octave-antonio: scripts/special-matrix/gallery.m comparison

comparison scripts/special-matrix/gallery.m @ 16634:2510fffc05e1

gallery: new function

author	Carnë Draug <carandraug@octave.org>
date	Fri, 10 May 2013 01:02:58 +0100
parents
children	67b67fc0969a

comparison

equal deleted inserted replaced

-:8d32a887754a
+:2510fffc05e1
+## Copyright (C) 1989-1995 Nicholas .J. Higham
+## Copyright (C) 2013 Carnë Draug
+##
+## 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/>.
+## -*- texinfo -*-
+## @deftypefn  {Function File} {} gallery (@var{name})
+## @deftypefnx {Function File} {} gallery (@var{name}, @var{args})
+## Create interesting matrices for testing.
+##
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{c} =} gallery ("cauchy", @var{x})
+## @deftypefnx {Function File} {@var{c} =} gallery ("cauchy", @var{x}, @var{y})
+## Create a Cauchy matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{c} =} gallery ("chebspec", @var{n})
+## @deftypefnx {Function File} {@var{c} =} gallery ("chebspec", @var{n}, @var{k})
+## Create a Chebyshev spectral differentiation matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{c} =} gallery ("chebvand", @var{p})
+## @deftypefnx {Function File} {@var{c} =} gallery ("chebvand", @var{m}, @var{p})
+## Create a Vandermonde-like matrix for the Chebyshev polynomials.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("chow", @var{n})
+## @deftypefnx {Function File} {@var{a} =} gallery ("chow", @var{n}, @var{alpha})
+## @deftypefnx {Function File} {@var{a} =} gallery ("chow", @var{n}, @var{alpha}, @var{delta})
+## Create a Chow matrix -- a singular Toeplitz lower Hessenberg matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{c} =} gallery ("circul", @var{v})
+## Create a circulant matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("clement", @var{n})
+## @deftypefnx {Function File} {@var{a} =} gallery ("clement", @var{n}, @var{k})
+## Create a tridiagonal matrix with zero diagonal entries.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{c} =} gallery ("compar", @var{a})
+## @deftypefnx {Function File} {@var{c} =} gallery ("compar", @var{a}, @var{k})
+## Create a comparison matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("condex", @var{n})
+## @deftypefnx {Function File} {@var{a} =} gallery ("condex", @var{n}, @var{k})
+## @deftypefnx {Function File} {@var{a} =} gallery ("condex", @var{n}, @var{k}, @var{theta})
+## Create a `counterexample' matrix to a condition estimator.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("cycol", [@var{m} @var{n}])
+## @deftypefnx {Function File} {@var{a} =} gallery ("cycol", @var{n})
+## @deftypefnx {Function File} {@var{a} =} gallery (@dots{}, @var{k})
+## Create a matrix whose columns repeat cyclically.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {[@var{c},@var{d}, @var{e}] =} gallery ("dorr", @var{n})
+## @deftypefnx {Function File} {[@var{c},@var{d}, @var{e}] =} gallery ("dorr", @var{n}, @var{theta})
+## @deftypefnx {Function File} {@var{a} =} gallery ("dorr", @dots{})
+## Create a diagonally dominant, ill conditioned, tridiagonal matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("dramadah", @var{n})
+## @deftypefnx {Function File} {@var{a} =} gallery ("dramadah", @var{n}, @var{k})
+## Create a (0, 1) matrix whose inverse has large integer entries.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("fiedler", @var{c})
+## Create a symmetric Fiedler matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("forsythe", @var{n})
+## @deftypefnx {Function File} {@var{a} =} gallery ("forsythe", @var{n}, @var{alpha})
+## @deftypefnx {Function File} {@var{a} =} gallery ("forsythe", @var{n}, @var{alpha}, @var{lambda})
+## Create a Forsythe matrix (a perturbed Jordan block).
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{f} =} gallery ("frank", @var{n})
+## @deftypefnx {Function File} {@var{f} =} gallery ("frank", @var{n}, @var{k})
+## Create a Frank matrix (ill conditioned eigenvalues).
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{c} =} gallery ("gcdmat", @var{n})
+## Create a greatest common divisor matrix.
+##
+## @var{c} is an @var{n}-by-@var{n} matrix whose values correspond to the
+## greatest common divisor of its coordinate values, i.e., @var{c}(i,j)
+## correspond @code{gcd (i, j)}.
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("gearmat", @var{n})
+## @deftypefnx {Function File} {@var{a} =} gallery ("gearmat", @var{n}, @var{i})
+## @deftypefnx {Function File} {@var{a} =} gallery ("gearmat", @var{n}, @var{i}, @var{j})
+## Create a Gear matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{g} =} gallery ("grcar", @var{n})
+## @deftypefnx {Function File} {@var{g} =} gallery ("grcar", @var{n}, @var{k})
+## Create a Toeplitz matrix with sensitive eigenvalues.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("hanowa", @var{n})
+## @deftypefnx {Function File} {@var{a} =} gallery ("hanowa", @var{n}, @var{d})
+## Create a matrix whose eigenvalues lie on a vertical line in the complex plane.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{v} =} gallery ("house", @var{x})
+## @deftypefnx {Function File} {[@var{v}, @var{beta}] =} gallery ("house", @var{x})
+## Create a householder matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("invhess", @var{x})
+## @deftypefnx {Function File} {@var{a} =} gallery ("invhess", @var{x}, @var{y})
+## Create the inverse of an upper Hessenberg matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("invol", @var{n})
+## Create an involutory matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("ipjfact", @var{n})
+## @deftypefnx {Function File} {@var{a} =} gallery ("ipjfact", @var{n}, @var{k})
+## Create an Hankel matrix with factorial elements.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("jordbloc", @var{n})
+## @deftypefnx {Function File} {@var{a} =} gallery ("jordbloc", @var{n}, @var{lambda})
+## Create a Jordan block.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{u} =} gallery ("kahan", @var{n})
+## @deftypefnx {Function File} {@var{u} =} gallery ("kahan", @var{n}, @var{theta})
+## @deftypefnx {Function File} {@var{u} =} gallery ("kahan", @var{n}, @var{theta}, @var{pert})
+## Create a Kahan matrix (upper trapezoidal).
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("kms", @var{n})
+## @deftypefnx {Function File} {@var{a} =} gallery ("kms", @var{n}, @var{rho})
+## Create a Kac-Murdock-Szego Toeplitz matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{b} =} gallery ("krylov", @var{a})
+## @deftypefnx {Function File} {@var{b} =} gallery ("krylov", @var{a}, @var{x})
+## @deftypefnx {Function File} {@var{b} =} gallery ("krylov", @var{a}, @var{x}, @var{j})
+## Create a Krylov matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("lauchli", @var{n})
+## @deftypefnx {Function File} {@var{a} =} gallery ("lauchli", @var{n}, @var{mu})
+## Create a Lauchli matrix (rectangular).
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("lehmer", @var{n})
+## Create a Lehmer matrix (symmetric positive definite).
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{t} =} gallery ("lesp", @var{n})
+## Create a tridiagonal matrix with real, sensitive eigenvalues.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("lotkin", @var{n})
+## Create a Lotkin matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("minij", @var{n})
+## Create a symmetric positive definite matrix MIN(i,j).
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("moler", @var{n})
+## @deftypefnx {Function File} {@var{a} =} gallery ("moler", @var{n}, @var{alpha})
+## Create a Moler matrix (symmetric positive definite).
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {[@var{a}, @var{t}] =} gallery ("neumann", @var{n})
+## Create a singular matrix from the discrete Neumann problem (sparse).
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{q} =} gallery ("orthog", @var{n})
+## @deftypefnx {Function File} {@var{q} =} gallery ("orthog", @var{n}, @var{k})
+## Create orthogonal and nearly orthogonal matrices.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("parter", @var{n})
+## Create a Parter matrix (a Toeplitz matrix with singular values near pi).
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{p} =} gallery ("pei", @var{n})
+## @deftypefnx {Function File} {@var{p} =} gallery ("pei", @var{n}, @var{alpha})
+## Create a Pei matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("poisson", @var{n})
+## Create a block tridiagonal matrix from Poisson's equation (sparse).
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("prolate", @var{n})
+## @deftypefnx {Function File} {@var{a} =} gallery ("prolate", @var{n}, @var{w})
+## Create a prolate matrix (symmetric, ill-conditioned Toeplitz matrix).
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{h} =} gallery ("randhess", @var{x})
+## Create a random, orthogonal upper Hessenberg matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("rando", @var{n})
+## @deftypefnx {Function File} {@var{a} =} gallery ("rando", @var{n}, @var{k})
+## Create a random matrix with elements -1, 0 or 1.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("randsvd", @var{n})
+## @deftypefnx {Function File} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa})
+## @deftypefnx {Function File} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa}, @var{mode})
+## @deftypefnx {Function File} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa}, @var{mode}, @var{kl})
+## @deftypefnx {Function File} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa}, @var{mode}, @var{kl}, @var{ku})
+## Create a random matrix with pre-assigned singular values.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("redheff", @var{n})
+## Create a zero and ones matrix of Redheffer associated with the Riemann
+## hypothesis.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("riemann", @var{n})
+## Create a matrix associated with the Riemann hypothesis.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("ris", @var{n})
+## Create a symmetric Hankel matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("smoke", @var{n})
+## @deftypefnx {Function File} {@var{a} =} gallery ("smoke", @var{n}, @var{k})
+## Create a complex matrix, with a `smoke ring' pseudospectrum.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{t} =} gallery ("toeppd", @var{n})
+## @deftypefnx {Function File} {@var{t} =} gallery ("toeppd", @var{n}, @var{m})
+## @deftypefnx {Function File} {@var{t} =} gallery ("toeppd", @var{n}, @var{m}, @var{w})
+## @deftypefnx {Function File} {@var{t} =} gallery ("toeppd", @var{n}, @var{m}, @var{w}, @var{theta})
+## Create a symmetric positive definite Toeplitz matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{p} =} gallery ("toeppen", @var{n})
+## @deftypefnx {Function File} {@var{p} =} gallery ("toeppen", @var{n}, @var{a})
+## @deftypefnx {Function File} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b})
+## @deftypefnx {Function File} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b}, @var{c})
+## @deftypefnx {Function File} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b}, @var{c}, @var{d})
+## @deftypefnx {Function File} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b}, @var{c}, @var{d}, @var{e})
+## Create a pentadiagonal Toeplitz matrix (sparse).
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("tridiag", @var{x}, @var{y}, @var{z})
+## @deftypefnx {Function File} {@var{a} =} gallery ("tridiag", @var{n})
+## @deftypefnx {Function File} {@var{a} =} gallery ("tridiag", @var{n}, @var{c}, @var{d}, @var{e})
+## Create a tridiagonal matrix (sparse).
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{t} =} gallery ("triw", @var{n})
+## @deftypefnx {Function File} {@var{t} =} gallery ("triw", @var{n}, @var{alpha})
+## @deftypefnx {Function File} {@var{t} =} gallery ("triw", @var{n}, @var{alpha}, @var{k})
+## Create an upper triangular matrix discussed by Kahan, Golub and Wilkinson.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {@var{a} =} gallery ("wathen", @var{nx}, @var{ny})
+## @deftypefnx {Function File} {@var{a} =} gallery ("wathen", @var{nx}, @var{ny}, @var{k})
+## Create the Wathen matrix.
+##
+## @end deftypefn
+##
+## @deftypefn  {Function File} {[@var{a}, @var{b}] =} gallery ("wilk", @var{n})
+## Create various specific matrices devised/discussed by Wilkinson.
+##
+## @end deftypefn
+## Code for most of the individual matrices (except binomial, gcdmat,
+## integerdata, leslie, normaldata, randcolu, randcorr, randjorth, sampling,
+## uniformdata) by Nicholas .J. Higham <Nicholas.J.Higham@manchester.ac.uk>
+## Adapted for Octave and into single gallery function by Carnë Draug
+function [varargout] = gallery (name, varargin)
+if (nargin < 1)
+print_usage ();
+elseif (! ischar (name))
+error ("gallery: NAME must be a string.");
+endif
+## NOTE: there isn't a lot of input check in the individual functions
+## that actually build the functions.  This is by design. The original
+## code by Higham did not perform it and was propagated to Matlab, so
+## for compatibility, we also don't make it. For example, arguments
+## that behave as switches, and in theory accepting a value of 0 or 1,
+## will use a value of 0, for any value other than 1 (only check made
+## is if the value is equal to 1). It will often also accept string
+## values instead of numeric. Only input check added was where it
+## would be causing an error anyway.
