view liboctave/numeric/oct-norm.cc @ 21301:40de9f8f23a6

Use '#include "config.h"' rather than <config.h>. * mk-octave-config-h.sh, mk-opts.pl, Backend.cc, BaseControl.cc, ButtonControl.cc, Canvas.cc, CheckBoxControl.cc, Container.cc, ContextMenu.cc, EditControl.cc, Figure.cc, FigureWindow.cc, GLCanvas.cc, KeyMap.cc, ListBoxControl.cc, Logger.cc, Menu.cc, MouseModeActionGroup.cc, Object.cc, ObjectFactory.cc, ObjectProxy.cc, Panel.cc, PopupMenuControl.cc, PushButtonControl.cc, PushTool.cc, QtHandlesUtils.cc, RadioButtonControl.cc, SliderControl.cc, TextControl.cc, TextEdit.cc, ToggleButtonControl.cc, ToggleTool.cc, ToolBar.cc, ToolBarButton.cc, __init_qt__.cc, annotation-dialog.cc, gl-select.cc, module.mk, kpty.cpp, color-picker.cc, dialog.cc, documentation-dock-widget.cc, files-dock-widget.cc, find-files-dialog.cc, find-files-model.cc, history-dock-widget.cc, file-editor-tab.cc, file-editor-tab.h, file-editor.cc, find-dialog.cc, marker.cc, octave-qscintilla.cc, octave-txt-lexer.cc, main-window.cc, octave-cmd.cc, octave-dock-widget.cc, octave-gui.cc, octave-interpreter.cc, octave-qt-link.cc, parser.cc, webinfo.cc, resource-manager.cc, settings-dialog.cc, shortcut-manager.cc, terminal-dock-widget.cc, thread-manager.cc, welcome-wizard.cc, workspace-model.cc, workspace-view.cc, build-env-features.sh, build-env.in.cc, Cell.cc, __contourc__.cc, __dispatch__.cc, __dsearchn__.cc, __ichol__.cc, __ilu__.cc, __lin_interpn__.cc, __pchip_deriv__.cc, __qp__.cc, balance.cc, besselj.cc, betainc.cc, bitfcns.cc, bsxfun.cc, c-file-ptr-stream.cc, cdisplay.c, cellfun.cc, coct-hdf5-types.c, colloc.cc, comment-list.cc, conv2.cc, daspk.cc, dasrt.cc, dassl.cc, data.cc, debug.cc, defaults.cc, defun.cc, det.cc, dirfns.cc, display.cc, dlmread.cc, dot.cc, dynamic-ld.cc, eig.cc, ellipj.cc, error.cc, errwarn.cc, event-queue.cc, fft.cc, fft2.cc, fftn.cc, file-io.cc, filter.cc, find.cc, ft-text-renderer.cc, gammainc.cc, gcd.cc, getgrent.cc, getpwent.cc, getrusage.cc, givens.cc, gl-render.cc, gl2ps-print.cc, graphics.cc, gripes.cc, hash.cc, help.cc, hess.cc, hex2num.cc, hook-fcn.cc, input.cc, inv.cc, jit-ir.cc, jit-typeinfo.cc, jit-util.cc, kron.cc, load-path.cc, load-save.cc, lookup.cc, ls-ascii-helper.cc, ls-hdf5.cc, ls-mat-ascii.cc, ls-mat4.cc, ls-mat5.cc, ls-oct-binary.cc, ls-oct-text.cc, ls-utils.cc, lsode.cc, lu.cc, luinc.cc, mappers.cc, matrix_type.cc, max.cc, mex.cc, mgorth.cc, nproc.cc, oct-errno.in.cc, oct-fstrm.cc, oct-hdf5-types.cc, oct-hist.cc, oct-iostrm.cc, oct-lvalue.cc, oct-map.cc, oct-prcstrm.cc, oct-procbuf.cc, oct-stream.cc, oct-strstrm.cc, oct-tex-lexer.in.ll, oct-tex-parser.in.yy, octave-link.cc, ordschur.cc, pager.cc, pinv.cc, pr-output.cc, procstream.cc, profiler.cc, psi.cc, pt-jit.cc, quad.cc, quadcc.cc, qz.cc, rand.cc, rcond.cc, regexp.cc, schur.cc, sighandlers.cc, siglist.c, sparse-xdiv.cc, sparse-xpow.cc, sparse.cc, spparms.cc, sqrtm.cc, str2double.cc, strfind.cc, strfns.cc, sub2ind.cc, svd.cc, sylvester.cc, symtab.cc, syscalls.cc, sysdep.cc, text-renderer.cc, time.cc, toplev.cc, tril.cc, tsearch.cc, txt-eng.cc, typecast.cc, urlwrite.cc, utils.cc, variables.cc, xdiv.cc, xgl2ps.c, xnorm.cc, xpow.cc, zfstream.cc, __delaunayn__.cc, __eigs__.cc, __fltk_uigetfile__.cc, __glpk__.cc, __init_fltk__.cc, __init_gnuplot__.cc, __magick_read__.cc, __osmesa_print__.cc, __voronoi__.cc, amd.cc, audiodevinfo.cc, audioread.cc, ccolamd.cc, chol.cc, colamd.cc, convhulln.cc, dmperm.cc, fftw.cc, qr.cc, symbfact.cc, symrcm.cc, mkbuiltins, mkops, ov-base-diag.cc, ov-base-int.cc, ov-base-mat.cc, ov-base-scalar.cc, ov-base-sparse.cc, ov-base.cc, ov-bool-mat.cc, ov-bool-sparse.cc, ov-bool.cc, ov-builtin.cc, ov-cell.cc, ov-ch-mat.cc, ov-class.cc, ov-classdef.cc, ov-colon.cc, ov-complex.cc, ov-cs-list.cc, ov-cx-diag.cc, ov-cx-mat.cc, ov-cx-sparse.cc, ov-dld-fcn.cc, ov-fcn-handle.cc, ov-fcn-inline.cc, ov-fcn.cc, ov-float.cc, ov-flt-complex.cc, ov-flt-cx-diag.cc, ov-flt-cx-mat.cc, ov-flt-re-diag.cc, ov-flt-re-mat.cc, ov-int16.cc, ov-int32.cc, ov-int64.cc, ov-int8.cc, ov-java.cc, ov-lazy-idx.cc, ov-mex-fcn.cc, ov-null-mat.cc, ov-oncleanup.cc, ov-perm.cc, ov-range.cc, ov-re-diag.cc, ov-re-mat.cc, ov-re-sparse.cc, ov-scalar.cc, ov-str-mat.cc, ov-struct.cc, ov-typeinfo.cc, ov-uint16.cc, ov-uint32.cc, ov-uint64.cc, ov-uint8.cc, ov-usr-fcn.cc, ov.cc, ovl.cc, octave.cc, 