Mercurial > octave-nkf
view scripts/polynomial/private/__splinefit__.m @ 19627:446c46af4b42 stable
strip trailing whitespace from most source files
* Makefile.am, NEWS, build-aux/common.mk, configure.ac,
doc/Makefile.am, doc/doxyhtml/Makefile.am,
doc/interpreter/Makefile.am, doc/interpreter/arith.txi,
doc/interpreter/audio.txi, doc/interpreter/basics.txi,
doc/interpreter/bugs.txi, doc/interpreter/container.txi,
doc/interpreter/cp-idx.txi, doc/interpreter/data.txi,
doc/interpreter/debug.txi, doc/interpreter/diagperm.txi,
doc/interpreter/diffeq.txi, doc/interpreter/doccheck/README,
doc/interpreter/doccheck/spellcheck, doc/interpreter/emacs.txi,
doc/interpreter/errors.txi, doc/interpreter/eval.txi,
doc/interpreter/expr.txi, doc/interpreter/external.txi,
doc/interpreter/fn-idx.txi, doc/interpreter/func.txi,
doc/interpreter/geometry.txi, doc/interpreter/geometryimages.m,
doc/interpreter/gpl.txi, doc/interpreter/grammar.txi,
doc/interpreter/gui.txi, doc/interpreter/image.txi,
doc/interpreter/install.txi, doc/interpreter/interp.txi,
doc/interpreter/interpimages.m, doc/interpreter/intro.txi,
doc/interpreter/io.txi, doc/interpreter/java.txi,
doc/interpreter/linalg.txi, doc/interpreter/macros.texi,
doc/interpreter/matrix.txi, doc/interpreter/munge-texi.pl,
doc/interpreter/nonlin.txi, doc/interpreter/numbers.txi,
doc/interpreter/obsolete.txi, doc/interpreter/octave-config.1,
doc/interpreter/octave.texi, doc/interpreter/oop.txi,
doc/interpreter/op-idx.txi, doc/interpreter/optim.txi,
doc/interpreter/package.txi, doc/interpreter/plot.txi,
doc/interpreter/poly.txi, doc/interpreter/preface.txi,
doc/interpreter/quad.txi, doc/interpreter/set.txi,
doc/interpreter/signal.txi, doc/interpreter/sparse.txi,
doc/interpreter/sparseimages.m, doc/interpreter/splineimages.m,
doc/interpreter/stats.txi, doc/interpreter/stmt.txi,
doc/interpreter/strings.txi, doc/interpreter/system.txi,
doc/interpreter/testfun.txi, doc/interpreter/tips.txi,
doc/interpreter/var.txi, doc/interpreter/vectorize.txi,
doc/liboctave/Makefile.am, doc/liboctave/array.texi,
doc/liboctave/bugs.texi, doc/liboctave/cp-idx.texi,
doc/liboctave/dae.texi, doc/liboctave/diffeq.texi,
doc/liboctave/error.texi, doc/liboctave/factor.texi,
doc/liboctave/fn-idx.texi, doc/liboctave/gpl.texi,
doc/liboctave/install.texi, doc/liboctave/intro.texi,
doc/liboctave/liboctave.texi, doc/liboctave/matvec.texi,
doc/liboctave/nleqn.texi, doc/liboctave/nlfunc.texi,
doc/liboctave/ode.texi, doc/liboctave/optim.texi,
doc/liboctave/preface.texi, doc/liboctave/quad.texi,
doc/liboctave/range.texi, doc/refcard/Makefile.am,
doc/refcard/refcard.tex, etc/HACKING, etc/NEWS.1, etc/NEWS.2,
etc/NEWS.3, etc/OLD-ChangeLogs/ChangeLog,
etc/OLD-ChangeLogs/doc-ChangeLog,
etc/OLD-ChangeLogs/scripts-ChangeLog,
etc/OLD-ChangeLogs/src-ChangeLog, etc/OLD-ChangeLogs/test-ChangeLog,
etc/PROJECTS, etc/README.Cygwin, etc/README.MacOS, etc/README.MinGW,
etc/README.gnuplot, etc/gdbinit, etc/icons/Makefile.am,
examples/@polynomial/end.m, examples/@polynomial/subsasgn.m,
examples/Makefile.am, examples/standalonebuiltin.cc,
libgui/Makefile.am, libgui/qterminal/libqterminal/README,
libgui/qterminal/libqterminal/unix/BlockArray.cpp,
libgui/qterminal/libqterminal/unix/BlockArray.h,
libgui/qterminal/libqterminal/unix/Character.h,
libgui/qterminal/libqterminal/unix/CharacterColor.h,
libgui/qterminal/libqterminal/unix/Emulation.cpp,
libgui/qterminal/libqterminal/unix/Emulation.h,