+## we will always want to return at least 1 output
+n_out = nargout;
+if (n_out == 0)
+n_out = 1;
+endif
+switch (tolower (name))
+case "binomial"
+error ("gallery: matrix %s not implemented.", name);
+case "cauchy"     , [varargout{1:n_out}] = cauchy      (varargin{:});
+case "chebspec"   , [varargout{1:n_out}] = chebspec    (varargin{:});
+case "chebvand"   , [varargout{1:n_out}] = chebvand    (varargin{:});
+case "chow"       , [varargout{1:n_out}] = chow        (varargin{:});
+case "circul"     , [varargout{1:n_out}] = circul      (varargin{:});
+case "clement"    , [varargout{1:n_out}] = clement     (varargin{:});
+case "compar"     , [varargout{1:n_out}] = compar      (varargin{:});
+case "condex"     , [varargout{1:n_out}] = condex      (varargin{:});
+case "cycol"      , [varargout{1:n_out}] = cycol       (varargin{:});
+case "dorr"       , [varargout{1:n_out}] = dorr        (varargin{:});
+case "dramadah"   , [varargout{1:n_out}] = dramadah    (varargin{:});
+case "fiedler"    , [varargout{1:n_out}] = fiedler     (varargin{:});
+case "forsythe"   , [varargout{1:n_out}] = forsythe    (varargin{:});
+case "frank"      , [varargout{1:n_out}] = frank       (varargin{:});
+case "gearmat"    , [varargout{1:n_out}] = gearmat     (varargin{:});
+case "gcdmat"     , [varargout{1:n_out}] = gcdmat      (varargin{:});
+case "grcar"      , [varargout{1:n_out}] = grcar       (varargin{:});
+case "hanowa"     , [varargout{1:n_out}] = hanowa      (varargin{:});
+case "house"      , [varargout{1:n_out}] = house       (varargin{:});
+case "integerdata"
+error ("gallery: matrix %s not implemented.", name);
+case "invhess"    , [varargout{1:n_out}] = invhess     (varargin{:});
+case "invol"      , [varargout{1:n_out}] = invol       (varargin{:});
+case "ipjfact"    , [varargout{1:n_out}] = ipjfact     (varargin{:});
+case "jordbloc"   , [varargout{1:n_out}] = jordbloc    (varargin{:});
+case "kahan"      , [varargout{1:n_out}] = kahan       (varargin{:});
+case "kms"        , [varargout{1:n_out}] = kms         (varargin{:});
+case "krylov"     , [varargout{1:n_out}] = krylov      (varargin{:});
+case "lauchli"    , [varargout{1:n_out}] = lauchli     (varargin{:});
+case "lehmer"     , [varargout{1:n_out}] = lehmer      (varargin{:});
+case "leslie"
+error ("gallery: matrix %s not implemented.", name);
+case "lesp"       , [varargout{1:n_out}] = lesp        (varargin{:});
+case "lotkin"     , [varargout{1:n_out}] = lotkin      (varargin{:});
+case "minij"      , [varargout{1:n_out}] = minij       (varargin{:});
+case "moler"      , [varargout{1:n_out}] = moler       (varargin{:});
+case "neumann"    , [varargout{1:n_out}] = neumann     (varargin{:});
+case "normaldata"
+error ("gallery: matrix %s not implemented.", name);
+case "orthog"     , [varargout{1:n_out}] = orthog      (varargin{:});
+case "parter"     , [varargout{1:n_out}] = parter      (varargin{:});
+case "pei"        , [varargout{1:n_out}] = pei         (varargin{:});
+case "poisson"    , [varargout{1:n_out}] = poisson     (varargin{:});
+case "prolate"    , [varargout{1:n_out}] = prolate     (varargin{:});
+case "randcolu"
+error ("gallery: matrix %s not implemented.", name);
+case "randcorr"
+error ("gallery: matrix %s not implemented.", name);
+case "randhess"    , [varargout{1:n_out}] = randhess    (varargin{:});
+case "randjorth"
+error ("gallery: matrix %s not implemented.", name);
+case "rando"       , [varargout{1:n_out}] = rando       (varargin{:});
+case "randsvd"     , [varargout{1:n_out}] = randsvd     (varargin{:});
+case "redheff"     , [varargout{1:n_out}] = redheff     (varargin{:});
+case "riemann"     , [varargout{1:n_out}] = riemann     (varargin{:});
+case "ris"         , [varargout{1:n_out}] = ris         (varargin{:});
+case "sampling"
+error ("gallery: matrix %s not implemented.", name);
+case "smoke"       , [varargout{1:n_out}] = smoke       (varargin{:});
+case "toeppd"      , [varargout{1:n_out}] = toeppd      (varargin{:});
+case "toeppen"     , [varargout{1:n_out}] = toeppen     (varargin{:});
+case "tridiag"     , [varargout{1:n_out}] = tridiag     (varargin{:});
+case "triw"        , [varargout{1:n_out}] = triw        (varargin{:});
+case "uniformdata"
+error ("gallery: matrix %s not implemented.", name);
+case "wathen"      , [varargout{1:n_out}] = wathen      (varargin{:});
+case "wilk"        , [varargout{1:n_out}] = wilk        (varargin{:});
+otherwise
+error ("gallery: unknown matrix with NAME %s", name);
+endswitch
+endfunction
+function C = cauchy (x, y)
+##CAUCHY  Cauchy matrix.
+##        C = CAUCHY(X, Y), where X, Y are N-vectors, is the N-by-N matrix
+##        with C(i,j) = 1/(X(i)+Y(j)).   By default, Y = X.
+##        Special case: if X is a scalar CAUCHY(X) is the same as CAUCHY(1:X).
+##        Explicit formulas are known for DET(C) (which is nonzero if X and Y
+##        both have distinct elements) and the elements of INV(C).
+##        C is totally positive if 0 < X(1) < ... < X(N) and
+##        0 < Y(1) < ... < Y(N).
+##
+##        References:
+##        N.J. Higham, Accuracy and Stability of Numerical Algorithms,
+##          Society for Industrial and Applied Mathematics, Philadelphia, PA,
+##          USA, 1996; sec. 26.1.
+##        D.E. Knuth, The Art of Computer Programming, Volume 1,
+##          Fundamental Algorithms, second edition, Addison-Wesley, Reading,
+##          Massachusetts, 1973, p. 36.
+##        E.E. Tyrtyshnikov, Cauchy-Toeplitz matrices and some applications,
+##          Linear Algebra and Appl., 149 (1991), pp. 1-18.
+##          O. Taussky and M. Marcus, Eigenvalues of finite matrices, in
+##          Survey of Numerical Analysis, J. Todd, ed., McGraw-Hill, New York,
+##          pp. 279-313, 1962. (States the totally positive property on p. 295.)
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 or 2 arguments are required for cauchy matrix.");
+elseif (! isnumeric (x))
+error ("gallery: X must be numeric for cauchy matrix.");
+elseif (nargin == 2 && ! isnumeric (y))
+error ("gallery: Y must be numeric for cauchy matrix.");
+endif
+n = numel (x);
+if (isscalar (x) && fix (x) == x)
+n = x;
+x = 1:n;
+elseif (n > 1 && isvector (x))
+## do nothing
+else
+error ("gallery: X be an integer or a vector for cauchy matrix.");
+endif
+if (nargin == 1)
+y = x;
+endif
+## Ensure x and y are column vectors
+x = x(:);
+y = y(:);
+if (numel (x) != numel (y))
+error ("gallery: X and Y must be vectors of same length for cauchy matrix.");
+endif
+C = x * ones (1, n) + ones (n, 1) * y.';
+C = ones (n) ./ C;
+endfunction
+function C = chebspec (n, k = 0)
+## CHEBSPEC  Chebyshev spectral differentiation matrix.
+##           C = CHEBSPEC(N, K) is a Chebyshev spectral differentiation
+##           matrix of order N.  K = 0 (the default) or 1.
+##           For K = 0 (`no boundary conditions'), C is nilpotent, with
+##               C^N = 0 and it has the null vector ONES(N,1).
+##               C is similar to a Jordan block of size N with eigenvalue zero.
+##           For K = 1, C is nonsingular and well-conditioned, and its eigenvalues
+##               have negative real parts.
+##           For both K, the computed eigenvector matrix X from EIG is
+##               ill-conditioned (MESH(REAL(X)) is interesting).
+##
+##           References:
+##           C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral
+##              Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988; p. 69.
+##           L.N. Trefethen and M.R. Trummer, An instability phenomenon in
+##              spectral methods, SIAM J. Numer. Anal., 24 (1987), pp. 1008-1023.
+##           D. Funaro, Computing the inverse of the Chebyshev collocation
+##              derivative, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 1050-1057.
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 to 2 arguments are required for chebspec matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for chebspec matrix.");
+elseif (! isnumeric (k) || ! isscalar (k))
+error ("gallery: K must be a scalar for chebspec matrix.");
+endif
+## k = 1 case obtained from k = 0 case with one bigger n.
+switch (k)
+case (0), # do nothing
+case (1), n = n + 1;
+otherwise
+error ("gallery: unknown K `%d' for chebspec matrix.", k);
+endswitch
+n = n-1;
+C = zeros (n+1);
+one    = ones (n+1, 1);
+x      = cos ((0:n)' * (pi/n));
+d      = ones (n+1, 1);
+d(1)   = 2;
+d(n+1) = 2;
+## eye(size(C)) on next line avoids div by zero.
+C = (d * (one./d)') ./ (x*one'-one*x' + eye (size (C)));
+##  Now fix diagonal and signs.
+C(1,1) = (2*n^2+1)/6;
+for i = 2:n+1
+if (rem (i, 2) == 0)
+C(:,i) = -C(:,i);
+C(i,:) = -C(i,:);
+endif
+if (i < n+1)
+C(i,i) = -x(i)/(2*(1-x(i)^2));
+else
+C(n+1,n+1) = -C(1,1);
+endif
+endfor
+if (k == 1)
+C = C(2:n+1,2:n+1);
+endif
+endfunction
+function C = chebvand (m, p)
+## CHEBVAND Vandermonde-like matrix for the Chebyshev polynomials.
+##          C = CHEBVAND(P), where P is a vector, produces the (primal)
+##          Chebyshev Vandermonde matrix based on the points P,
+##          i.e., C(i,j) = T_{i-1}(P(j)), where T_{i-1} is the Chebyshev
+##          polynomial of degree i-1.
+##          CHEBVAND(M,P) is a rectangular version of CHEBVAND(P) with M rows.
+##          Special case: If P is a scalar then P equally spaced points on
+##                        [0,1] are used.
+##
+##          Reference:
+##          N.J. Higham, Stability analysis of algorithms for solving confluent
+##            Vandermonde-like systems, SIAM J. Matrix Anal. Appl., 11 (1990),
+##            pp. 23-41.
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 or 2 arguments are required for chebvand matrix.");
+endif
+## because the order of the arguments changes if nargin is 1 or 2 ...
+if (nargin == 1)
+p = m;
+endif
+n = numel (p);
+if (! isnumeric (p))
+error ("gallery: P must be numeric for chebvand matrix.");
+elseif (isscalar (p) && fix (p) == p)
+n = p;
+p = linspace (0, 1, n);
+elseif (n > 1 && isvector (p))
+## do nothing
+endif
+p = p(:).'; # Ensure p is a row vector.
+if (nargin == 1)
+m = n;
+elseif (! isnumeric (m) || ! isscalar (m))
+error ("gallery: M must be a scalar for chebvand matrix.");
+endif
+C = ones (m, n);
+if (m != 1)
+C(2,:) = p;
+##      Use Chebyshev polynomial recurrence.
+for i = 3:m
+C(i,:) = 2.*p.*C(i-1,:) - C(i-2,:);
+endfor
+endif
+endfunction
+function A = chow (n, alpha = 1, delta = 0)
+## CHOW    Chow matrix - a singular Toeplitz lower Hessenberg matrix.
+##         A = CHOW(N, ALPHA, DELTA) is a Toeplitz lower Hessenberg matrix
+##         A = H(ALPHA) + DELTA*EYE, where H(i,j) = ALPHA^(i-j+1).
+##         H(ALPHA) has p = FLOOR(N/2) zero eigenvalues, the rest being
+##         4*ALPHA*COS( k*PI/(N+2) )^2, k=1:N-p.
+##         Defaults: ALPHA = 1, DELTA = 0.
+##
+##         References:
+##         T.S. Chow, A class of Hessenberg matrices with known
+##            eigenvalues and inverses, SIAM Review, 11 (1969), pp. 391-395.
+##         G. Fairweather, On the eigenvalues and eigenvectors of a class of
+##            Hessenberg matrices, SIAM Review, 13 (1971), pp. 220-221.
+if (nargin < 1 || nargin > 3)
+error ("gallery: 1 to 3 arguments are required for chow matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for chow matrix.");
+elseif (! isnumeric (alpha) || ! isscalar (alpha))
+error ("gallery: ALPHA must be a scalar for chow matrix.");
+elseif (! isnumeric (delta) || ! isscalar (delta))
+error ("gallery: DELTA must be a scalar for chow matrix.");
+endif
+A = toeplitz (alpha.^(1:n), [alpha 1 zeros(1, n-2)]) + delta * eye (n);
+endfunction
+function C = circul (v)
+## CIRCUL  Circulant matrix.