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-cell.cc, op-chm.cc, op-class.cc, op-cm-cm.cc, op-cm-cs.cc, op-cm-m.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-dm.cc, op-dm-scm.cc, op-dm-sm.cc, op-dm-template.cc, op-dms-template.cc, op-double-conv.cc, op-fcdm-fcdm.cc, op-fcdm-fdm.cc, op-fcm-fcm.cc, op-fcm-fcs.cc, op-fcm-fm.cc, op-fcm-fs.cc, op-fcn.cc, op-fcs-fcm.cc, op-fcs-fcs.cc, op-fcs-fm.cc, op-fcs-fs.cc, op-fdm-fdm.cc, op-float-conv.cc, op-fm-fcm.cc, op-fm-fcs.cc, op-fm-fm.cc, op-fm-fs.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-conv.cc, op-m-cm.cc, op-m-cs.cc, op-m-m.cc, op-m-s.cc, op-m-scm.cc, op-m-sm.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, lex.ll, oct-parse.in.yy, pt-arg-list.cc, pt-array-list.cc, pt-assign.cc, pt-binop.cc, pt-bp.cc, pt-cbinop.cc, pt-cell.cc, pt-check.cc, pt-classdef.cc, pt-cmd.cc, pt-colon.cc, pt-const.cc, pt-decl.cc, pt-eval.cc, pt-except.cc, pt-exp.cc, pt-fcn-handle.cc, pt-funcall.cc, pt-id.cc, pt-idx.cc, pt-jump.cc, pt-loop.cc, pt-mat.cc, pt-misc.cc, pt-pr-code.cc, pt-select.cc, pt-stmt.cc, pt-unop.cc, pt.cc, token.cc, Array-jit.cc, Array-os.cc, Array-sym.cc, Array-tc.cc, version.cc, 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-voidp.cc, Array.cc, CColVector.cc, CDiagMatrix.cc, CMatrix.cc, CNDArray.cc, CRowVector.cc, CSparse.cc, DiagArray2.cc, MArray-C.cc, MArray-d.cc, MArray-f.cc, MArray-fC.cc, MArray-i.cc, MArray-s.cc, MArray.cc, MDiagArray2.cc, MSparse-C.cc, MSparse-d.cc, MatrixType.cc, PermMatrix.cc, Range.cc, Sparse-C.cc, Sparse-b.cc, Sparse-d.cc, Sparse.cc, boolMatrix.cc, boolNDArray.cc, boolSparse.cc, chMatrix.cc, chNDArray.cc, dColVector.cc, dDiagMatrix.cc, dMatrix.cc, dNDArray.cc, dRowVector.cc, dSparse.cc, dim-vector.cc, fCColVector.cc, fCDiagMatrix.cc, fCMatrix.cc, fCNDArray.cc, fCRowVector.cc, fColVector.cc, fDiagMatrix.cc, fMatrix.cc, fNDArray.cc, fRowVector.cc, idx-vector.cc, int16NDArray.cc, int32NDArray.cc, int64NDArray.cc, int8NDArray.cc, intNDArray.cc, uint16NDArray.cc, uint32NDArray.cc, uint64NDArray.cc, uint8NDArray.cc, blaswrap.c, cquit.c, f77-extern.cc, f77-fcn.c, lo-error.c, quit.cc, CollocWt.cc, DASPK.cc, DASRT.cc, DASSL.cc, EIG.cc, LSODE.cc, ODES.cc, Quad.cc, aepbalance.cc, chol.cc, eigs-base.cc, fEIG.cc, gepbalance.cc, hess.cc, lo-mappers.cc, lo-specfun.cc, lu.cc, oct-convn.cc, oct-fftw.cc, oct-norm.cc, oct-rand.cc, oct-spparms.cc, qr.cc, qrp.cc, randgamma.c, randmtzig.c, randpoisson.c, schur.cc, sparse-chol.cc, sparse-dmsolve.cc, sparse-lu.cc, sparse-qr.cc, svd.cc, mk-ops.awk, dir-ops.cc, file-ops.cc, file-stat.cc, lo-sysdep.cc, mach-info.cc, oct-env.cc, oct-group.cc, oct-passwd.cc, oct-syscalls.cc, oct-time.cc, oct-uname.cc, cmd-edit.cc, cmd-hist.cc, data-conv.cc, f2c-main.c, glob-match.cc, kpse.cc, lo-array-errwarn.cc, lo-array-gripes.cc, lo-cutils.c, lo-ieee.cc, lo-regexp.cc, lo-utils.cc, oct-base64.cc, oct-glob.cc, oct-inttypes.cc, oct-locbuf.cc, oct-mutex.cc, oct-rl-edit.c, oct-rl-hist.c, oct-shlib.cc, oct-sort.cc, pathsearch.cc, singleton-cleanup.cc, sparse-sort.cc, sparse-util.cc, str-vec.cc, unwind-prot.cc, url-transfer.cc, display-available.c, main-cli.cc, main-gui.cc, main.in.cc, mkoctfile.in.cc, octave-config.in.cc: Use '#include "config.h"' rather than <config.h>.
author Rik <rik@octave.org>
date Thu, 18 Feb 2016 13:34:50 -0800
parents 85d8280c64f4
children bcc30b45a225
line wrap: on
line source

