (exp))
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2939 %!
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2940 %! exp = 1 ./ [
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2941 %! -1 0 1 2 3
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2942 %! 0 1 2 3 4
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2943 %! 1 2 3 4 5
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2944 %! 2 3 4 5 6
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2945 %! 3 4 5 6 7];
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2946 %! assert (gallery ("cauchy", 1:5, -2:2), exp)
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2947 %!
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2948 %! exp = 1 ./ [
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2949 %! 8 18 -4 2
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2950 %! 13 23 1 7
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2951 %! 9 19 -3 3
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2952 %! 15 25 3 9];
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2953 %! assert (gallery ("cauchy", [-2 3 -1 5], [10 20 -2 4]), exp)
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2954 %! assert (gallery ("cauchy", [-2 3 -1 5], [10 20 -2 4]'), exp)
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2955 %! assert (gallery ("cauchy", [-2 3 -1 5]', [10 20 -2 4]), exp)
20299
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2956
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2957 %!assert (size (gallery ("chebspec", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2958 %!assert (size (gallery ("chebspec", 5, 1)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2959 %!assert (size (gallery ("chebspec", 5, 0)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2960
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2961 %!assert (size (gallery ("chebvand", 7)), [7 7])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2962 %!assert (size (gallery ("chebvand", 1:7)), [7 7])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2963 %!assert (size (gallery ("chebvand", 5, 7)), [5 7])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2964
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2965 %!assert (size (gallery ("chow", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2966 %!assert (size (gallery ("chow", 5, 6)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2967 %!assert (size (gallery ("chow", 5, 6, 7)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2968
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2969 %!assert (gallery ("circul", 3), [1 2 3; 3 1 2; 2 3 1])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2970 %!assert (gallery ("circul", [1 3 6]), [1 3 6; 6 1 3; 3 6 1])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2971
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2972 %!assert (size (gallery ("clement", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2973 %!assert (size (gallery ("clement", 5, 1)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2974 %!assert (size (gallery ("clement", 5, 0)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2975
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2976 %!assert (size (gallery ("compar", ones (5))), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2977 %!assert (size (gallery ("compar", ones (5), 0)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2978 %!assert (size (gallery ("compar", ones (5), 1)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2979
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2980 %!assert (size (gallery ("condex", 4)), [4 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2981 %!assert (size (gallery ("condex", 4, 1)), [4 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2982 %!assert (size (gallery ("condex", 4, 1, 50)), [4 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2983
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2984 %!assert (size (gallery ("cycol", [4 5])), [4 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2985 %!assert (size (gallery ("cycol", [4 5], 1)), [4 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2986 %!assert (size (gallery ("cycol", 4)), [4 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2987 %!assert (size (gallery ("cycol", 4, 1)), [4 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2988
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2989 %!assert (size (gallery ("dorr", 4)), [4 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2990 %!assert (cellfun (@rows, nthargout (1:3, @gallery, "dorr", 4)), [3 4 3])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2991
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2992 %!assert (size (gallery ("dramadah", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2993 %!assert (size (gallery ("dramadah", 5, 2)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
2994
23381
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2995 %!test
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2996 %! exp = [
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2997 %! 0 1 2 3 4
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2998 %! 1 0 1 2 3
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
2999 %! 2 1 0 1 2
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
3000 %! 3 2 1 0 1
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
3001 %! 4 3 2 1 0];
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
3002 %! assert (gallery ("fiedler", 5), exp)
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
3003 %! assert (gallery ("fiedler", 1:5), exp)
e15ffa2b2262 gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents: 23084
diff changeset
3004 %! assert (gallery ("fiedler", -2:2), exp)
23384
d20b385d8c99 gallery.m: fix failing test due to typo on test unit.
Carnë Draug <carandraug@octave.org>
parents: 23381
diff changeset
3005 %! assert (gallery ("fiedler", 2:5), exp(1:4,1:4))
20299
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3006
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3007 %!assert (size (gallery ("forsythe", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3008 %!assert (size (gallery ("forsythe", 5, 1, 0.5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3009 %!assert (size (gallery ("forsythe", 5, 4, 7)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3010
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3011 %!assert (size (gallery ("frank", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3012 %!assert (size (gallery ("frank", 5, 1)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3013
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3014 %!assert (size (gallery ("gcdmat", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3015
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3016 %!assert (size (gallery ("gearmat", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3017 %!assert (size (gallery ("gearmat", 5, 4)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3018 %!assert (size (gallery ("gearmat", 5, 4, 3)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3019
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3020 %!assert (size (gallery ("grcar", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3021 %!assert (size (gallery ("grcar", 5, 2)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3022
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3023 %!error <N must be even> gallery ("hanowa", 5)
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3024 %!assert (size (gallery ("hanowa", 6, 5)), [6 6])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3025 %!assert (size (gallery ("hanowa", 6, 5)), [6 6])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3026
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3027 %!assert (size (gallery ("house", [1:5]')), [5 1])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3028 %!assert (cellfun (@rows, nthargout (1:2, @gallery, "house", [1:5]')), [5 1])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3029
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3030 %!assert (size (gallery ("integerdata", 5, [3 2], 5)), [3 2])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3031 %!assert (size (gallery ("integerdata", 5, [3 2 6], 5)), [3 2 6])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3032
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3033 %!assert (size (gallery ("invhess", 1:4, 1:3)), [4 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3034
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3035 %!assert (size (gallery ("invol", 4)), [4 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3036
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3037 %!assert (size (gallery ("ipjfact", 4)), [4 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3038 %!assert (size (gallery ("ipjfact", 4, 0)), [4 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3039 %!assert (size (gallery ("ipjfact", 4, 1)), [4 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3040
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3041 %!assert (size (gallery ("jordbloc", 4)), [4 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3042 %!assert (size (gallery ("jordbloc", 4, 1)), [4 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3043 %!assert (size (gallery ("jordbloc", 4, 3)), [4 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3044
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3045 %!assert (size (gallery ("kahan", 4)), [4 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3046 %!assert (size (gallery ("kahan", [4 5])), [4 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3047 %!assert (size (gallery ("kahan", [4 5], 1)), [4 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3048 %!assert (size (gallery ("kahan", [4 5], 1, 30)), [4 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3049
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3050 %!assert (size (gallery ("kms", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3051 %!assert (size (gallery ("kms", 5, 0.2)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3052
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3053 %!assert (size (gallery ("krylov", 4)), [4 4])
22021
e67ff2b11147 Check that second argument to size () is a scalar (bug #48368).