+##         C = CIRCUL(V) is the circulant matrix whose first row is V.
+##         (A circulant matrix has the property that each row is obtained
+##         from the previous one by cyclically permuting the entries one step
+##         forward; it is a special Toeplitz matrix in which the diagonals
+##         `wrap round'.)
+##         Special case: if V is a scalar then C = CIRCUL(1:V).
+##         The eigensystem of C (N-by-N) is known explicitly.   If t is an Nth
+##         root of unity, then the inner product of V with W = [1 t t^2 ... t^N]
+##         is an eigenvalue of C, and W(N:-1:1) is an eigenvector of C.
+##
+##         Reference:
+##         P.J. Davis, Circulant Matrices, John Wiley, 1977.
+if (nargin != 1)
+error ("gallery: 1 argument is required for circul matrix.");
+elseif (! isnumeric (v))
+error ("gallery: V must be numeric for circul matrix.");
+endif
+n = numel (x);
+if (isscalar (x) && fix (x) == x)
+n = v;
+v = 1:n;
+elseif (n > 1 && isvector (x))
+## do nothing
+else
+error ("gallery: X must be a scalar or a vector for circul matrix.");
+endif
+v = v(:).';   # Make sure v is a row vector
+C = toeplitz ([v(1) v(n:-1:2)], v);
+endfunction
+function A = clement (n, k = 0)
+## CLEMENT   Clement matrix - tridiagonal with zero diagonal entries.
+##           CLEMENT(N, K) is a tridiagonal matrix with zero diagonal entries
+##           and known eigenvalues.  It is singular if N is odd.  About 64
+##           percent of the entries of the inverse are zero.  The eigenvalues
+##           are plus and minus the numbers N-1, N-3, N-5, ..., (1 or 0).
+##           For K = 0 (the default) the matrix is unsymmetric, while for
+##           K = 1 it is symmetric.
+##           CLEMENT(N, 1) is diagonally similar to CLEMENT(N).
+##
+##           Similar properties hold for TRIDIAG(X,Y,Z) where Y = ZEROS(N,1).
+##           The eigenvalues still come in plus/minus pairs but they are not
+##           known explicitly.
+##
+##           References:
+##           P.A. Clement, A class of triple-diagonal matrices for test
+##              purposes, SIAM Review, 1 (1959), pp. 50-52.
+##           A. Edelman and E. Kostlan, The road from Kac's matrix to Kac's
+##              random polynomials. In John~G. Lewis, editor, Proceedings of
+##              the Fifth SIAM Conference on Applied Linear Algebra Society
+##              for Industrial and Applied Mathematics, Philadelphia, 1994,
+##              pp. 503-507.
+##           O. Taussky and J. Todd, Another look at a matrix of Mark Kac,
+##              Linear Algebra and Appl., 150 (1991), pp. 341-360.
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 or 2 arguments are required for clement matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for clement matrix.");
+elseif (! isnumeric (k) || ! isscalar (k))
+error ("gallery: K must be a numeric scalar for clement matrix.");
+endif
+n = n-1;
+x = n:-1:1;
+z = 1:n;
+if (k == 0)
+A = diag (x, -1) + diag (z, 1);
+elseif (k == 1)
+y = sqrt (x.*z);
+A = diag (y, -1) + diag (y, 1);
+else
+error ("gallery: K must have a value of 0 or 1 for clement matrix.");
+endif
+endfunction
+function C = compar (A, k = 0)
+## COMP    Comparison matrices.
+##         COMP(A) is DIAG(B) - TRIL(B,-1) - TRIU(B,1), where B = ABS(A).
+##         COMP(A, 1) is A with each diagonal element replaced by its
+##         absolute value, and each off-diagonal element replaced by minus
+##         the absolute value of the largest element in absolute value in
+##         its row.  However, if A is triangular COMP(A, 1) is too.
+##         COMP(A, 0) is the same as COMP(A).
+##         COMP(A) is often denoted by M(A) in the literature.
+##
+##         Reference (e.g.):
+##         N.J. Higham, A survey of condition number estimation for
+##         triangular matrices, SIAM Review, 29 (1987), pp. 575-596.
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 or 2 arguments are required for compar matrix.");
+elseif (! isnumeric (A) || ndims (A) != 2)
+error ("gallery: A must be a 2D matrix for compar matrix.");
+elseif (! isnumeric (k) || ! isscalar (k))
+error ("gallery: K must be a numeric scalar for compar matrix.");
+endif
+[m, n] = size (A);
+p = min (m, n);
+if (k == 0)
+## This code uses less temporary storage than
+## the `high level' definition above.
+C = -abs (A);
+for j = 1:p
+C(j,j) = abs (A(j,j));
+endfor
+elseif (k == 1)
+C = A';
+for j = 1:p
+C(k,k) = 0;
+endfor
+mx = max (abs (C));
+C  = -mx'*ones (1, n);
+for j = 1:p
+C(j,j) = abs (A(j,j));
+endfor
+if (all (A == tril (A))), C = tril (C); endif
+if (all (A == triu (A))), C = triu (C); endif
+else
+error ("gallery: K must have a value of 0 or 1 for compar matrix.");
+endif
+endfunction
+function A = condex (n, k = 4, theta = 100)
+## CONDEX   `Counterexamples' to matrix condition number estimators.
+##          CONDEX(N, K, THETA) is a `counterexample' matrix to a condition
+##          estimator.  It has order N and scalar parameter THETA (default 100).
+##          If N is not equal to the `natural' size of the matrix then
+##          the matrix is padded out with an identity matrix to order N.
+##          The matrix, its natural size, and the estimator to which it applies
+##          are specified by K (default K = 4) as follows:
+##              K = 1:   4-by-4,     LINPACK (RCOND)
+##              K = 2:   3-by-3,     LINPACK (RCOND)
+##              K = 3:   arbitrary,  LINPACK (RCOND) (independent of THETA)
+##              K = 4:   N >= 4,     SONEST (Higham 1988)
+##          (Note that in practice the K = 4 matrix is not usually a
+##           counterexample because of the rounding errors in forming it.)
+##
+##          References:
+##          A.K. Cline and R.K. Rew, A set of counter-examples to three
+##             condition number estimators, SIAM J. Sci. Stat. Comput.,
+##             4 (1983), pp. 602-611.
+##          N.J. Higham, FORTRAN codes for estimating the one-norm of a real or
+##             complex matrix, with applications to condition estimation
+##             (Algorithm 674), ACM Trans. Math. Soft., 14 (1988), pp. 381-396.
+if (nargin < 1 || nargin > 3)
+error ("gallery: 1 to 3 arguments are required for condex matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for condex matrix.");
+elseif (! isnumeric (k) || ! isscalar (k))
+error ("gallery: K must be a numeric scalar for condex matrix.");
+elseif (! isnumeric (theta) || ! isscalar (theta))
+error ("gallery: THETA must be a numeric scalar for condex matrix.");
+endif
+if (k == 1)       # Cline and Rew (1983), Example B.
+A = [1  -1  -2*theta     0
+0   1     theta  -theta
+0   1   1+theta  -(theta+1)
+0   0   0         theta];
+elseif (k == 2)   # Cline and Rew (1983), Example C.
+A = [1   1-2/theta^2  -2
+0   1/theta      -1/theta
+0   0             1];
+elseif (k == 3)   # Cline and Rew (1983), Example D.
+A = gallery ("triw", n, -1)';
+A(n,n) = -1;
+elseif (k == 4)   # Higham (1988), p. 390.
+x = ones (n, 3);            #  First col is e
+x(2:n,2) = zeros (n-1, 1);  #  Second col is e(1)
+## Third col is special vector b in SONEST
+x(:, 3) = (-1).^[0:n-1]' .* ( 1 + [0:n-1]'/(n-1) );
+Q = orth (x);  #  Q*Q' is now the orthogonal projector onto span(e(1),e,b)).
+P = eye (n) - Q*Q';
+A = eye (n) + theta*P;
+else
+error ("gallery: unknown estimator K `%d' for condex matrix.", k);
+endif
+## Pad out with identity as necessary.
+m = columns (A);
+if (m < n)
+for i = n:-1:m+1
+A(i,i) = 1;
+endfor
+endif
+endfunction
+function A = cycol (n, k)
+## CYCOL   Matrix whose columns repeat cyclically.
+##         A = CYCOL([M N], K) is an M-by-N matrix of the form A = B(1:M,1:N)
+##         where B = [C C C...] and C = RANDN(M, K).  Thus A's columns repeat
+##         cyclically, and A has rank at most K.   K need not divide N.
+##         K defaults to ROUND(N/4).
+##         CYCOL(N, K), where N is a scalar, is the same as CYCOL([N N], K).
+##
+##         This type of matrix can lead to underflow problems for Gaussian
+##         elimination: see NA Digest Volume 89, Issue 3 (January 22, 1989).
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 or 2 arguments are required for cycol matrix.");
+elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n)
+error ("gallery: N must be a 1 or 2 element integer for cycol matrix.");
+elseif (! isnumeric (k) || ! isscalar (k))
+error ("gallery: K must be a scalar for cycol matrix.");
+endif
+## Parameter n specifies dimension: m-by-n
+m = n(1);
+n = n(end);
+if (nargin < 2)
+k = max (round (n/4), 1);
+endif
+A = randn (m, k);
+for i = 2:ceil (n/k)
+A = [A A(:,1:k)];
+endfor
+A = A(:,1:n);
+endfunction
+function [c, d, e] = dorr (n, theta = 0.01)
+## DORR  Dorr matrix - diagonally dominant, ill conditioned, tridiagonal.
+##       [C, D, E] = DORR(N, THETA) returns the vectors defining a row diagonally
+##       dominant, tridiagonal M-matrix that is ill conditioned for small
+##       values of the parameter THETA >= 0.
+##       If only one output parameter is supplied then
+##       C = FULL(TRIDIAG(C,D,E)), i.e., the matrix iself is returned.
+##       The columns of INV(C) vary greatly in norm.  THETA defaults to 0.01.
+##       The amount of diagonal dominance is given by (ignoring rounding errors):
+##             COMP(C)*ONES(N,1) = THETA*(N+1)^2 * [1 0 0 ... 0 1]'.
+##
+##       Reference:
+##       F.W. Dorr, An example of ill-conditioning in the numerical
+##       solution of singular perturbation problems, Math. Comp., 25 (1971),
+##       pp. 271-283.
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 or 2 arguments are required for dorr matrix.");
+elseif (! isscalar (n) || ! isnumeric (n) || fix (n) != n)
+error ("gallery: N must be an integer for dorr matrix.");
+elseif (! isscalar (theta) || ! isnumeric (theta))
+error ("gallery: THETA must be a numeric scalar for dorr matrix.");
+endif
+c = zeros (n, 1);
+e = c;
+d = c;
+##  All length n for convenience.  Make c, e of length n-1 later.
+h = 1/(n+1);
+m = floor ((n+1)/2);
+term = theta/h^2;
+i = (1:m)';
+c(i) = -term * ones (m, 1);
+e(i) = c(i) - (0.5-i*h)/h;
+d(i) = -(c(i) + e(i));
+i = (m+1:n)';
+e(i) = -term * ones (n-m, 1);
+c(i) = e(i) + (0.5-i*h)/h;
+d(i) = -(c(i) + e(i));
+c = c(2:n);
+e = e(1:n-1);
+if (nargout <= 1)
+c = tridiag (c, d, e);
+endif
+endfunction
+function A = dramadah (n, k = 1)
+## DRAMADAH  A (0,1) matrix whose inverse has large integer entries.
+##           An anti-Hadamard matrix A is a matrix with elements 0 or 1 for
+##           which MU(A) := NORM(INV(A),'FRO') is maximal.
+##           A = DRAMADAH(N, K) is an N-by-N (0,1) matrix for which MU(A) is
+##           relatively large, although not necessarily maximal.
+##           Available types (the default is K = 1):
+##           K = 1: A is Toeplitz, with ABS(DET(A)) = 1, and MU(A) > c(1.75)^N,
+##                  where c is a constant.
+##           K = 2: A is upper triangular and Toeplitz.
+##           The inverses of both types have integer entries.
+##
+##           Another interesting (0,1) matrix:
+##           K = 3: A has maximal determinant among (0,1) lower Hessenberg
+##           matrices: det(A) = the n'th Fibonacci number.  A is Toeplitz.
+##           The eigenvalues have an interesting distribution in the complex
+##           plane.
+##
+##           References:
+##           R.L. Graham and N.J.A. Sloane, Anti-Hadamard matrices,
+##              Linear Algebra and Appl., 62 (1984), pp. 113-137.
+##           L. Ching, The maximum determinant of an nxn lower Hessenberg
+##              (0,1) matrix, Linear Algebra and Appl., 183 (1993), pp. 147-153.