/*

Copyright (C) 2008-2015 VZLU Prague, a.s.

This file is part of Octave.

Octave is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by the
Free Software Foundation; either version 3 of the License, or (at your
option) any later version.

Octave is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
for more details.

You should have received a copy of the GNU General Public License
along with Octave; see the file COPYING.  If not, see
<http://www.gnu.org/licenses/>.

*/

// author: Jaroslav Hajek <highegg@gmail.com>

#ifdef HAVE_CONFIG_H
#  include "config.h"
#endif

#include <cassert>
#include <cfloat>
#include <cmath>

#include <iostream>
#include <vector>

#include "Array-util.h"
#include "Array.h"
#include "CColVector.h"
#include "CMatrix.h"
#include "CRowVector.h"
#include "CSparse.h"
#include "dColVector.h"
#include "dDiagMatrix.h"
#include "dMatrix.h"
#include "dRowVector.h"
#include "dSparse.h"
#include "fCColVector.h"
#include "fCMatrix.h"
#include "fCRowVector.h"
#include "fColVector.h"
#include "fDiagMatrix.h"
#include "fMatrix.h"
#include "fRowVector.h"
#include "lo-error.h"
#include "lo-ieee.h"
#include "mx-cm-s.h"
#include "mx-fcm-fs.h"
#include "mx-fs-fcm.h"
#include "mx-s-cm.h"
#include "oct-cmplx.h"
#include "svd.h"

// Theory: norm accumulator is an object that has an accum method able
// to handle both real and complex element, and a cast operator
// returning the intermediate norm. Reference: Higham, N. "Estimating
// the Matrix p-Norm." Numer. Math. 62, 539-555, 1992.