libgui/qterminal/libqterminal/unix/Filter.cpp,
libgui/qterminal/libqterminal/unix/Filter.h,
libgui/qterminal/libqterminal/unix/History.cpp,
libgui/qterminal/libqterminal/unix/History.h,
libgui/qterminal/libqterminal/unix/KeyboardTranslator.cpp,
libgui/qterminal/libqterminal/unix/KeyboardTranslator.h,
libgui/qterminal/libqterminal/unix/LineFont.h,
libgui/qterminal/libqterminal/unix/QUnixTerminalImpl.cpp,
libgui/qterminal/libqterminal/unix/QUnixTerminalImpl.h,
libgui/qterminal/libqterminal/unix/Screen.cpp,
libgui/qterminal/libqterminal/unix/Screen.h,
libgui/qterminal/libqterminal/unix/ScreenWindow.cpp,
libgui/qterminal/libqterminal/unix/ScreenWindow.h,
libgui/qterminal/libqterminal/unix/TerminalCharacterDecoder.cpp,
libgui/qterminal/libqterminal/unix/TerminalCharacterDecoder.h,
libgui/qterminal/libqterminal/unix/Vt102Emulation.h,
libgui/qterminal/libqterminal/win32/QWinTerminalImpl.cpp,
libgui/qterminal/qterminal/main.cpp,
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libinterp/corefcn/graphics.in.h, libinterp/corefcn/ls-mat5.cc,
libinterp/corefcn/lu.cc, libinterp/corefcn/oct-tex-parser.yy,
libinterp/corefcn/oct-tex-symbols.in, libinterp/corefcn/quadcc.cc,
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libinterp/dldfcn/__voronoi__.cc, libinterp/gendoc.pl,
libinterp/genprops.awk, libinterp/mk-errno-list, libinterp/mk-pkg-add,
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libinterp/parse-tree/lex.ll, libinterp/parse-tree/oct-parse.in.yy,
libinterp/parse-tree/octave.gperf, liboctave/Makefile.am,
liboctave/array/Array.cc, liboctave/array/module.mk,
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liboctave/cruft/daspk/dfnrmk.f, liboctave/cruft/daspk/dhels.f,
liboctave/cruft/daspk/dheqr.f, liboctave/cruft/daspk/dinvwt.f,
liboctave/cruft/daspk/dlinsd.f, liboctave/cruft/daspk/dlinsk.f,
liboctave/cruft/daspk/dmatd.f, liboctave/cruft/daspk/dnedd.f,
liboctave/cruft/daspk/dnedk.f, liboctave/cruft/daspk/dnsd.f,
liboctave/cruft/daspk/dnsid.f, liboctave/cruft/daspk/dnsik.f,
liboctave/cruft/daspk/dnsk.f, liboctave/cruft/daspk/dorth.f,
liboctave/cruft/daspk/dslvd.f, liboctave/cruft/daspk/dslvk.f,
liboctave/cruft/daspk/dspigm.f, liboctave/cruft/daspk/dyypnw.f,
liboctave/cruft/dasrt/ddasrt.f, liboctave/cruft/dasrt/drchek.f,
liboctave/cruft/dassl/ddaslv.f, liboctave/cruft/dassl/ddassl.f,
liboctave/cruft/misc/blaswrap.c, liboctave/cruft/misc/module.mk,
liboctave/cruft/odepack/cfode.f, liboctave/cruft/odepack/dlsode.f,
liboctave/cruft/odepack/ewset.f, liboctave/cruft/odepack/intdy.f,
liboctave/cruft/odepack/prepj.f, liboctave/cruft/odepack/sintdy.f,
liboctave/cruft/odepack/slsode.f, liboctave/cruft/odepack/solsy.f,
liboctave/cruft/odepack/ssolsy.f, liboctave/cruft/odepack/stode.f,
liboctave/cruft/odepack/vnorm.f, liboctave/cruft/ranlib/Basegen.doc,
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liboctave/cruft/slatec-fn/xsgmainc.f, liboctave/numeric/module.mk,
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liboctave/operators/sparse-mk-ops.awk,
liboctave/operators/sparse-mx-ops, liboctave/operators/vx-ops,
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scripts/geometry/convhull.m, scripts/geometry/delaunay.m,
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scripts/geometry/griddata.m, scripts/geometry/griddatan.m,
scripts/geometry/voronoi.m, scripts/geometry/voronoin.m,
scripts/gui/guihandles.m, scripts/gui/inputdlg.m,