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 to 2 arguments are required for dramadah matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for dramadah matrix.");
+elseif (! isnumeric (k) || ! isscalar (k))
+error ("gallery: K must be a numeric scalar for dramadah matrix.");
+endif
+switch (k)
+case (1)  # Toeplitz
+c = ones (n, 1);
+for i = 2:4:n
+m = min (1, n-i);
+c(i:i+m) = zeros (m+1, 1);
+endfor
+r = zeros (n, 1);
+r(1:4) = [1 1 0 1];
+if (n < 4)
+r = r(1:n);
+endif
+A = toeplitz (c, r);
+case (2)  # Upper triangular and Toeplitz
+c = zeros (n, 1);
+c(1) = 1;
+r = ones (n, 1);
+for i= 3:2:n
+r(i) = 0;
+endfor
+A = toeplitz (c, r);
+case (3)  # Lower Hessenberg
+c = ones (n, 1);
+for i= 2:2:n
+c(i) = 0;
+endfor
+A = toeplitz (c, [1 1 zeros(1,n-2)]);
+otherwise
+error ("gallery: unknown K `%d' for dramadah matrix.", k);
+endswitch
+endfunction
+function A = fiedler (c)
+## FIEDLER  Fiedler matrix - symmetric.
+##          A = FIEDLER(C), where C is an n-vector, is the n-by-n symmetric
+##          matrix with elements ABS(C(i)-C(j)).
+##          Special case: if C is a scalar, then A = FIEDLER(1:C)
+##                        (i.e. A(i,j) = ABS(i-j)).
+##          Properties:
+##            FIEDLER(N) has a dominant positive eigenvalue and all the other
+##                       eigenvalues are negative (Szego, 1936).
+##            Explicit formulas for INV(A) and DET(A) are given by Todd (1977)
+##            and attributed to Fiedler.  These indicate that INV(A) is
+##            tridiagonal except for nonzero (1,n) and (n,1) elements.
+##            [I think these formulas are valid only if the elements of
+##            C are in increasing or decreasing order---NJH.]
+##
+##            References:
+##            G. Szego, Solution to problem 3705, Amer. Math. Monthly,
+##               43 (1936), pp. 246-259.
+##            J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra,
+##               Birkhauser, Basel, and Academic Press, New York, 1977, p. 159.
+if (nargin != 1)
+error ("gallery: 1 argument is required for fiedler matrix.");
+elseif (! isnumeric (c))
+error ("gallery: C must be numeric for fiedler matrix.");
+endif
+n = numel (c);
+if (isscalar (c) && fix (c) == c)
+n = c;
+c = 1:n;
+elseif (n > 1 && isvector (c))
+## do nothing
+else
+error ("gallery: C must be an integer or a vector for fiedler matrix.");
+endif
+c = c(:).';           # Ensure c is a row vector.
+A = ones (n, 1) * c;
+A = abs (A - A.');    # NB. array transpose.
+endfunction
+function A = forsythe (n, alpha = sqrt (eps), lambda = 0)
+## FORSYTHE  Forsythe matrix - a perturbed Jordan block.
+##           FORSYTHE(N, ALPHA, LAMBDA) is the N-by-N matrix equal to
+##           JORDBLOC(N, LAMBDA) except it has an ALPHA in the (N,1) position.
+##           It has the characteristic polynomial
+##                   DET(A-t*EYE) = (LAMBDA-t)^N - (-1)^N ALPHA.
+##           ALPHA defaults to SQRT(EPS) and LAMBDA to 0.
+if (nargin < 1 || nargin > 3)
+error ("gallery: 1 to 3 arguments are required for forsythe matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for forsythe matrix.");
+elseif (! isnumeric (alpha) || ! isscalar (alpha))
+error ("gallery: ALPHA must be a numeric scalar for forsythe matrix.");
+elseif (! isnumeric (lambda) || ! isscalar (lambda))
+error ("gallery: LAMBDA must be a numeric scalar for forsythe matrix.");
+endif
+A = jordbloc (n, lambda);
+A(n,1) = alpha;
+endfunction
+function F = frank (n, k = 0)
+## FRANK   Frank matrix---ill conditioned eigenvalues.
+##         F = FRANK(N, K) is the Frank matrix of order N.  It is upper
+##         Hessenberg with determinant 1.  K = 0 is the default; if K = 1 the
+##         elements are reflected about the anti-diagonal (1,N)--(N,1).
+##         F has all positive eigenvalues and they occur in reciprocal pairs
+##         (so that 1 is an eigenvalue if N is odd).
+##         The eigenvalues of F may be obtained in terms of the zeros of the
+##         Hermite polynomials.
+##         The FLOOR(N/2) smallest eigenvalues of F are ill conditioned,
+##         the more so for bigger N.
+##
+##         DET(FRANK(N)') comes out far from 1 for large N---see Frank (1958)
+##         and Wilkinson (1960) for discussions.
+##
+##         This version incorporates improvements suggested by W. Kahan.
+##
+##         References:
+##         W.L. Frank, Computing eigenvalues of complex matrices by determinant
+##            evaluation and by methods of Danilewski and Wielandt, J. Soc.
+##            Indust. Appl. Math., 6 (1958), pp. 378-392 (see pp. 385, 388).
+##         G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the
+##            computation of the Jordan canonical form, SIAM Review, 18 (1976),
+##              pp. 578-619 (Section 13).
+##         H. Rutishauser, On test matrices, Programmation en Mathematiques
+##            Numeriques, Editions Centre Nat. Recherche Sci., Paris, 165,
+##            1966, pp. 349-365.  Section 9.
+##         J.H. Wilkinson, Error analysis of floating-point computation,
+##            Numer. Math., 2 (1960), pp. 319-340 (Section 8).
+##         J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University
+##            Press, 1965 (pp. 92-93).
+##         The next two references give details of the eigensystem, as does
+##         Rutishauser (see above).
+##         P.J. Eberlein, A note on the matrices denoted by B_n, SIAM J. Appl.
+##            Math., 20 (1971), pp. 87-92.
+##         J.M. Varah, A generalization of the Frank matrix, SIAM J. Sci. Stat.
+##            Comput., 7 (1986), pp. 835-839.
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 to 2 arguments are required for frank matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for frank matrix.");
+elseif (! isnumeric (k) || ! isscalar (k))
+error ("gallery: K must be a numeric scalar for frank matrix.");
+endif
+p = n:-1:1;
+F = triu (p(ones (n, 1), :) - diag (ones (n-1, 1), -1), -1);
+switch (k)
+case (0), # do nothing
+case (1), F = F(p,p)';
+otherwise
+error ("gallery: K must have a value of 0 or 1 for frank matrix.");
+endswitch
+endfunction
+function c = gcdmat (n)
+if (nargin != 1)
+error ("gallery: 1 argument is required for gcdmat matrix.");
+elseif (! isscalar (n) || ! isnumeric (n) || fix (n) != n)
+error ("gallery: N must be an integer for gcdmat matrix.");
+endif
+c = gcd (repmat ((1:n)', [1 n]), repmat (1:n, [n 1]));
+endfunction
+function A = gearmat (n, i = n, j = -n)
+## NOTE: this function was named gearm in the original Test Matrix Toolbox
+## GEARMAT   Gear matrix.
+##         A = GEARMAT(N,I,J) is the N-by-N matrix with ones on the sub- and
+##         super-diagonals, SIGN(I) in the (1,ABS(I)) position, SIGN(J)
+##         in the (N,N+1-ABS(J)) position, and zeros everywhere else.
+##         Defaults: I = N, j = -N.
+##         All eigenvalues are of the form 2*COS(a) and the eigenvectors
+##         are of the form [SIN(w+a), SIN(w+2a), ..., SIN(w+Na)].
+##         The values of a and w are given in the reference below.
+##         A can have double and triple eigenvalues and can be defective.
+##         GEARMAT(N) is singular.
+##
+##         (GEAR is a Simulink function, hence GEARMAT for Gear matrix.)
+##         Reference:
+##         C.W. Gear, A simple set of test matrices for eigenvalue programs,
+##         Math. Comp., 23 (1969), pp. 119-125.
+if (nargin < 1 || nargin > 3)
+error ("gallery: 1 to 3 arguments are required for gearmat matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for gearmat matrix.");
+elseif (! isnumeric (i) || ! isscalar (i) || i == 0 || abs (i) <= n)
+error ("gallery: I must be a non-zero scalar, and abs (I) <= N for gearmat matrix.");
+elseif (! isnumeric (j) || ! isscalar (j) || i == 0 || abs (j) <= n)
+error ("gallery: J must be a non-zero scalar, and abs (J) <= N for gearmat matrix.");
+endif
+A = diag (ones (n-1, 1), -1) + diag (ones (n-1, 1), 1);
+A(1, abs (i)) = sign (i);
+A(n, n+1 - abs (j)) = sign (j);
+endfunction
+function G = grcar (n, k = 3)
+## GRCAR     Grcar matrix - a Toeplitz matrix with sensitive eigenvalues.
+##           GRCAR(N, K) is an N-by-N matrix with -1s on the
+##           subdiagonal, 1s on the diagonal, and K superdiagonals of 1s.
+##           The default is K = 3.  The eigenvalues of this matrix form an
+##           interesting pattern in the complex plane (try PS(GRCAR(32))).
+##
+##           References:
+##           J.F. Grcar, Operator coefficient methods for linear equations,
+##                Report SAND89-8691, Sandia National Laboratories, Albuquerque,
+##                New Mexico, 1989 (Appendix 2).
+##           N.M. Nachtigal, L. Reichel and L.N. Trefethen, A hybrid GMRES
+##                algorithm for nonsymmetric linear systems, SIAM J. Matrix Anal.
+##                Appl., 13 (1992), pp. 796-825.
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 to 2 arguments are required for grcar matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for grcar matrix.");
+elseif (! isnumeric (k) || ! isscalar (k))
+error ("gallery: K must be a numeric scalar for grcar matrix.");
+endif
+G = tril (triu (ones (n)), k) - diag (ones (n-1, 1), -1);
+endfunction
+function A = hanowa (n, d = -1)
+## HANOWA  A matrix whose eigenvalues lie on a vertical line in the complex plane.
+##         HANOWA(N, d) is the N-by-N block 2x2 matrix (thus N = 2M must be even)
+##                       [d*EYE(M)   -DIAG(1:M)
+##                        DIAG(1:M)   d*EYE(M)]
+##         It has complex eigenvalues lambda(k) = d +/- k*i  (1 <= k <= M).
+##         Parameter d defaults to -1.
+##
+##         Reference:
+##         E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary
+##         Differential Equations I: Nonstiff Problems, Springer-Verlag,
+##         Berlin, 1987. (pp. 86-87)
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 to 2 arguments are required for hanowa matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for hanowa matrix.");
+elseif (rem (n, 2) != 0)
+error ("gallery: N must be even for hanowa matrix.");
+elseif (! isnumeric (lambda) || ! isscalar (lambda))
+error ("gallery: D must be a numeric scalar for hanowa matrix.");
+endif
+m = n/2;
+A = [ d*eye(m)  -diag(1:m)
+diag(1:m)  d*eye(m) ];
+endfunction
+function [v, beta] = house (x)
+## HOUSE   Householder matrix.
+##         If [v, beta] = HOUSE(x) then H = EYE - beta*v*v' is a Householder
+##         matrix such that Hx = -sign(x(1))*norm(x)*e_1.
+##         NB: If x = 0 then v = 0, beta = 1 is returned.
+##             x can be real or complex.
+##             sign(x) := exp(i*arg(x)) ( = x./abs(x) when x ~= 0).
+##
+##         Theory: (textbook references Golub & Van Loan 1989, 38-43;
+##                  Stewart 1973, 231-234, 262; Wilkinson 1965, 48-50).
+##         Hx = y: (I - beta*v*v')x = -s*e_1.
+##         Must have |s| = norm(x), v = x+s*e_1, and
+##         x'y = x'Hx =(x'Hx)' real => arg(s) = arg(x(1)).
+##         So take s = sign(x(1))*norm(x) (which avoids cancellation).
+##         v'v = (x(1)+s)^2 + x(2)^2 + ... + x(n)^2
+##             = 2*norm(x)*(norm(x) + |x(1)|).
+##
+##         References:
+##         G.H. Golub and C.F. Van Loan, Matrix Computations, second edition,
+##            Johns Hopkins University Press, Baltimore, Maryland, 1989.
+##         G.W. Stewart, Introduction to Matrix Computations, Academic Press,
+##            New York, 1973,
+##         J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University
+##            Press, 1965.
+if (nargin != 1)
+error ("gallery: 1 argument is required for house matrix.");
+elseif (! isnumeric (x) || ! isvector (x) || numel (x) <= 1)
+error ("gallery: X must be a vector for house matrix.");
+endif
+## must be a column vector
+x = x(:);
+s = norm (x) * (sign (x(1)) + (x(1) == 0)); # Modification for sign (0) == 1.