// norm accumulator for the p-norm
template <typename R>
class norm_accumulator_p
{
  R p,scl,sum;
public:
  norm_accumulator_p () {} // we need this one for Array
  norm_accumulator_p (R pp) : p(pp), scl(0), sum(1) {}

  template <typename U>
  void accum (U val)
  {
    octave_quit ();
    R t = std::abs (val);
    if (scl == t) // we need this to handle Infs properly
      sum += 1;
    else if (scl < t)
      {
        sum *= std::pow (scl/t, p);
        sum += 1;
        scl = t;
      }
    else if (t != 0)
      sum += std::pow (t/scl, p);
  }
  operator R () { return scl * std::pow (sum, 1/p); }
};

// norm accumulator for the minus p-pseudonorm
template <typename R>
class norm_accumulator_mp
{
  R p,scl,sum;
public:
  norm_accumulator_mp () {} // we need this one for Array
  norm_accumulator_mp (R pp) : p(pp), scl(0), sum(1) {}

  template <typename U>
  void accum (U val)
  {
    octave_quit ();
    R t = 1 / std::abs (val);
    if (scl == t)
      sum += 1;
    else if (scl < t)
      {
        sum *= std::pow (scl/t, p);
        sum += 1;
        scl = t;
      }
    else if (t != 0)
      sum += std::pow (t/scl, p);
  }
  operator R () { return scl * std::pow (sum, -1/p); }
};

// norm accumulator for the 2-norm (euclidean)
template <typename R>
class norm_accumulator_2
{
  R scl,sum;
  static R pow2 (R x) { return x*x; }
public:
  norm_accumulator_2 () : scl(0), sum(1) {}

  void accum (R val)
  {
    R t = std::abs (val);
    if (scl == t)
      sum += 1;
    else if (scl < t)
      {
        sum *= pow2 (scl/t);
        sum += 1;
        scl = t;
      }
    else if (t != 0)
      sum += pow2 (t/scl);
  }

  void accum (std::complex<R> val)
  {
    accum (val.real ());
    accum (val.imag ());
  }

  operator R () { return scl * std::sqrt (sum); }
};

// norm accumulator for the 1-norm (city metric)
template <typename R>
class norm_accumulator_1
{
  R sum;
public:
  norm_accumulator_1 () : sum (0) {}
  template <typename U>
  void accum (U val)
  {
    sum += std::abs (val);
  }
  operator R () { return sum; }
};

// norm accumulator for the inf-norm (max metric)
template <typename R>
class norm_accumulator_inf
{
  R max;
public:
  norm_accumulator_inf () : max (0) {}
  template <typename U>
  void accum (U val)
  {
    if (xisnan (val))
      max = octave_NaN;
    else
      max = std::max (max, std::abs (val));
  }
  operator R () { return max; }
};

// norm accumulator for the -inf pseudonorm (min abs value)
template <typename R>
class norm_accumulator_minf
{
  R min;
public:
  norm_accumulator_minf () : min (octave_Inf) {}
  template <typename U>
  void accum (U val)
  {
    if (xisnan (val))
      min = octave_NaN;
    else
      min = std::min (min, std::abs (val));
  }
  operator R () { return min; }
};

// norm accumulator for the 0-pseudonorm (hamming distance)
template <typename R>
class norm_accumulator_0
{
  unsigned int num;
public:
  norm_accumulator_0 () : num (0) {}
  template <typename U>
  void accum (U val)
  {
    if (val != static_cast<U> (0)) ++num;
  }
  operator R () { return num; }
};