scripts/gui/listdlg.m, scripts/gui/msgbox.m, scripts/gui/questdlg.m,
scripts/gui/uigetfile.m, scripts/gui/waitbar.m, scripts/gui/warndlg.m,
scripts/help/doc.m, scripts/help/help.m, scripts/help/type.m,
scripts/image/bone.m, scripts/image/cmpermute.m,
scripts/image/cmunique.m, scripts/image/colorcube.m,
scripts/image/colormap.m, scripts/image/contrast.m,
scripts/image/gray2ind.m, scripts/image/image.m,
scripts/image/imshow.m, scripts/image/ind2gray.m, scripts/image/jet.m,
scripts/image/rgb2ntsc.m, scripts/image/spinmap.m,
scripts/io/importdata.m, scripts/io/strread.m, scripts/io/textread.m,
scripts/io/textscan.m, scripts/java/java_get.m,
scripts/java/java_set.m, scripts/java/javaaddpath.m,
scripts/java/javaclasspath.m, scripts/java/javamem.m,
scripts/linear-algebra/linsolve.m, scripts/linear-algebra/qzhess.m,
scripts/miscellaneous/debug.m, scripts/miscellaneous/desktop.m,
scripts/miscellaneous/dir.m, scripts/miscellaneous/dos.m,
scripts/miscellaneous/edit.m, scripts/miscellaneous/fact.m,
scripts/miscellaneous/getappdata.m, scripts/miscellaneous/inputname.m,
scripts/miscellaneous/license.m, scripts/miscellaneous/ls_command.m,
scripts/miscellaneous/run.m, scripts/miscellaneous/setfield.m,
scripts/miscellaneous/unix.m, scripts/miscellaneous/ver.m,
scripts/mk-pkg-add, scripts/mkdoc.pl,
scripts/optimization/fminsearch.m, scripts/optimization/optimset.m,
scripts/optimization/sqp.m, scripts/pkg/pkg.m,
scripts/pkg/private/create_pkgadddel.m,
scripts/pkg/private/fix_depends.m, scripts/pkg/private/install.m,
scripts/plot/appearance/axis.m, scripts/plot/appearance/box.m,
scripts/plot/appearance/clabel.m, scripts/plot/appearance/daspect.m,
scripts/plot/appearance/datetick.m, scripts/plot/appearance/grid.m,
scripts/plot/appearance/legend.m, scripts/plot/appearance/orient.m,
scripts/plot/appearance/shading.m, scripts/plot/appearance/text.m,
scripts/plot/appearance/title.m, scripts/plot/appearance/xlabel.m,
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scripts/plot/draw/area.m, scripts/plot/draw/bar.m,
scripts/plot/draw/barh.m, scripts/plot/draw/colorbar.m,
scripts/plot/draw/contour.m, scripts/plot/draw/contour3.m,
scripts/plot/draw/contourf.m, scripts/plot/draw/ellipsoid.m,
scripts/plot/draw/errorbar.m, scripts/plot/draw/ezcontour.m,
scripts/plot/draw/ezcontourf.m, scripts/plot/draw/ezmesh.m,
scripts/plot/draw/ezpolar.m, scripts/plot/draw/fill.m,
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scripts/plot/draw/meshc.m, scripts/plot/draw/meshz.m,
scripts/plot/draw/pareto.m, scripts/plot/draw/patch.m,
scripts/plot/draw/peaks.m, scripts/plot/draw/pie.m,
scripts/plot/draw/pie3.m, scripts/plot/draw/plot.m,
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scripts/plot/draw/semilogx.m, scripts/plot/draw/shrinkfaces.m,
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scripts/plot/draw/surfl.m, scripts/plot/draw/surfnorm.m,
scripts/plot/draw/tetramesh.m, scripts/plot/draw/trimesh.m,
scripts/plot/draw/triplot.m, scripts/plot/draw/trisurf.m,
scripts/plot/util/__gnuplot_drawnow__.m,
scripts/plot/util/__plt_get_axis_arg__.m, scripts/plot/util/axes.m,
scripts/plot/util/clf.m, scripts/plot/util/copyobj.m,
scripts/plot/util/figure.m, scripts/plot/util/gcbo.m,
scripts/plot/util/graphics_toolkit.m, scripts/plot/util/hggroup.m,
scripts/plot/util/meshgrid.m, scripts/plot/util/newplot.m,
scripts/plot/util/print.m,