+v = x;
+if (s == 0)
+## Quit if x is the zero vector.
+beta = 1;
+else
+v(1) = v(1) + s;
+beta = 1/(s'*v(1));                       # NB the conjugated s.
+##  beta = 1/(abs (s) * (abs (s) +abs(x(1)) would guarantee beta real.
+##  But beta as above can be non-real (due to rounding) only when x is complex.
+endif
+endfunction
+function A = invhess (x, y)
+## INVHESS  Inverse of an upper Hessenberg matrix.
+##          INVHESS(X, Y), where X is an N-vector and Y an N-1 vector,
+##          is the matrix whose lower triangle agrees with that of
+##          ONES(N,1)*X' and whose strict upper triangle agrees with
+##          that of [1 Y]*ONES(1,N).
+##          The matrix is nonsingular if X(1) ~= 0 and X(i+1) ~= Y(i)
+##          for all i, and its inverse is an upper Hessenberg matrix.
+##          If Y is omitted it defaults to -X(1:N-1).
+##          Special case: if X is a scalar INVHESS(X) is the same as
+##          INVHESS(1:X).
+##
+##          References:
+##          F.N. Valvi and V.S. Geroyannis, Analytic inverses and
+##              determinants for a class of matrices, IMA Journal of Numerical
+##              Analysis, 7 (1987), pp. 123-128.
+##          W.-L. Cao and W.J. Stewart, A note on inverses of Hessenberg-like
+##              matrices, Linear Algebra and Appl., 76 (1986), pp. 233-240.
+##          Y. Ikebe, On inverses of Hessenberg matrices, Linear Algebra and
+##              Appl., 24 (1979), pp. 93-97.
+##          P. Rozsa, On the inverse of band matrices, Integral Equations and
+##              Operator Theory, 10 (1987), pp. 82-95.
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 to 2 arguments are required for invhess matrix.");
+elseif (! isnumeric (x))
+error ("gallery: X must be numeric for invhess matrix.");
+endif
+if (isscalar (x) && fix (x) == x)
+n = x;
+x = 1:n;
+elseif (! isscalar (x) && isvector (x))
+n = numel (n);
+else
+error ("gallery: X must be an integer scalar, or a vector for invhess matrix.");
+endif
+if (nargin < 2)
+y = -x(1:end-1);
+elseif (! isvector (y) || numel (y) != numel (x) -1)
+error ("gallery: Y must be a vector of length -1 than X for invhess matrix.");
+endif
+x = x(:);
+y = y(:);
+##  On next line, z = x'; A = z(ones(n,1),:) would be more efficient.
+A = ones (n, 1) * x';
+for j = 2:n
+A(1:j-1,j) = y(1:j-1);
+endfor
+endfunction
+function A = invol (n)
+## INVOL   An involutory matrix.
+##         A = INVOL(N) is an N-by-N involutory (A*A = EYE(N)) and
+##         ill-conditioned matrix.
+##         It is a diagonally scaled version of HILB(N).
+##         NB: B = (EYE(N)-A)/2 and B = (EYE(N)+A)/2 are idempotent (B*B = B).
+##
+##         Reference:
+##         A.S. Householder and J.A. Carpenter, The singular values
+##         of involutory and of idempotent matrices, Numer. Math. 5 (1963),
+##         pp. 234-237.
+if (nargin != 1)
+error ("gallery: 1 argument is required for invol matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for invol matrix.");
+endif
+A = hilb (n);
+d = -n;
+A(:, 1) = d * A(:, 1);
+for i = 1:n-1
+d = -(n+i)*(n-i)*d/(i*i);
+A(i+1,:) = d * A(i+1,:);
+endfor
+endfunction
+function [A, detA] = ipjfact (n, k = 0)
+## IPJFACT   A Hankel matrix with factorial elements.
+##           A = IPJFACT(N, K) is the matrix with
+##                     A(i,j) = (i+j)!    (K = 0, default)
+##                     A(i,j) = 1/(i+j)!  (K = 1)
+##           Both are Hankel matrices.
+##           The determinant and inverse are known explicitly.
+##           If a second output argument is present, d = DET(A) is returned:
+##           [A, d] = IPJFACT(N, K);
+##
+##           Suggested by P. R. Graves-Morris.
+##
+##           Reference:
+##           M.J.C. Gover, The explicit inverse of factorial Hankel matrices,
+##           Dept. of Mathematics, University of Bradford, 1993.
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 to 2 arguments are required for ipjfact matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for ipjfact matrix.");
+elseif (! isnumeric (k) || ! isscalar (k))
+error ("gallery: K must be a numeric scalar for ipjfact matrix.");
+endif
+c = cumprod (2:n+1);
+d = cumprod (n+1:2*n) * c(n-1);
+A = hankel (c, d);
+switch (k)
+case (0), # do nothing
+case (1), A = ones (n) ./ A;
+otherwise
+error ("gallery: K must have a value of 0 or 1 for ipjfact matrix.");
+endswitch
+if (nargout == 2)
+d = 1;
+if (k == 0)
+for i = 1:n-1
+d = d * prod (1:i+1) * prod (1:n-i);
+endfor
+d = d * prod (1:n+1);
+elseif (k == 1)
+for i = 0:n-1
+d = d * prod (1:i) / prod (1:n+1+i);
+endfor
+if (rem (n*(n-1)/2, 2))
+d = -d;
+endif
+else
+error ("gallery: K must have a value of 0 or 1 for ipjfact matrix.");
+endif
+detA = d;
+endif
+endfunction
+function J = jordbloc (n, lambda = 1)
+## JORDBLOC  Jordan block.
+##           JORDBLOC(N, LAMBDA) is the N-by-N Jordan block with eigenvalue
+##           LAMBDA.  LAMBDA = 1 is the default.
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 to 2 arguments are required for jordbloc matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for jordbloc matrix.");
+elseif (! isnumeric (lambda) || ! isscalar (lambda))
+error ("gallery: LAMBDA must be a numeric scalar for jordbloc matrix.");
+endif
+J = lambda * eye (n) + diag (ones (n-1, 1), 1);
+endfunction
+function U = kahan (n, theta = 1.2, pert = 25)
+## KAHAN  Kahan matrix - upper trapezoidal.
+##        KAHAN(N, THETA) is an upper trapezoidal matrix
+##        that has some interesting properties regarding estimation of
+##        condition and rank.
+##        The matrix is N-by-N unless N is a 2-vector, in which case it
+##        is N(1)-by-N(2).
+##        The parameter THETA defaults to 1.2.
+##        The useful range of THETA is 0 < THETA < PI.
+##
+##        To ensure that the QR factorization with column pivoting does not
+##        interchange columns in the presence of rounding errors, the diagonal
+##        is perturbed by PERT*EPS*diag( [N:-1:1] ).
+##        The default is PERT = 25, which ensures no interchanges for KAHAN(N)
+##        up to at least N = 90 in IEEE arithmetic.
+##        KAHAN(N, THETA, PERT) uses the given value of PERT.
+##
+##        The inverse of KAHAN(N, THETA) is known explicitly: see
+##        Higham (1987, p. 588), for example.
+##        The diagonal perturbation was suggested by Christian Bischof.
+##
+##        References:
+##        W. Kahan, Numerical linear algebra, Canadian Math. Bulletin,
+##           9 (1966), pp. 757-801.
+##        N.J. Higham, A survey of condition number estimation for
+##           triangular matrices, SIAM Review, 29 (1987), pp. 575-596.
+if (nargin < 1 || nargin > 3)
+error ("gallery: 1 to 3 arguments are required for kahan matrix.");
+elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n)
+error ("gallery: N must be a 1 or 2 element integer for kahan matrix.");
+elseif (! isnumeric (theta) || ! isscalar (theta))
+error ("gallery: THETA must be a numeric scalar for kahan matrix.");
+elseif (! isnumeric (pert) || ! isscalar (pert))
+error ("gallery: PERT must be a numeric scalar for kahan matrix.");
+endif
+## Parameter n specifies dimension: r-by-n
+r = n(1);
+n = n(end);
+s = sin (theta);
+c = cos (theta);
+U = eye (n) - c * triu (ones (n), 1);
+U = diag (s.^[0:n-1]) * U + pert*eps* diag ([n:-1:1]);
+if (r > n)
+U(r,n) = 0;     # Extend to an r-by-n matrix
+else
+U = U(1:r,:);   # Reduce to an r-by-n matrix
+endif
+endfunction
+function A = kms (n, rho = 0.5)
+## KMS   Kac-Murdock-Szego Toeplitz matrix.
+##       A = KMS(N, RHO) is the N-by-N Kac-Murdock-Szego Toeplitz matrix with
+##       A(i,j) = RHO^(ABS((i-j))) (for real RHO).
+##       If RHO is complex, then the same formula holds except that elements
+##       below the diagonal are conjugated.
+##       RHO defaults to 0.5.
+##       Properties:
+##          A has an LDL' factorization with
+##                   L = INV(TRIW(N,-RHO,1)'),
+##                   D(i,i) = (1-ABS(RHO)^2)*EYE(N) except D(1,1) = 1.
+##          A is positive definite if and only if 0 < ABS(RHO) < 1.
+##          INV(A) is tridiagonal.
+##
+##        Reference:
+##        W.F. Trench, Numerical solution of the eigenvalue problem
+##        for Hermitian Toeplitz matrices, SIAM J. Matrix Analysis and Appl.,
+##        10 (1989), pp. 135-146 (and see the references therein).
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 to 2 arguments are required for lauchli matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error("gallery: N must be an integer for lauchli matrix.")
+elseif (! isscalar (mu))
+error("gallery: MU must be a scalar for lauchli matrix.")
+endif
+A = (1:n)'*ones(1,n);
+A = abs(A - A');
+A = rho .^ A;
+if imag(rho)
+A = conj(tril(A,-1)) + triu(A);
+endif
+endfunction
+function B = krylov (A, x, j)
+## KRYLOV    Krylov matrix.
+##           KRYLOV(A, x, j) is the Krylov matrix
+##                [x, Ax, A^2x, ..., A^(j-1)x],
+##           where A is an n-by-n matrix and x is an n-vector.
+##           Defaults: x = ONES(n,1), j = n.
+##           KRYLOV(n) is the same as KRYLOV(RANDN(n)).
+##
+##           Reference:
+##           G.H. Golub and C.F. Van Loan, Matrix Computations, second edition,
+##           Johns Hopkins University Press, Baltimore, Maryland, 1989, p. 369.
+if (nargin < 1 || nargin > 3)
+error ("gallery: 1 to 3 arguments are required for krylov matrix.");
+elseif (! isnumeric (A) || ! issquare (A) || ndims (A) != 2)
+error ("gallery: A must be a square 2D matrix for krylov matrix.");
+endif
+n = length (A);
+if (isscalar (A))
+n = A;
+A = randn (n);
+endif
+if (nargin < 2)
+x = ones (n, 1);
+elseif (! isvector (x) || numel (x) != n)
+error ("gallery: X must be a vector of length equal to A for krylov matrix.");
+endif
+if (nargin < 3)
+j = n;
+elseif (! isnumeric (j) || ! isscalar (j) || fix (j) != j)
+error ("gallery: J must be an integer for krylov matrix.");
+endif
+B = ones (n, j);
+B(:,1) = x(:);
+for i = 2:j
+B(:,i) = A*B(:,i-1);
+endfor
+endfunction
+function A = lauchli (n, mu = sqrt (eps))
+## LAUCHLI   Lauchli matrix - rectangular.
+##           LAUCHLI(N, MU) is the (N+1)-by-N matrix [ONES(1,N); MU*EYE(N))].
+##           It is a well-known example in least squares and other problems
+##           that indicates the dangers of forming A'*A.
+##           MU defaults to SQRT(EPS).
+##
+##           Reference:
+##           P. Lauchli, Jordan-Elimination und Ausgleichung nach
+##           kleinsten Quadraten, Numer. Math, 3 (1961), pp. 226-240.
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 to 2 arguments are required for lauchli matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for lauchli matrix.");
+elseif (! isscalar (mu))
+error ("gallery: MU must be a scalar for lauchli matrix.");
+endif
+A = [ones(1, n)
+mu*eye(n) ];
+endfunction
+function A = lehmer (n)
+## LEHMER  Lehmer matrix - symmetric positive definite.
+##         A = LEHMER(N) is the symmetric positive definite N-by-N matrix with
+##                          A(i,j) = i/j for j >= i.
+##         A is totally nonnegative.  INV(A) is tridiagonal, and explicit
+##         formulas are known for its entries.
+##         N <= COND(A) <= 4*N*N.
+##
+##         References:
+##         M. Newman and J. Todd, The evaluation of matrix inversion
+##            programs, J. Soc. Indust. Appl. Math., 6 (1958), pp. 466-476.