// OK, we're armed :) Now let's go for the fun

template <typename T, typename R, typename ACC>
inline void vector_norm (const Array<T>& v, R& res, ACC acc)
{
  for (octave_idx_type i = 0; i < v.numel (); i++)
    acc.accum (v(i));

  res = acc;
}

// dense versions
template <typename T, typename R, typename ACC>
void column_norms (const MArray<T>& m, MArray<R>& res, ACC acc)
{
  res = MArray<R> (dim_vector (1, m.columns ()));
  for (octave_idx_type j = 0; j < m.columns (); j++)
    {
      ACC accj = acc;
      for (octave_idx_type i = 0; i < m.rows (); i++)
        accj.accum (m(i, j));

      res.xelem (j) = accj;
    }
}

template <typename T, typename R, typename ACC>
void row_norms (const MArray<T>& m, MArray<R>& res, ACC acc)
{
  res = MArray<R> (dim_vector (m.rows (), 1));
  std::vector<ACC> acci (m.rows (), acc);
  for (octave_idx_type j = 0; j < m.columns (); j++)
    {
      for (octave_idx_type i = 0; i < m.rows (); i++)
        acci[i].accum (m(i, j));
    }

  for (octave_idx_type i = 0; i < m.rows (); i++)
    res.xelem (i) = acci[i];
}

// sparse versions
template <typename T, typename R, typename ACC>
void column_norms (const MSparse<T>& m, MArray<R>& res, ACC acc)
{
  res = MArray<R> (dim_vector (1, m.columns ()));
  for (octave_idx_type j = 0; j < m.columns (); j++)
    {
      ACC accj = acc;
      for (octave_idx_type k = m.cidx (j); k < m.cidx (j+1); k++)
        accj.accum (m.data (k));

      res.xelem (j) = accj;
    }
}

template <typename T, typename R, typename ACC>
void row_norms (const MSparse<T>& m, MArray<R>& res, ACC acc)
{
  res = MArray<R> (dim_vector (m.rows (), 1));
  std::vector<ACC> acci (m.rows (), acc);
  for (octave_idx_type j = 0; j < m.columns (); j++)
    {
      for (octave_idx_type k = m.cidx (j); k < m.cidx (j+1); k++)
        acci[m.ridx (k)].accum (m.data (k));
    }

  for (octave_idx_type i = 0; i < m.rows (); i++)
    res.xelem (i) = acci[i];
}

// now the dispatchers
#define DEFINE_DISPATCHER(FUNC_NAME, ARG_TYPE, RES_TYPE) \
template <typename T, typename R> \
RES_TYPE FUNC_NAME (const ARG_TYPE& v, R p) \
{ \
  RES_TYPE res; \
  if (p == 2) \
    FUNC_NAME (v, res, norm_accumulator_2<R> ()); \
  else if (p == 1) \
    FUNC_NAME (v, res, norm_accumulator_1<R> ()); \
  else if (lo_ieee_isinf (p)) \
    { \
      if (p > 0) \
        FUNC_NAME (v, res, norm_accumulator_inf<R> ()); \
      else \
        FUNC_NAME (v, res, norm_accumulator_minf<R> ()); \
    } \
  else if (p == 0) \
    FUNC_NAME (v, res, norm_accumulator_0<R> ()); \
  else if (p > 0) \
    FUNC_NAME (v, res, norm_accumulator_p<R> (p)); \
  else \
    FUNC_NAME (v, res, norm_accumulator_mp<R> (p)); \
  return res; \
}

DEFINE_DISPATCHER (vector_norm, MArray<T>, R)
DEFINE_DISPATCHER (column_norms, MArray<T>, MArray<R>)
DEFINE_DISPATCHER (row_norms, MArray<T>, MArray<R>)
DEFINE_DISPATCHER (column_norms, MSparse<T>, MArray<R>)
DEFINE_DISPATCHER (row_norms, MSparse<T>, MArray<R>)

// The approximate subproblem in Higham's method. Find lambda and mu such that
// norm ([lambda, mu], p) == 1 and norm (y*lambda + col*mu, p) is maximized.
// Real version. As in Higham's paper.
template <typename ColVectorT, typename R>
static void
higham_subp (const ColVectorT& y, const ColVectorT& col,
             octave_idx_type nsamp, R p, R& lambda, R& mu)
{
  R nrm = 0;
  for (octave_idx_type i = 0; i < nsamp; i++)
    {
      octave_quit ();
      R fi = i * static_cast<R> (M_PI) / nsamp;
      R lambda1 = cos (fi);
      R mu1 = sin (fi);
      R lmnr = std::pow (std::pow (std::abs (lambda1), p) +
                         std::pow (std::abs (mu1), p), 1/p);
      lambda1 /= lmnr; mu1 /= lmnr;
      R nrm1 = vector_norm (lambda1 * y + mu1 * col, p);
      if (nrm1 > nrm)
        {
          lambda = lambda1;
          mu = mu1;
          nrm = nrm1;
        }
    }
}