scripts/plot/util/private/__add_default_menu__.m,
scripts/plot/util/private/__fltk_print__.m,
scripts/plot/util/private/__gnuplot_print__.m,
scripts/plot/util/private/__print_parse_opts__.m,
scripts/plot/util/refreshdata.m, scripts/plot/util/subplot.m,
scripts/polynomial/conv.m, scripts/polynomial/poly.m,
scripts/polynomial/polyeig.m, scripts/polynomial/polyfit.m,
scripts/polynomial/polyval.m,
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scripts/polynomial/spline.m, scripts/prefs/prefdir.m,
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scripts/sparse/private/__sprand_impl__.m, scripts/sparse/sprand.m,
scripts/sparse/sprandn.m, scripts/sparse/spy.m, scripts/sparse/svds.m,
scripts/specfun/expint.m, scripts/specfun/factor.m,
scripts/special-matrix/gallery.m, scripts/special-matrix/hankel.m,
scripts/special-matrix/toeplitz.m, scripts/startup/inputrc,
scripts/statistics/base/kurtosis.m, scripts/statistics/base/moment.m,
scripts/statistics/base/qqplot.m, scripts/statistics/base/var.m,
scripts/statistics/distributions/betarnd.m,
scripts/statistics/distributions/binoinv.m,
scripts/statistics/distributions/binopdf.m,
scripts/statistics/distributions/binornd.m,
scripts/statistics/distributions/cauchy_rnd.m,
scripts/statistics/distributions/chi2rnd.m,
scripts/statistics/distributions/discrete_pdf.m,
scripts/statistics/distributions/discrete_rnd.m,
scripts/statistics/distributions/empirical_rnd.m,
scripts/statistics/distributions/exprnd.m,
scripts/statistics/distributions/frnd.m,
scripts/statistics/distributions/gamrnd.m,
scripts/statistics/distributions/geornd.m,
scripts/statistics/distributions/hygernd.m,
scripts/statistics/distributions/kolmogorov_smirnov_cdf.m,
scripts/statistics/distributions/laplace_cdf.m,
scripts/statistics/distributions/laplace_pdf.m,
scripts/statistics/distributions/logistic_cdf.m,
scripts/statistics/distributions/logistic_pdf.m,
scripts/statistics/distributions/lognrnd.m,
scripts/statistics/distributions/nbincdf.m,
scripts/statistics/distributions/nbininv.m,
scripts/statistics/distributions/nbinpdf.m,
scripts/statistics/distributions/nbinrnd.m,
scripts/statistics/distributions/normrnd.m,
scripts/statistics/distributions/poissinv.m,
scripts/statistics/distributions/poissrnd.m,
scripts/statistics/distributions/tinv.m,
scripts/statistics/distributions/trnd.m,
scripts/statistics/distributions/unidcdf.m,
scripts/statistics/distributions/unidpdf.m,
scripts/statistics/distributions/unidrnd.m,
scripts/statistics/distributions/unifrnd.m,
scripts/statistics/distributions/wblrnd.m,
scripts/statistics/models/module.mk,
scripts/statistics/tests/kruskal_wallis_test.m,
scripts/strings/base2dec.m, scripts/strings/deblank.m,
scripts/strings/dec2base.m, scripts/strings/dec2bin.m,
scripts/strings/dec2hex.m, scripts/strings/mat2str.m,
scripts/strings/ostrsplit.m, scripts/strings/regexptranslate.m,
scripts/strings/str2num.m, scripts/strings/strcat.m,
scripts/strings/strjoin.m, scripts/strings/strsplit.m,
scripts/strings/strtok.m, scripts/strings/strtrim.m,
scripts/strings/strtrunc.m, scripts/strings/substr.m,
scripts/testfun/__run_test_suite__.m, scripts/testfun/speed.m,
scripts/testfun/test.m, scripts/time/asctime.m,
scripts/time/datenum.m, scripts/time/datevec.m,
scripts/time/weekday.m, src/Makefile.am, test/Makefile.am,
test/build-bc-overload-tests.sh, test/build-sparse-tests.sh,
test/jit.tst, test/line-continue.tst: Strip trailing whitespace.