+##         Solutions to problem E710 (proposed by D.H. Lehmer): The inverse
+##            of a matrix, Amer. Math. Monthly, 53 (1946), pp. 534-535.
+##         J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra,
+##            Birkhauser, Basel, and Academic Press, New York, 1977, p. 154.
+if (nargin != 1)
+error ("gallery: 1 argument is required for lehmer matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for lehmer matrix.");
+endif
+A = ones (n, 1) * (1:n);
+A = A./A';
+A = tril (A) + tril (A, -1)';
+endfunction
+function T = lesp (n)
+## LESP   A tridiagonal matrix with real, sensitive eigenvalues.
+##        LESP(N) is an N-by-N matrix whose eigenvalues are real and smoothly
+##        distributed in the interval approximately [-2*N-3.5, -4.5].
+##        The sensitivities of the eigenvalues increase exponentially as
+##        the eigenvalues grow more negative.
+##        The matrix is similar to the symmetric tridiagonal matrix with
+##        the same diagonal entries and with off-diagonal entries 1,
+##        via a similarity transformation with D = diag(1!,2!,...,N!).
+##
+##        References:
+##        H.W.J. Lenferink and M.N. Spijker, On the use of stability regions in
+##             the numerical analysis of initial value problems,
+##             Math. Comp., 57 (1991), pp. 221-237.
+##        L.N. Trefethen, Pseudospectra of matrices, in Numerical Analysis 1991,
+##             Proceedings of the 14th Dundee Conference,
+##             D.F. Griffiths and G.A. Watson, eds, Pitman Research Notes in
+##             Mathematics, volume 260, Longman Scientific and Technical, Essex,
+##             UK, 1992, pp. 234-266.
+if (nargin != 1)
+error ("gallery: 1 argument is required for lesp matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for lesp matrix.");
+endif
+x = 2:n;
+T = full (tridiag (ones (size (x)) ./x, -(2*[x n+1]+1), x));
+endfunction
+function A = lotkin (n)
+## LOTKIN  Lotkin matrix.
+##         A = LOTKIN(N) is the Hilbert matrix with its first row altered to
+##         all ones.  A is unsymmetric, ill-conditioned, and has many negative
+##         eigenvalues of small magnitude.
+##         The inverse has integer entries and is known explicitly.
+##
+##         Reference:
+##         M. Lotkin, A set of test matrices, MTAC, 9 (1955), pp. 153-161.
+if (nargin != 1)
+error ("gallery: 1 argument is required for lotkin matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for lotkin matrix.");
+endif
+A = hilb (n);
+A(1,:) = ones (1, n);
+endfunction
+function A = minij (n)
+## MINIJ   Symmetric positive definite matrix MIN(i,j).
+##         A = MINIJ(N) is the N-by-N symmetric positive definite matrix with
+##         A(i,j) = MIN(i,j).
+##         Properties, variations:
+##         INV(A) is tridiagonal: it is minus the second difference matrix
+##                     except its (N,N) element is 1.
+##         2*A-ONES(N) (Givens' matrix) has tridiagonal inverse and
+##                     eigenvalues .5*sec^2([2r-1)PI/4N], r=1:N.
+##         (N+1)*ONES(N)-A also has a tridiagonal inverse.
+##
+##         References:
+##         J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra,
+##            Birkhauser, Basel, and Academic Press, New York, 1977, p. 158.
+##         D.E. Rutherford, Some continuant determinants arising in physics and
+##            chemistry---II, Proc. Royal Soc. Edin., 63, A (1952), pp. 232-241.
+##            (For the eigenvalues of Givens' matrix.)
+if (nargin != 1)
+error ("gallery: 1 argument is required for minij matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for minij matrix.");
+endif
+A = min (ones (n, 1) * (1:n), (1:n)' * ones (1, n));
+endfunction
+function A = moler (n, alpha = -1)
+## MOLER   Moler matrix - symmetric positive definite.
+##         A = MOLER(N, ALPHA) is the symmetric positive definite N-by-N matrix
+##         U'*U where U = TRIW(N, ALPHA).
+##         For ALPHA = -1 (the default) A(i,j) = MIN(i,j)-2, A(i,i) = i.
+##         A has one small eigenvalue.
+##
+##         Nash (1990) attributes the ALPHA = -1 matrix to Moler.
+##
+##         Reference:
+##         J.C. Nash, Compact Numerical Methods for Computers: Linear
+##         Algebra and Function Minimisation, second edition, Adam Hilger,
+##         Bristol, 1990 (Appendix 1).
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 to 2 arguments are required for moler matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for moler matrix.");
+elseif (! isscalar (alpha))
+error ("gallery: ALPHA must be a scalar for moler matrix.");
+endif
+A = triw (n, alpha)' * triw (n, alpha);
+endfunction
+function [A, T] = neumann (n)
+## NEUMANN  Singular matrix from the discrete Neumann problem (sparse).
+##          NEUMANN(N) is the singular, row diagonally dominant matrix resulting
+##          from discretizing the Neumann problem with the usual five point
+##          operator on a regular mesh.
+##          It has a one-dimensional null space with null vector ONES(N,1).
+##          The dimension N should be a perfect square, or else a 2-vector,
+##          in which case the dimension of the matrix is N(1)*N(2).
+##
+##          Reference:
+##          R.J. Plemmons, Regular splittings and the discrete Neumann
+##          problem, Numer. Math., 25 (1976), pp. 153-161.
+if (nargin != 1)
+error ("gallery: 1 argument is required for neumann matrix.");
+elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n)
+error ("gallery: N must be a 1 or 2 element integer for neumann matrix.");
+endif
+if (isscalar (n))
+m = sqrt (n);
+if (m^2 != n)
+error ("gallery: N must be a perfect square for neumann matrix.");
+endif
+n(1) = m;
+n(2) = m;
+endif
+T = tridiag (n(1), -1, 2, -1);
+T(1,2) = -2;
+T(n(1),n(1)-1) = -2;
+A = kron (T, eye (n(2))) + kron (eye (n(2)), T);
+endfunction
+function Q = orthog (n, k = 1)
+## ORTHOG Orthogonal and nearly orthogonal matrices.
+##        Q = ORTHOG(N, K) selects the K'th type of matrix of order N.
+##        K > 0 for exactly orthogonal matrices, K < 0 for diagonal scalings of
+##        orthogonal matrices.
+##        Available types: (K = 1 is the default)
+##        K = 1:  Q(i,j) = SQRT(2/(n+1)) * SIN( i*j*PI/(n+1) )
+##                Symmetric eigenvector matrix for second difference matrix.
+##        K = 2:  Q(i,j) = 2/SQRT(2*n+1)) * SIN( 2*i*j*PI/(2*n+1) )
+##                Symmetric.
+##        K = 3:  Q(r,s) = EXP(2*PI*i*(r-1)*(s-1)/n) / SQRT(n)  (i=SQRT(-1))
+##                Unitary, the Fourier matrix.  Q^4 is the identity.
+##                This is essentially the same matrix as FFT(EYE(N))/SQRT(N)!
+##        K = 4:  Helmert matrix: a permutation of a lower Hessenberg matrix,
+##                whose first row is ONES(1:N)/SQRT(N).
+##        K = 5:  Q(i,j) = SIN( 2*PI*(i-1)*(j-1)/n ) + COS( 2*PI*(i-1)*(j-1)/n ).
+##                Symmetric matrix arising in the Hartley transform.
+##        K = -1: Q(i,j) = COS( (i-1)*(j-1)*PI/(n-1) )
+##                Chebyshev Vandermonde-like matrix, based on extrema of T(n-1).
+##        K = -2: Q(i,j) = COS( (i-1)*(j-1/2)*PI/n) )
+##                Chebyshev Vandermonde-like matrix, based on zeros of T(n).
+##
+##        References:
+##        N.J. Higham and D.J. Higham, Large growth factors in Gaussian
+##             elimination with pivoting, SIAM J. Matrix Analysis and  Appl.,
+##             10 (1989), pp. 155-164.
+##        P. Morton, On the eigenvectors of Schur's matrix, J. Number Theory,
+##             12 (1980), pp. 122-127. (Re. ORTHOG(N, 3))
+##        H.O. Lancaster, The Helmert Matrices, Amer. Math. Monthly, 72 (1965),
+##             pp. 4-12.
+##        D. Bini and P. Favati, On a matrix algebra related to the discrete
+##             Hartley transform, SIAM J. Matrix Anal. Appl., 14 (1993),
+##             pp. 500-507.
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 to 2 arguments are required for orthog matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for orthog matrix.");
+elseif (! isnumeric (k) || ! isscalar (k))
+error ("gallery: K must be a numeric scalar for orthog matrix.");
+endif
+switch (k)
+case (1)
+## E'vectors second difference matrix
+m = (1:n)'*(1:n) * (pi/(n+1));
+Q = sin (m) * sqrt (2/(n+1));
+case (2)
+m = (1:n)'*(1:n) * (2*pi/(2*n+1));
+Q = sin (m) * (2/ sqrt (2*n+1));
+case (3)
+## Vandermonde based on roots of unity
+m = 0:n-1;
+Q = exp (m'*m*2*pi* sqrt (-1) / n) / sqrt (n);
+case (4)
+## Helmert matrix
+Q = tril (ones (n));
+Q(1,2:n) = ones (1, n-1);
+for i = 2:n
+Q(i,i) = -(i-1);
+end
+Q = diag (sqrt ([n 1:n-1] .* [1:n])) \ Q;
+case (5)
+## Hartley matrix
+m = (0:n-1)'*(0:n-1) * (2*pi/n);
+Q = (cos (m) + sin (m)) / sqrt (n);
+case (-1)
+##  extrema of T(n-1)
+m = (0:n-1)'*(0:n-1) * (pi/(n-1));
+Q = cos (m);
+case (-2)
+## zeros of T(n)
+m = (0:n-1)'*(.5:n-.5) * (pi/n);
+Q = cos (m);
+otherwise
+error ("gallery: unknown K `%d' for orthog matrix.", k);
+endswitch
+endfunction
+function A = parter (n)
+## PARTER    Parter matrix - a Toeplitz matrix with singular values near PI.
+##           PARTER(N) is the matrix with (i,j) element 1/(i-j+0.5).
+##           It is a Cauchy matrix and a Toeplitz matrix.
+##
+##           At the Second SIAM Conference on Linear Algebra, Raleigh, N.C.,
+##           1985, Cleve Moler noted that most of the singular values of
+##           PARTER(N) are very close to PI.  An explanation of the phenomenon
+##           was given by Parter; see also the paper by Tyrtyshnikov.
+##
+##           References:
+##           The MathWorks Newsletter, Volume 1, Issue 1, March 1986, page 2.
+##           S.V. Parter, On the distribution of the singular values of Toeplitz
+##                matrices, Linear Algebra and Appl., 80 (1986), pp. 115-130.
+##           E.E. Tyrtyshnikov, Cauchy-Toeplitz matrices and some applications,
+##                Linear Algebra and Appl., 149 (1991), pp. 1-18.
+if (nargin != 1)
+error ("gallery: 1 argument is required for parter matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for parter matrix.");
+endif
+A = cauchy ((1:n) + 0.5, -(1:n));
+endfunction
+function P = pei (n, alpha = 1)
+## PEI    Pei matrix.
+##        PEI(N, ALPHA), where ALPHA is a scalar, is the symmetric matrix
+##        ALPHA*EYE(N) + ONES(N).
+##        If ALPHA is omitted then ALPHA = 1 is used.
+##        The matrix is singular for ALPHA = 0, -N.
+##
+##        Reference:
+##        M.L. Pei, A test matrix for inversion procedures,
+##        Comm. ACM, 5 (1962), p. 508.
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 to 2 arguments are required for pei matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for pei matrix.");
+elseif (! isnumeric (w) || ! isscalar (w))
+error ("gallery: ALPHA must be a scalar for pei matrix.");
+endif
+P = alpha * eye (n) + ones (n);
+endfunction
+function A = poisson (n)
+## POISSON   Block tridiagonal matrix from Poisson's equation (sparse).
+##           POISSON(N) is the block tridiagonal matrix of order N^2
+##           resulting from discretizing Poisson's equation with the
+##           5-point operator on an N-by-N mesh.
+##
+##           Reference:
+##           G.H. Golub and C.F. Van Loan, Matrix Computations, second edition,
+##           Johns Hopkins University Press, Baltimore, Maryland, 1989
+##           (Section 4.5.4).
+if (nargin != 1)
+error ("gallery: 1 argument is required for poisson matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for poisson matrix.");
+endif
+S = tridiag (n, -1, 2, -1);
+I = speye (n);
+A = kron (I, S) + kron (S, I);
+endfunction
+function A = prolate (n, w = 0.25)
+## PROLATE   Prolate matrix - symmetric, ill-conditioned Toeplitz matrix.
+##           A = PROLATE(N, W) is the N-by-N prolate matrix with parameter W.
+##           It is a symmetric Toeplitz matrix.