Lachlan Andrew <lachlanbis@gmail.com>
parents: 21759
diff changeset
3054 %!assert (size (gallery ("krylov", ones (4))), [4 4])
e67ff2b11147 Check that second argument to size () is a scalar (bug #48368).
Lachlan Andrew <lachlanbis@gmail.com>
parents: 21759
diff changeset
3055 %!assert (size (gallery ("krylov", ones (4), [.2 .3 .4 .5], 3)), [4 3])
20299
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3056
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3057 %!assert (size (gallery ("lauchli", 5)), [6 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3058 %!assert (size (gallery ("lauchli", 5, 3)), [6 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3059
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3060 %!assert (size (gallery ("lehmer", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3061
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3062 %!assert (size (gallery ("lesp", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3063
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3064 %!assert (size (gallery ("lotkin", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3065
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3066 %!assert (size (gallery ("minij", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3067
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3068 %!assert (size (gallery ("moler", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3069 %!assert (size (gallery ("moler", 5, 0.2)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3070
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3071 %!assert (size (gallery ("neumann", 4)), [4 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3072
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3073 %!assert (size (gallery ("normaldata", [5 4 6], 3)), [5 4 6])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3074
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3075 %!assert (size (gallery ("orthog", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3076 %!assert (size (gallery ("orthog", 5, 2)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3077 %!assert (size (gallery ("orthog", 5, -2)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3078
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3079 %!assert (size (gallery ("parter", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3080
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3081 %!assert (size (gallery ("pei", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3082 %!assert (size (gallery ("pei", 5, 4)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3083
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3084 %!assert (size (gallery ("poisson", 1)), [1 1])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3085 %!assert (size (gallery ("poisson", 4)), [16 16])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3086 %!assert (size (gallery ("poisson", 5)), [25 25])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3087
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3088 %!assert (size (gallery ("prolate", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3089 %!assert (size (gallery ("prolate", 5, 0.5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3090
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3091 %!assert (size (gallery ("randhess", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3092 %!assert (size (gallery ("randhess", 2:5)), [4 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3093
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3094 %!assert (size (gallery ("rando", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3095 %!assert (size (gallery ("rando", 5, 2)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3096
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3097 %!assert (size (gallery ("randsvd", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3098 %!assert (size (gallery ("randsvd", [5 3])), [5 3])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3099
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3100 %!assert (size (gallery ("redheff", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3101 %!assert (size (gallery ("riemann", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3102 %!assert (size (gallery ("ris", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3103
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3104 %!assert (size (gallery ("smoke", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3105 %!assert (size (gallery ("smoke", 5, 1)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3106 %!assert (gallery ("smoke", 5, 1)(5, 1), 0)
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3107
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3108 %!assert (size (gallery ("toeppd", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3109
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3110 %!assert (size (gallery ("toeppen", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3111
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3112 %!assert (size (gallery ("tridiag", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3113 %!assert (size (gallery ("tridiag", 1:4, 1:5, 1:4)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3114 %!assert (gallery ("tridiag", 5), gallery ("tridiag", 5, -1, 2, -1))
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3115
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3116 %!assert (size (gallery ("triw", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3117
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3118 %!assert (size (gallery ("uniformdata", [5 3 4], 3)), [5 3 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3119
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3120 %!assert (size (gallery ("wathen", 2, 3)), [29 29])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3121
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3122 %!assert (cellfun (@rows, nthargout (1:2, @gallery, "wilk", 3)), [3 3])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3123 %!assert (cellfun (@rows, nthargout (1:2, @gallery, "wilk", 4)), [4 4])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3124 %!assert (size (gallery ("wilk", 5)), [5 5])
c5a8eff5a05d gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents: 20298
diff changeset
3125 %!assert (size (gallery ("wilk", 21)), [21 21])