// Complex version. Higham's paper does not deal with complex case, so we use a
// simple extension. First, guess the magnitudes as in real version, then try
// to rotate lambda to improve further.
template <typename ColVectorT, typename R>
static void
higham_subp (const ColVectorT& y, const ColVectorT& col,
             octave_idx_type nsamp, R p,
             std::complex<R>& lambda, std::complex<R>& mu)
{
  typedef std::complex<R> CR;
  R nrm = 0;
  lambda = 1.0;
  CR lamcu = lambda / std::abs (lambda);
  // Probe magnitudes
  for (octave_idx_type i = 0; i < nsamp; i++)
    {
      octave_quit ();
      R fi = i * static_cast<R> (M_PI) / nsamp;
      R lambda1 = cos (fi);
      R mu1 = sin (fi);
      R lmnr = std::pow (std::pow (std::abs (lambda1), p) +
                         std::pow (std::abs (mu1), p), 1/p);
      lambda1 /= lmnr; mu1 /= lmnr;
      R nrm1 = vector_norm (lambda1 * lamcu * y + mu1 * col, p);
      if (nrm1 > nrm)
        {
          lambda = lambda1 * lamcu;
          mu = mu1;
          nrm = nrm1;
        }
    }
  R lama = std::abs (lambda);
  // Probe orientation
  for (octave_idx_type i = 0; i < nsamp; i++)
    {
      octave_quit ();
      R fi = i * static_cast<R> (M_PI) / nsamp;
      lamcu = CR (cos (fi), sin (fi));
      R nrm1 = vector_norm (lama * lamcu * y + mu * col, p);
      if (nrm1 > nrm)
        {
          lambda = lama * lamcu;
          nrm = nrm1;
        }
    }
}

// the p-dual element (should work for both real and complex)
template <typename T, typename R>
inline T elem_dual_p (T x, R p)
{
  return signum (x) * std::pow (std::abs (x), p-1);
}

// the VectorT is used for vectors, but actually it has to be
// a Matrix type to allow all the operations. For instance SparseMatrix
// does not support multiplication with column/row vectors.
// the dual vector
template <typename VectorT, typename R>
VectorT dual_p (const VectorT& x, R p, R q)
{
  VectorT res (x.dims ());
  for (octave_idx_type i = 0; i < x.numel (); i++)
    res.xelem (i) = elem_dual_p (x(i), p);
  return res / vector_norm (res, q);
}

// Higham's hybrid method
template <typename MatrixT, typename VectorT, typename R>
R higham (const MatrixT& m, R p, R tol, int maxiter,
          VectorT& x)
{
  x.resize (m.columns (), 1);
  // the OSE part
  VectorT y(m.rows (), 1, 0), z(m.rows (), 1);
  typedef typename VectorT::element_type RR;
  RR lambda = 0;
  RR mu = 1;
  for (octave_idx_type k = 0; k < m.columns (); k++)
    {
      octave_quit ();
      VectorT col (m.column (k));
      if (k > 0)
        higham_subp (y, col, 4*k, p, lambda, mu);
      for (octave_idx_type i = 0; i < k; i++)
        x(i) *= lambda;
      x(k) = mu;
      y = lambda * y + mu * col;
    }

  // the PM part
  x = x / vector_norm (x, p);
  R q = p/(p-1);

  R gamma = 0, gamma1;
  int iter = 0;
  while (iter < maxiter)
    {
      octave_quit ();
      y = m*x;
      gamma1 = gamma;
      gamma = vector_norm (y, p);
      z = dual_p (y, p, q);
      z = z.hermitian ();
      z = z * m;

      if (iter > 0 && (vector_norm (z, q) <= gamma
                       || (gamma - gamma1) <= tol*gamma))
        break;

      z = z.hermitian ();
      x = dual_p (z, q, p);
      iter++;
    }

  return gamma;
}

// derive column vector and SVD types

static const char *p_less1_gripe = "xnorm: p must be at least 1";

// Static constant to control the maximum number of iterations.  100 seems to
// be a good value.  Eventually, we can provide a means to change this
// constant from Octave.
static int max_norm_iter = 100;