+##           If 0 < W < 0.5 then
+##              - A is positive definite
+##              - the eigenvalues of A are distinct, lie in (0, 1), and
+##                tend to cluster around 0 and 1.
+##           W defaults to 0.25.
+##
+##           Reference:
+##           J.M. Varah. The Prolate matrix. Linear Algebra and Appl.,
+##           187:269--278, 1993.
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 to 2 arguments are required for prolate matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for prolate matrix.");
+elseif (! isnumeric (w) || ! isscalar (w))
+error ("gallery: W must be a scalar for prolate matrix.");
+endif
+a      = zeros (n, 1);
+a(1)   = 2*w;
+a(2:n) = sin (2*pi*w*(1:n-1)) ./ (pi*(1:n-1));
+A = toeplitz(a);
+endfunction
+function H = randhess (x)
+## NOTE: this function was named ohess in the original Test Matrix Toolbox
+## RANDHESS  Random, orthogonal upper Hessenberg matrix.
+##        H = RANDHESS(N) is an N-by-N real, random, orthogonal
+##        upper Hessenberg matrix.
+##        Alternatively, H = RANDHESS(X), where X is an arbitrary real
+##        N-vector (N > 1) constructs H non-randomly using the elements
+##        of X as parameters.
+##        In both cases H is constructed via a product of N-1 Givens rotations.
+##
+##        Note: See Gragg (1986) for how to represent an N-by-N (complex)
+##        unitary Hessenberg matrix with positive subdiagonal elements in terms
+##        of 2N-1 real parameters (the Schur parametrization).
+##        This M-file handles the real case only and is intended simply as a
+##        convenient way to generate random or non-random orthogonal Hessenberg
+##        matrices.
+##
+##        Reference:
+##        W.B. Gragg, The QR algorithm for unitary Hessenberg matrices,
+##        J. Comp. Appl. Math., 16 (1986), pp. 1-8.
+if (nargin != 1)
+error ("gallery: 1 argument is required for randhess matrix.");
+elseif (! isnumeric (x) || ! isreal (x))
+error ("gallery: N or X must be numeric real values for randhess matrix.");
+endif
+if (isscalar (x))
+n = x;
+x = rand (n-1, 1) * 2*pi;
+H = eye (n);
+H(n,n) = sign (randn);
+elseif (isvector (x))
+n = numel (x);
+H = eye (n);
+H(n,n) = sign (x(n)) + (x(n) == 0); # Second term ensures H(n,n) nonzero.
+else
+error ("gallery: N or X must be a scalar or a vector for randhess matrix.");
+endif
+for i = n:-1:2
+## Apply Givens rotation through angle x(i-1).
+theta = x(i-1);
+c = cos (theta);
+s = sin (theta);
+H([i-1 i], :) = [ c*H(i-1,:)+s*H(i,:)
+-s*H(i-1,:)+c*H(i,:) ];
+endfor
+endfunction
+function A = rando (n, k = 1)
+## RANDO   Random matrix with elements -1, 0 or 1.
+##         A = RANDO(N, K) is a random N-by-N matrix with elements from
+##         one of the following discrete distributions (default K = 1):
+##           K = 1:  A(i,j) =  0 or 1    with equal probability,
+##           K = 2:  A(i,j) = -1 or 1    with equal probability,
+##           K = 3:  A(i,j) = -1, 0 or 1 with equal probability.
+##         N may be a 2-vector, in which case the matrix is N(1)-by-N(2).
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 to 2 arguments are required for rando matrix.");
+elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n)
+error ("gallery: N must be an integer for rando matrix.");
+elseif (! isnumeric (k) || ! isscalar (k))
+error ("gallery: K must be a numeric scalar for smoke matrix.");
+endif
+## Parameter n specifies dimension: m-by-n.
+m = n(1);
+n = n(end);
+switch (k)
+case (1), A =   floor (  rand(m, n) + 0.5);     # {0, 1}
+case (2), A = 2*floor (  rand(m, n) + 0.5) -1;  # {-1, 1}
+case (3), A =   round (3*rand(m, n) - 1.5);     # {-1, 0, 1}
+otherwise
+error ("gallery: unknown K `%d' for smoke matrix.", k);
+endswitch
+endfunction
+function A = randsvd (n, kappa = sqrt (1/eps), mode = 3, kl = n-1, ku = kl)
+## RANDSVD  Random matrix with pre-assigned singular values.
+##       RANDSVD(N, KAPPA, MODE, KL, KU) is a (banded) random matrix of order N
+##       with COND(A) = KAPPA and singular values from the distribution MODE.
+##       N may be a 2-vector, in which case the matrix is N(1)-by-N(2).
+##       Available types:
+##              MODE = 1:   one large singular value,
+##              MODE = 2:   one small singular value,
+##              MODE = 3:   geometrically distributed singular values,
+##              MODE = 4:   arithmetically distributed singular values,
+##              MODE = 5:   random singular values with unif. dist. logarithm.
+##       If omitted, MODE defaults to 3, and KAPPA defaults to SQRT(1/EPS).
+##       If MODE < 0 then the effect is as for ABS(MODE) except that in the
+##       original matrix of singular values the order of the diagonal entries
+##       is reversed: small to large instead of large to small.
+##       KL and KU are the lower and upper bandwidths respectively; if they
+##       are omitted a full matrix is produced.
+##       If only KL is present, KU defaults to KL.
+##       Special case: if KAPPA < 0 then a random full symmetric positive
+##                     definite matrix is produced with COND(A) = -KAPPA and
+##                     eigenvalues distributed according to MODE.
+##                     KL and KU, if present, are ignored.
+##
+##       Reference:
+##       N.J. Higham, Accuracy and Stability of Numerical Algorithms,
+##          Society for Industrial and Applied Mathematics, Philadelphia, PA,
+##          USA, 1996; sec. 26.3.
+##
+##       This routine is similar to the more comprehensive Fortran routine xLATMS
+##       in the following reference:
+##       J.W. Demmel and A. McKenney, A test matrix generation suite,
+##       LAPACK Working Note #9, Courant Institute of Mathematical Sciences,
+##       New York, 1989.
+if (nargin < 1 || nargin > 5)
+error ("gallery: 1 to 5 arguments are required for randsvd matrix.");
+elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n)
+error ("gallery: N must be a 1 or 2 element integer vector for randsvd matrix.");
+elseif (! isnumeric (kappa) || ! isscalar (kappa))
+error ("gallery: KAPPA must be a numeric scalar for randsvd matrix.");
+elseif (abs (kappa) < 1)
+error ("gallery: KAPPA must larger than or equal to 1 for randsvd matrix.");
+elseif (! isnumeric (mode) || ! isscalar (mode))
+error ("gallery: MODE must be a numeric scalar for randsvd matrix.");
+elseif (! isnumeric (kl) || ! isscalar (kl))
+error ("gallery: KL must be a numeric scalar for randsvd matrix.");
+elseif (! isnumeric (ku) || ! isscalar (ku))
+error ("gallery: KU must be a numeric scalar for randsvd matrix.");
+endif
+posdef = 0;
+if (kappa < 0)
+posdef = 1;
+kappa  = -kappa;
+endif
+## Parameter n specifies dimension: m-by-n.
+m = n(1);
+n = n(end);
+p = min ([m n]);
+## If A will be a vector
+if (p == 1)
+A = randn (m, n);
+A = A / norm (A);
+return
+end
+##  Set up vector sigma of singular values.
+switch (abs (mode))
+case (1)
+sigma = ones (p, 1) ./ kappa;
+sigma(1) = 1;
+case (2)
+sigma = ones (p, 1);
+sigma(p) = 1 / kappa;
+case (3)
+factor = kappa^(-1/(p-1));
+sigma  = factor.^[0:p-1];
+case (4)
+sigma = ones (p, 1) - (0:p-1)'/(p-1)*(1-1/kappa);
+case (5)
+## In this case cond (A) <= kappa.
+rand ("uniform");
+sigma = exp (-rand (p, 1) * log (kappa));
+otherwise
+error ("gallery: unknown MODE `%d' for randsvd matrix.", mode);
+endswitch
+##  Convert to diagonal matrix of singular values.
+if (mode < 0)
+sigma = sigma (p:-1:1);
+end
+sigma = diag (sigma);
+if (posdef)
+## handle case where KAPPA was negative
+Q = qmult (p);
+A = Q' * sigma * Q;
+A = (A + A') / 2;  # Ensure matrix is symmetric.
+return
+endif
+if (m != n)
+## Expand to m-by-n diagonal matrix
+sigma(m, n) = 0;
+end
+if (kl == 0 && ku == 0)
+## Diagonal matrix requested - nothing more to do.
+A = sigma;
+else
+##  A = U*sigma*V, where U, V are random orthogonal matrices from the
+##  Haar distribution.
+A = qmult (sigma');
+A = qmult (A');
+if (kl < n-1 || ku < n-1)
+## Bandwidth reduction
+A = bandred (A, kl, ku);
+endif
+endif
+endfunction
+function A = redheff (n)
+## REDHEFF    A (0,1) matrix of Redheffer associated with the Riemann hypothesis.
+##            A = REDHEFF(N) is an N-by-N matrix of 0s and 1s defined by
+##                A(i,j) = 1 if j = 1 or if i divides j,
+##                A(i,j) = 0 otherwise.
+##            It has N - FLOOR(LOG2(N)) - 1 eigenvalues equal to 1,
+##            a real eigenvalue (the spectral radius) approximately SQRT(N),
+##            a negative eigenvalue approximately -SQRT(N),
+##            and the remaining eigenvalues are provably ``small''.
+##            Barrett and Jarvis (1992) conjecture that
+##              ``the small eigenvalues all lie inside the unit circle
+##                ABS(Z) = 1'',
+##            and a proof of this conjecture, together with a proof that some
+##            eigenvalue tends to zero as N tends to infinity, would yield
+##            a new proof of the prime number theorem.
+##            The Riemann hypothesis is true if and only if
+##            DET(A) = O( N^(1/2+epsilon) ) for every epsilon > 0
+##                                              (`!' denotes factorial).
+##            See also RIEMANN.
+##
+##            Reference:
+##            W.W. Barrett and T.J. Jarvis,
+##            Spectral Properties of a Matrix of Redheffer,
+##            Linear Algebra and Appl., 162 (1992), pp. 673-683.
+if (nargin != 1)
+error ("gallery: 1 argument is required for redheff matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for redheff matrix.");
+endif
+i = (1:n)' * ones (1, n);
+A = ! rem (i', i);
+A(:,1) = ones (n, 1);
+endfunction
+function A = riemann (n)
+## RIEMANN    A matrix associated with the Riemann hypothesis.
+##            A = RIEMANN(N) is an N-by-N matrix for which the
+##            Riemann hypothesis is true if and only if
+##            DET(A) = O( N! N^(-1/2+epsilon) ) for every epsilon > 0
+##                                              (`!' denotes factorial).
+##            A = B(2:N+1, 2:N+1), where
+##            B(i,j) = i-1 if i divides j and -1 otherwise.
+##            Properties include, with M = N+1:
+##               Each eigenvalue E(i) satisfies ABS(E(i)) <= M - 1/M.
+##               i <= E(i) <= i+1 with at most M-SQRT(M) exceptions.
+##               All integers in the interval (M/3, M/2] are eigenvalues.
+##
+##            See also REDHEFF.
+##
+##            Reference:
+##            F. Roesler, Riemann's hypothesis as an eigenvalue problem,
+##            Linear Algebra and Appl., 81 (1986), pp. 153-198.
+if (nargin != 1)
+error ("gallery: 1 argument is required for riemann matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for riemann matrix.");
+endif
+n = n+1;
+i = (2:n)' * ones (1, n-1);
+j = i';
+A = i .* (! rem (j, i)) - ones (n-1);
+endfunction
+function A = ris (n)
+## NOTE: this function was named dingdong in the original Test Matrix Toolbox
+## RIS       Dingdong matrix - a symmetric Hankel matrix.
+##           A = RIS(N) is the symmetric N-by-N Hankel matrix with
+##                          A(i,j) = 0.5/(N-i-j+1.5).
+##           The eigenvalues of A cluster around PI/2 and -PI/2.
+##
+##           Invented by F.N. Ris.
+##
+##           Reference:
+##           J.C. Nash, Compact Numerical Methods for Computers: Linear
+##           Algebra and Function Minimisation, second edition, Adam Hilger,
+##           Bristol, 1990 (Appendix 1).
+if (nargin != 1)
+error ("gallery: 1 argument is required for ris matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for ris matrix.");
+endif
+p = -2*(1:n) + (n+1.5);
+A = cauchy (p);
+endfunction
+function A = smoke (n, k = 0)
+## SMOKE     Smoke matrix - complex, with a `smoke ring' pseudospectrum.
+##           SMOKE(N) is an N-by-N matrix with 1s on the
+##           superdiagonal, 1 in the (N,1) position, and powers of
+##           roots of unity along the diagonal.