author | John W. Eaton <jwe@octave.org> |
---|---|
date | Tue, 20 Jan 2015 08:26:57 -0500 |
parents | 9deb214ae9d5 |
children | 0e1f5a750d00 |
line wrap: on
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## This function is private because it is maintained by Jonas Lundgren ## separtely from Octave. Please do not reformat to match Octave coding ## conventions as that would make it harder to integrate upstream ## changes. % Copyright (c) 2010, Jonas Lundgren % All rights reserved. % % Redistribution and use in source and binary forms, with or without % modification, are permitted provided that the following conditions are % met: % % * Redistributions of source code must retain the above copyright % notice, this list of conditions and the following disclaimer. % * Redistributions in binary form must reproduce the above copyright % notice, this list of conditions and the following disclaimer in % the documentation and/or other materials provided with the distribution % % THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" % AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE % IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE % ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE % LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR % CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF % SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS % INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN % CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) % ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE % POSSIBILITY OF SUCH DAMAGE. function pp = __splinefit__(varargin) %SPLINEFIT Fit a spline to noisy data. % PP = SPLINEFIT(X,Y,BREAKS) fits a piecewise cubic spline with breaks % (knots) BREAKS to the noisy data (X,Y). X is a vector and Y is a vector % or an ND array. If Y is an ND array, then X(j) and Y(:,...,:,j) are % matched. Use PPVAL to evaluate PP. % % PP = SPLINEFIT(X,Y,P) where P is a positive integer interpolates the % breaks linearly from the sorted locations of X. P is the number of % spline pieces and P+1 is the number of breaks. % % OPTIONAL INPUT % Argument places 4 to 8 are reserved for optional input. % These optional arguments can be given in any order: % % PP = SPLINEFIT(...,'p') applies periodic boundary conditions to % the spline. The period length is MAX(BREAKS)-MIN(BREAKS). % % PP = SPLINEFIT(...,'r') uses robust fitting to reduce the influence % from outlying data points. Three iterations of weighted least squares % are performed. Weights are computed from previous residuals. % % PP = SPLINEFIT(...,BETA), where 0 < BETA < 1, sets the robust fitting % parameter BETA and activates robust fitting ('r' can be omitted). % Default is BETA = 1/2. BETA close to 0 gives all data equal weighting. % Increase BETA to reduce the influence from outlying data. BETA close % to 1 may cause instability or rank deficiency. % % PP = SPLINEFIT(...,N) sets the spline order to N. Default is a cubic % spline with order N = 4. A spline with P pieces has P+N-1 degrees of % freedom. With periodic boundary conditions the degrees of freedom are % reduced to P. % % PP = SPLINEFIT(...,CON) applies linear constraints to the spline. % CON is a structure with fields 'xc', 'yc' and 'cc': % 'xc', x-locations (vector) % 'yc', y-values (vector or ND array) % 'cc', coefficients (matrix). % % Constraints are linear combinations of derivatives of order 0 to N-2 % according to % % cc(1,j)*y(x) + cc(2,j)*y'(x) + ... = yc(:,...,:,j), x = xc(j). % % The maximum number of rows for 'cc' is N-1. If omitted or empty 'cc' % defaults to a single row of ones. Default for 'yc' is a zero array. % % EXAMPLES % % % Noisy data % x = linspace(0,2*pi,100); % y = sin(x) + 0.1*randn(size(x)); % % Breaks % breaks = [0:5,2*pi]; % % % Fit a spline of order 5 % pp = splinefit(x,y,breaks,5); % % % Fit a spline of order 3 with periodic boundary conditions % pp = splinefit(x,y,breaks,3,'p'); % % % Constraints: y(0) = 0, y'(0) = 1 and y(3) + y"(3) = 0 % xc = [0 0 3]; % yc = [0 1 0]; % cc = [1 0 1; 0 1 0; 0 0 1]; % con = struct('xc',xc,'yc',yc,'cc',cc); % % % Fit a cubic spline with 8 pieces and constraints % pp = splinefit(x,y,8,con); % % % Fit a spline of order 6 with constraints and periodicity % pp = splinefit(x,y,breaks,con,6,'p'); % % See also SPLINE, PPVAL, PPDIFF, PPINT % Author: Jonas Lundgren <splinefit@gmail.com> 2010 % 2009-05-06 Original SPLINEFIT. % 2010-06-23 New version of SPLINEFIT based on B-splines. % 2010-09-01 Robust fitting scheme added. % 2010-09-01 Support for data containing NaNs. % 2011-07-01 Robust fitting parameter added. % Check number of arguments error(nargchk(3,7,nargin)); % Check arguments [x,y,dim,breaks,n,periodic,beta,constr] = arguments(varargin{:}); % Evaluate B-splines base = splinebase(breaks,n); pieces = base.pieces; A = ppval(base,x); % Bin data [junk,ibin] = histc(x,[-inf,breaks(2:end-1),inf]); %#ok % Sparse system matrix mx = numel(x); ii = [ibin; ones(n-1,mx)]; ii = cumsum(ii,1); jj = repmat(1:mx,n,1); if periodic ii = mod(ii-1,pieces) + 1; A = sparse(ii,jj,A,pieces,mx); else A = sparse(ii,jj,A,pieces+n-1,mx); end % Don't use the sparse solver for small problems if pieces < 20*n/log(1.7*n) A = full(A); end % Solve if isempty(constr) % Solve Min norm(u*A-y) u = lsqsolve(A,y,beta); else % Evaluate constraints B = evalcon(base,constr,periodic); % Solve constraints [Z,u0] = solvecon(B,constr); % Solve Min norm(u*A-y), subject to u*B = yc y = y - u0*A; A = Z*A; v = lsqsolve(A,y,beta); u = u0 + v*Z; end % Periodic expansion of solution if periodic jj = mod(0:pieces+n-2,pieces) + 1; u = u(:,jj); end % Compute polynomial coefficients ii = [repmat(1:pieces,1,n); ones(n-1,n*pieces)]; ii = cumsum(ii,1); jj = repmat(1:n*pieces,n,1); C = sparse(ii,jj,base.coefs,pieces+n-1,n*pieces); coefs = u*C; coefs = reshape(coefs,[],n); % Make piecewise polynomial pp = mkpp(breaks,coefs,dim); %-------------------------------------------------------------------------- function [x,y,dim,breaks,n,periodic,beta,constr] = arguments(varargin) %ARGUMENTS Lengthy input checking % x Noisy data x-locations (1 x mx) % y Noisy data y-values (prod(dim) x mx) % dim Leading dimensions of y % breaks Breaks (1 x (pieces+1)) % n Spline order % periodic True if periodic boundary conditions % beta Robust fitting parameter, no robust fitting if beta = 0 % constr Constraint structure % constr.xc x-locations (1 x nx) % constr.yc y-values (prod(dim) x nx) % constr.cc Coefficients (?? x nx) % Reshape x-data x = varargin{1}; mx = numel(x); x = reshape(x,1,mx); % Remove trailing singleton dimensions from y y = varargin{2}; dim = size(y); while numel(dim) > 1 && dim(end) == 1 dim(end) = []; end my = dim(end); % Leading dimensions of y if numel(dim) > 1 dim(end) = []; else dim = 1; end % Reshape y-data pdim = prod(dim); y = reshape(y,pdim,my); % Check data size if mx ~= my mess = 'Last dimension of array y must equal length of vector x.'; error('arguments:datasize',mess) end % Treat NaNs in x-data inan = find(isnan(x)); if ~isempty(inan) x(inan) = []; y(:,inan) = []; mess = 'All data points with NaN as x-location will be ignored.'; warning('arguments:nanx',mess) end % Treat NaNs in y-data inan = find(any(isnan(y),1)); if ~isempty(inan) x(inan) = []; y(:,inan) = []; mess = 'All data points with NaN in their y-value will be ignored.'; warning('arguments:nany',mess) end % Check number of data points mx = numel(x); if mx == 0 error('arguments:nodata','There must be at least one data point.') end % Sort data if any(diff(x) < 0) [x,isort] = sort(x); y = y(:,isort); end % Breaks if isscalar(varargin{3}) % Number of pieces p = varargin{3}; if ~isreal(p) || ~isfinite(p) || p < 1 || fix(p) < p mess = 'Argument #3 must be a vector or a positive integer.'; error('arguments:breaks1',mess) end if x(1) < x(end) % Interpolate breaks linearly from x-data dx = diff(x); ibreaks = linspace(1,mx,p+1); [junk,ibin] = histc(ibreaks,[0,2:mx-1,mx+1]); %#ok breaks = x(ibin) + dx(ibin).