// version with SVD for dense matrices
template <typename MatrixT, typename VectorT, typename R>
R svd_matrix_norm (const MatrixT& m, R p, VectorT)
{
  R res = 0;
  if (p == 2)
    {
      svd<MatrixT> fact (m, svd<MatrixT>::sigma_only);
      res = fact.singular_values () (0,0);
    }
  else if (p == 1)
    res = xcolnorms (m, 1).max ();
  else if (lo_ieee_isinf (p))
    res = xrownorms (m, 1).max ();
  else if (p > 1)
    {
      VectorT x;
      const R sqrteps = std::sqrt (std::numeric_limits<R>::epsilon ());
      res = higham (m, p, sqrteps, max_norm_iter, x);
    }
  else
    (*current_liboctave_error_handler) (p_less1_gripe);

  return res;
}

// SVD-free version for sparse matrices
template <typename MatrixT, typename VectorT, typename R>
R matrix_norm (const MatrixT& m, R p, VectorT)
{
  R res = 0;
  if (p == 1)
    res = xcolnorms (m, 1).max ();
  else if (lo_ieee_isinf (p))
    res = xrownorms (m, 1).max ();
  else if (p > 1)
    {
      VectorT x;
      const R sqrteps = std::sqrt (std::numeric_limits<R>::epsilon ());
      res = higham (m, p, sqrteps, max_norm_iter, x);
    }
  else
    (*current_liboctave_error_handler) (p_less1_gripe);

  return res;
}

// and finally, here's what we've promised in the header file

#define DEFINE_XNORM_FUNCS(PREFIX, RTYPE) \
  OCTAVE_API RTYPE xnorm (const PREFIX##ColumnVector& x, RTYPE p) \
  { return vector_norm (x, p); } \
  OCTAVE_API RTYPE xnorm (const PREFIX##RowVector& x, RTYPE p) \
  { return vector_norm (x, p); } \
  OCTAVE_API RTYPE xnorm (const PREFIX##Matrix& x, RTYPE p) \
  { return svd_matrix_norm (x, p, PREFIX##Matrix ()); } \
  OCTAVE_API RTYPE xfrobnorm (const PREFIX##Matrix& x) \
  { return vector_norm (x, static_cast<RTYPE> (2)); }

DEFINE_XNORM_FUNCS(, double)
DEFINE_XNORM_FUNCS(Complex, double)
DEFINE_XNORM_FUNCS(Float, float)
DEFINE_XNORM_FUNCS(FloatComplex, float)

// this is needed to avoid copying the sparse matrix for xfrobnorm
template <typename T, typename R>
inline void array_norm_2 (const T* v, octave_idx_type n, R& res)
{
  norm_accumulator_2<R> acc;
  for (octave_idx_type i = 0; i < n; i++)
    acc.accum (v[i]);

  res = acc;
}

#define DEFINE_XNORM_SPARSE_FUNCS(PREFIX, RTYPE) \
  OCTAVE_API RTYPE xnorm (const Sparse##PREFIX##Matrix& x, RTYPE p) \
  { return matrix_norm (x, p, PREFIX##Matrix ()); } \
  OCTAVE_API RTYPE xfrobnorm (const Sparse##PREFIX##Matrix& x) \
  { \
    RTYPE res; \
    array_norm_2 (x.data (), x.nnz (), res); \
    return res; \
  }

DEFINE_XNORM_SPARSE_FUNCS(, double)
DEFINE_XNORM_SPARSE_FUNCS(Complex, double)

#define DEFINE_COLROW_NORM_FUNCS(PREFIX, RPREFIX, RTYPE) \
  extern OCTAVE_API RPREFIX##RowVector xcolnorms (const PREFIX##Matrix& m, RTYPE p) \
  { return column_norms (m, p); } \
  extern OCTAVE_API RPREFIX##ColumnVector xrownorms (const PREFIX##Matrix& m, RTYPE p) \
  { return row_norms (m, p); } \

DEFINE_COLROW_NORM_FUNCS(, , double)
DEFINE_COLROW_NORM_FUNCS(Complex, , double)
DEFINE_COLROW_NORM_FUNCS(Float, Float, float)
DEFINE_COLROW_NORM_FUNCS(FloatComplex, Float, float)

DEFINE_COLROW_NORM_FUNCS(Sparse, , double)
DEFINE_COLROW_NORM_FUNCS(SparseComplex, , double)