*(ibreaks-ibin); else breaks = x(1) + linspace(0,1,p+1); end else % Vector of breaks breaks = reshape(varargin{3},1,[]); if isempty(breaks) || min(breaks) == max(breaks) mess = 'At least two unique breaks are required.'; error('arguments:breaks2',mess); end end % Unique breaks if any(diff(breaks) <= 0) breaks = unique(breaks); end % Optional input defaults n = 4; % Cubic splines periodic = false; % No periodic boundaries robust = false; % No robust fitting scheme beta = 0.5; % Robust fitting parameter constr = []; % No constraints % Loop over optional arguments for k = 4:nargin a = varargin{k}; if ischar(a) && isscalar(a) && lower(a) == 'p' % Periodic conditions periodic = true; elseif ischar(a) && isscalar(a) && lower(a) == 'r' % Robust fitting scheme robust = true; elseif isreal(a) && isscalar(a) && isfinite(a) && a > 0 && a < 1 % Robust fitting parameter beta = a; robust = true; elseif isreal(a) && isscalar(a) && isfinite(a) && a > 0 && fix(a) == a % Spline order n = a; elseif isstruct(a) && isscalar(a) % Constraint structure constr = a; else error('arguments:nonsense','Failed to interpret argument #%d.',k) end end % No robust fitting if ~robust beta = 0; end % Check exterior data h = diff(breaks); xlim1 = breaks(1) - 0.01*h(1); xlim2 = breaks(end) + 0.01*h(end); if x(1) < xlim1 || x(end) > xlim2 if periodic % Move data inside domain P = breaks(end) - breaks(1); x = mod(x-breaks(1),P) + breaks(1); % Sort [x,isort] = sort(x); y = y(:,isort); else mess = 'Some data points are outside the spline domain.'; warning('arguments:exteriordata',mess) end end % Return if isempty(constr) return end % Unpack constraints xc = []; yc = []; cc = []; names = fieldnames(constr); for k = 1:numel(names) switch names{k} case {'xc'} xc = constr.xc; case {'yc'} yc = constr.yc; case {'cc'} cc = constr.cc; otherwise mess = 'Unknown field ''%s'' in constraint structure.'; warning('arguments:unknownfield',mess,names{k}) end end % Check xc if isempty(xc) mess = 'Constraints contains no x-locations.'; error('arguments:emptyxc',mess) else nx = numel(xc); xc = reshape(xc,1,nx); end % Check yc if isempty(yc) % Zero array yc = zeros(pdim,nx); elseif numel(yc) == 1 % Constant array yc = zeros(pdim,nx) + yc; elseif numel(yc) ~= pdim*nx % Malformed array error('arguments:ycsize','Cannot reshape yc to size %dx%d.',pdim,nx) else % Reshape array yc = reshape(yc,pdim,nx); end % Check cc if isempty(cc) cc = ones(size(xc)); elseif numel(size(cc)) ~= 2 error('arguments:ccsize1','Constraint coefficients cc must be 2-D.') elseif size(cc,2) ~= nx mess = 'Last dimension of cc must equal length of xc.'; error('arguments:ccsize2',mess) end % Check high order derivatives if size(cc,1) >= n if any(any(cc(n:end,:))) mess = 'Constraints involve derivatives of order %d or larger.'; error('arguments:difforder',mess,n-1) end cc = cc(1:n-1,:); end % Check exterior constraints if min(xc) < xlim1 || max(xc) > xlim2 if periodic % Move constraints inside domain P = breaks(end) - breaks(1); xc = mod(xc-breaks(1),P) + breaks(1); else mess = 'Some constraints are outside the spline domain.'; warning('arguments:exteriorconstr',mess) end end % Pack constraints constr = struct('xc',xc,'yc',yc,'cc',cc); %-------------------------------------------------------------------------- function pp = splinebase(breaks,n) %SPLINEBASE Generate B-spline base PP of order N for breaks BREAKS breaks = breaks(:); % Breaks breaks0 = breaks'; % Initial breaks h = diff(breaks); % Spacing pieces = numel(h); % Number of pieces deg = n - 1; % Polynomial degree % Extend breaks periodically if deg > 0 if deg <= pieces hcopy = h; else hcopy = repmat(h,ceil(deg/pieces),1); end % to the left hl = hcopy(end:-1:end-deg+1); bl = breaks(1) - cumsum(hl); % and to the right hr = hcopy(1:deg); br = breaks(end) + cumsum(hr); % Add breaks breaks = [bl(deg:-1:1); breaks; br]; h = diff(breaks); pieces = numel(h); end % Initiate polynomial coefficients coefs = zeros(n*pieces,n); coefs(1:n:end,1) = 1; % Expand h ii = [1:pieces; ones(deg,pieces)]; ii = cumsum(ii,1); ii = min(ii,pieces); H = h(ii(:)); % Recursive generation of B-splines for k = 2:n % Antiderivatives of splines for j = 1:k-1 coefs(:,j) = coefs(:,j).*H/(k-j); end Q = sum(coefs,2); Q = reshape(Q,n,pieces); Q = cumsum(Q,1); c0 = [zeros(1,pieces); Q(1:deg,:)]; coefs(:,k) = c0(:); % Normalize antiderivatives by max value fmax = repmat(Q(n,:),n,1); fmax = fmax(:); for j = 1:k coefs(:,j) = coefs(:,j)./fmax; end % Diff of adjacent antiderivatives coefs(1:end-deg,1:k) = coefs(1:end-deg,1:k) - coefs(n:end,1:k); coefs(1:n:end,k) = 0; end % Scale coefficients scale = ones(size(H)); for k = 1:n-1 scale = scale./H; coefs(:,n-k) = scale.*coefs(:,n-k); end % Reduce number of pieces pieces = pieces - 2*deg; % Sort coefficients by interval number ii = [n*(1:pieces); deg*ones(deg,pieces)]; ii = cumsum(ii,1); coefs = coefs(ii(:),:); % Make piecewise polynomial pp = mkpp(breaks0,coefs,n); %-------------------------------------------------------------------------- function B = evalcon(base,constr,periodic) %EVALCON Evaluate linear constraints % Unpack structures breaks = base.breaks; pieces = base.pieces; n = base.order; xc = constr.xc; cc = constr.cc; % Bin data [junk,ibin] = histc(xc,[-inf,breaks(2:end-1),inf]); %#ok % Evaluate constraints nx = numel(xc); B0 = zeros(n,nx); for k = 1:size(cc,1) if any(cc(k,:)) B0 = B0 + repmat(cc(k,:),n,1).*ppval(base,xc); end % Differentiate base coefs = base.coefs(:,1:n-k); for j = 1:n-k-1 coefs(:,j) = (n-k-j+1)*coefs(:,j); end base.coefs = coefs; base.order = n-k; end % Sparse output ii = [ibin; ones(n-1,nx)]; ii = cumsum(ii,1); jj = repmat(1:nx,n,1); if periodic ii = mod(ii-1,pieces) + 1; B = sparse(ii,jj,B0,pieces,nx); else B = sparse(ii,jj,B0,pieces+n-1,nx); end %-------------------------------------------------------------------------- function [Z,u0] = solvecon(B,constr) %SOLVECON Find a particular solution u0 and null space Z (Z*B = 0) % for constraint equation u*B = yc. yc = constr.yc; tol = 1000*eps; % Remove blank rows ii = any(B,2); B2 = full(B(ii,:)); % Null space of B2 if isempty(B2) Z2 = []; else % QR decomposition with column permutation [Q,R,dummy] = qr(B2); %#ok R = abs(R); jj = all(R < R(1)*tol, 2); Z2 = Q(:,jj)'; end % Sizes [m,ncon] = size(B); m2 = size(B2,1); nz = size(Z2,1); % Sparse null space of B Z = sparse(nz+1:nz+m-m2,find(~ii),1,nz+m-m2,m); Z(1:nz,ii) = Z2; % Warning rank deficient if nz + ncon > m2 mess = 'Rank deficient constraints, rank = %d.'; warning('solvecon:deficient',mess,m2-nz); end % Particular solution u0 = zeros(size(yc,1),m); if any(yc(:)) % Non-homogeneous case u0(:,ii) = yc/B2; % Check solution if norm(u0*B - yc,'fro') > norm(yc,'fro')*tol mess = 'Inconsistent constraints. No solution within tolerance.'; error('solvecon:inconsistent',mess) end end %-------------------------------------------------------------------------- function u = lsqsolve(A,y,beta) %LSQSOLVE Solve Min norm(u*A-y) % Avoid sparse-complex limitations if issparse(A) && ~isreal(y) A = full(A); end % Solution u = y/A; % Robust fitting if beta > 0 [m,n] = size(y); alpha = 0.5*beta/(1-beta)/m; for k = 1:3 % Residual r = u*A - y; rr = r.*conj(r); rrmean = sum(rr,2)/n; rrmean(~rrmean) = 1; rrhat = (alpha./rrmean)'*rr; % Weights w = exp(-rrhat); spw = spdiags(w',0,n,n); % Solve weighted problem u = (y*spw)/(A*spw); end end