+##           SMOKE(N, 1) is the same except for a zero (N,1) element.
+##           The eigenvalues of SMOKE(N, 1) are the N'th roots of unity;
+##           those of SMOKE(N) are the N'th roots of unity times 2^(1/N).
+##
+##           Try PS(SMOKE(32)).  For SMOKE(N, 1) the pseudospectrum looks
+##           like a sausage folded back on itself.
+##           GERSH(SMOKE(N, 1)) is interesting.
+##
+##           Reference:
+##           L. Reichel and L.N. Trefethen, Eigenvalues and pseudo-eigenvalues of
+##           Toeplitz matrices, Linear Algebra and Appl., 162-164:153-185, 1992.
+if (nargin < 1 || nargin > 2)
+error ("gallery: 1 to 2 arguments are required for smoke matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be an integer for smoke matrix.");
+elseif (! isnumeric (n) || ! isscalar (n))
+error ("gallery: K must be a numeric scalar for smoke matrix.");
+endif
+w = exp(2*pi*i/n);
+A = diag( [w.^(1:n-1) 1] ) + diag(ones(n-1,1),1);
+switch (k)
+case (0), A(n,1) = 1;
+case (1), # do nothing
+otherwise,
+error ("gallery: K must have a value of 0 or 1 for smoke matrix.");
+endswitch
+endfunction
+function T = toeppd (n, m = n, w = rand (m,1), theta = rand (m,1))
+## NOTE: this function was named pdtoep in the original Test Matrix Toolbox
+## TOEPPD   Symmetric positive definite Toeplitz matrix.
+##          TOEPPD(N, M, W, THETA) is an N-by-N symmetric positive (semi-)
+##          definite (SPD) Toeplitz matrix, comprised of the sum of M rank 2
+##          (or, for certain THETA, rank 1) SPD Toeplitz matrices.
+##          Specifically,
+##                  T = W(1)*T(THETA(1)) + ... + W(M)*T(THETA(M)),
+##          where T(THETA(k)) has (i,j) element COS(2*PI*THETA(k)*(i-j)).
+##          Defaults: M = N, W = RAND(M,1), THETA = RAND(M,1).
+##
+##          Reference:
+##          G. Cybenko and C.F. Van Loan, Computing the minimum eigenvalue of
+##          a symmetric positive definite Toeplitz matrix, SIAM J. Sci. Stat.
+##          Comput., 7 (1986), pp. 123-131.
+if (nargin < 1 || nargin > 4)
+error ("gallery: 1 to 4 arguments are required for toeppd matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be a numeric integer for toeppd matrix.");
+elseif (! isnumeric (m) || ! isscalar (m) || fix (m) != m)
+error ("gallery: M must be a numeric integer for toeppd matrix.");
+elseif (numel (w) != m || numel (theta) != m)
+error ("gallery: W and THETA must be vectors of length M for toeppd matrix.");
+endif
+T = zeros (n);
+E = 2*pi * ((1:n)' * ones (1, n) - ones (n, 1) * (1:n));
+for i = 1:m
+T = T + w(i) * cos (theta(i)*E);
+endfor
+endfunction
+function P = toeppen (n, a = 1, b = -10, c = 0, d = 10, e = 1)
+## NOTE: this function was named pentoep in the original Test Matrix Toolbox
+## TOEPPEN   Pentadiagonal Toeplitz matrix (sparse).
+##           P = TOEPPEN(N, A, B, C, D, E) is the N-by-N pentadiagonal
+##           Toeplitz matrix with diagonals composed of the numbers
+##           A =: P(3,1), B =: P(2,1), C =: P(1,1), D =: P(1,2), E =: P(1,3).
+##           Default: (A,B,C,D,E) = (1,-10,0,10,1) (a matrix of Rutishauser).
+##                     This matrix has eigenvalues lying approximately on
+##                     the line segment 2*cos(2*t) + 20*i*sin(t).
+##
+##           Interesting plots are
+##           PS(FULL(TOEPPEN(32,0,1,0,0,1/4)))  - `triangle'
+##           PS(FULL(TOEPPEN(32,0,1/2,0,0,1)))  - `propeller'
+##           PS(FULL(TOEPPEN(32,0,1/2,1,1,1)))  - `fish'
+##
+##           References:
+##           R.M. Beam and R.F. Warming, The asymptotic spectra of
+##              banded Toeplitz and quasi-Toeplitz matrices, SIAM J. Sci.
+##              Comput. 14 (4), 1993, pp. 971-1006.
+##           H. Rutishauser, On test matrices, Programmation en Mathematiques
+##              Numeriques, Editions Centre Nat. Recherche Sci., Paris, 165,
+##              1966, pp. 349-365.
+if (nargin < 1 || nargin > 6)
+error ("gallery: 1 to 6 arguments are required for toeppen matrix.");
+elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n)
+error ("gallery: N must be a numeric integer for toeppen matrix.");
+elseif (any (cellfun (@(x) ! isnumeric (x) || ! isscalar (x), {a b c d e})))
+error ("gallery: A, B, C, D and E must be numeric scalars for toeppen matrix.");
+endif
+P = spdiags ([a*ones(n,1) b*ones(n,1) c*ones(n,1) d*ones(n,1) e*ones(n,1)],
+-2:2, n, n);
+endfunction
+function T = tridiag (n, x = -1, y = 2, z = -1)
+## TRIDIAG  Tridiagonal matrix (sparse).
+##          TRIDIAG(X, Y, Z) is the tridiagonal matrix with subdiagonal X,
+##          diagonal Y, and superdiagonal Z.
+##          X and Z must be vectors of dimension one less than Y.
+##          Alternatively TRIDIAG(N, C, D, E), where C, D, and E are all
+##          scalars, yields the Toeplitz tridiagonal matrix of order N
+##          with subdiagonal elements C, diagonal elements D, and superdiagonal
+##          elements E.   This matrix has eigenvalues (Todd 1977)
+##                   D + 2*SQRT(C*E)*COS(k*PI/(N+1)), k=1:N.
+##          TRIDIAG(N) is the same as TRIDIAG(N,-1,2,-1), which is
+##          a symmetric positive definite M-matrix (the negative of the
+##          second difference matrix).
+##
+##          References:
+##          J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra,
+##            Birkhauser, Basel, and Academic Press, New York, 1977, p. 155.
+##          D.E. Rutherford, Some continuant determinants arising in physics and
+##            chemistry---II, Proc. Royal Soc. Edin., 63, A (1952), pp. 232-241.
+if (nargin != 1 && nargin != 3 && nargin != 4)
+error ("gallery: 1, 3, or 4 arguments are required for tridiag matrix.");
+elseif (nargin == 3)
+z = y;
+y = x;
+x = n;
+endif
+## Force column vectors
+x = x(:);
+y = y(:);
+z = z(:);
+if (isscalar (x) && isscalar (y) && isscalar (z))
+x *= ones (n-1, 1);
+z *= ones (n-1, 1);
+y *= ones (n,   1);
+elseif (numel (y) != numel (x) + 1)
+error ("gallery: X must have one element less than Y for tridiag matrix.");
+elseif (numel (y) != numel (z) + 1)
+error ("gallery: Z must have one element less than Y for tridiag matrix.");
+endif
+##  T = diag (x, -1) + diag (y) + diag (z, 1);  # For non-sparse matrix.
+n = numel (y);
+T = spdiags ([[x;0] y [0;z]], -1:1, n, n);
+endfunction
+function t = triw (n, alpha = -1, k = -1)
+## TRIW   Upper triangular matrix discussed by Wilkinson and others.
+##        TRIW(N, ALPHA, K) is the upper triangular matrix with ones on
+##        the diagonal and ALPHAs on the first K >= 0 superdiagonals.
+##        N may be a 2-vector, in which case the matrix is N(1)-by-N(2) and
+##        upper trapezoidal.
+##        Defaults: ALPHA = -1,
+##                  K = N - 1     (full upper triangle).
+##        TRIW(N) is a matrix discussed by Kahan, Golub and Wilkinson.
+##
+##        Ostrowski (1954) shows that
+##          COND(TRIW(N,2)) = COT(PI/(4*N))^2,
+##        and for large ABS(ALPHA),
+##          COND(TRIW(N,ALPHA)) is approximately ABS(ALPHA)^N*SIN(PI/(4*N-2)).
+##
+##        Adding -2^(2-N) to the (N,1) element makes TRIW(N) singular,
+##        as does adding -2^(1-N) to all elements in the first column.
+##
+##        References:
+##        G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the
+##           computation of the Jordan canonical form, SIAM Review,
+##           18(4), 1976, pp. 578-619.
+##        W. Kahan, Numerical linear algebra, Canadian Math. Bulletin,
+##           9 (1966), pp. 757-801.
+##        A.M. Ostrowski, On the spectrum of a one-parametric family of
+##           matrices, J. Reine Angew. Math., 193 (3/4), 1954, pp. 143-160.
+##        J.H. Wilkinson, Singular-value decomposition---basic aspects,
+##           in D.A.H. Jacobs, ed., Numerical Software---Needs and Availability,
+##           Academic Press, London, 1978, pp. 109-135.
+if (nargin < 1 || nargin > 3)
+error ("gallery: 1 to 3 arguments are required for triw matrix.");
+elseif (! isnumeric (n) || all (numel (n) != [1 2]))
+error ("gallery: N must be a 1 or 2 elements vector for triw matrix.");
+elseif (! isscalar (alpha))
+error ("gallery: ALPHA must be a scalar for triw matrix.");
+elseif (! isscalar (k) || ! isnumeric (k) || fix (k) != k)
+error ("gallery: K must be a numeric integer for triw matrix.");
+endif
+m = n(1);              # Parameter n specifies dimension: m-by-n.
+n = n(end);
+t = tril (eye (m, n) + alpha * triu (ones (m, n), 1), k);
+endfunction
+function A = wathen (nx, ny, k = 0)
+## # WATHEN returns the Wathen matrix.
+##
+##   Discussion:
+##
+##     The Wathen matrix is a finite element matrix which is sparse.
+##
+##     The entries of the matrix depend in part on a physical quantity
+##     related to density.  That density is here assigned random values between
+##     0 and 100.
+##
+##     A = WATHEN ( NX, NY ) is a sparse random N-by-N finite element matrix
+##     where N = 3*NX*NY + 2*NX + 2*NY + 1.
+##
+##     A is the consistent mass matrix for a regular NX-by-NY
+##     grid of 8-node (serendipity) elements in 2 space dimensions.
+##
+##     Here is an illustration for NX = 3, NX = 2:
+##
+##      23-24-25-26-27-28-29
+##       |     |     |     |
+##      19    20    21    22
+##       |     |     |     |
+##      12-13-14-15-16-17-18
+##       |     |     |     |
+##       8     9    10    11
+##       |     |     |     |
+##       1--2--3--4--5--6--7
+##
+##     For this example, the total number of nodes is, as expected,
+##
+##       N = 3 * 3 * 2 + 2 * 2 + 2 * 3 + 1 = 29.
+##
+##     A is symmetric positive definite for any (positive) values of
+##     the density, RHO(NX,NY), which is chosen randomly in this routine.
+##
+##     In particular, if D = DIAG(DIAG(A)), then
+##       0.25 <= EIG(INV(D)*A) <= 4.5
+##     for any positive integers NX and NY and any densities RHO(NX,NY).
+##
+##     A = WATHEN ( NX, NY, 1 ) returns the diagonally scaled matrix.
+##
+##   Modified:
+##
+##     17 September 2007
+##
+##   Author:
+##
+##     Nicholas Higham
+##
+##   Reference:
+##
+##     Nicholas Higham,
+##     Algorithm 694: A Collection of Test Matrices in MATLAB,
+##     ACM Transactions on Mathematical Software,
+##     Volume 17, Number 3, September 1991, pages 289-305.
+##
+##     Andrew Wathen,
+##     Realistic eigenvalue bounds for the Galerkin mass matrix,
+##     IMA Journal of Numerical Analysis,
+##     Volume 7, 1987, pages 449-457.
+##
+##   Parameters:
+##
+##     Input, integer NX, NY, the number of elements in the X and Y directions
+##     of the finite element grid.  NX and NY must each be at least 1.
+##
+##     Optional input, integer K, is used to request that the diagonally scaled
+##     version of the matrix be returned.  This happens if K is specified with
+##     the value 1.
+##
+##     Output, sparse real A(N,N), the matrix.  The dimension N is determined by
+##     NX and NY, as described above.  A is stored in the MATLAB sparse matrix
+##     format.
+if (nargin < 2 || nargin > 3)
+error ("gallery: 2 or 3 arguments are required for wathen matrix.");
+elseif (! isnumeric (nx) || ! isscalar (nx) || nx < 1)
+error ("gallery: NX must be a positive scalar for wathen matrix.");
+elseif (! isnumeric (ny) || ! isscalar (ny) || ny < 1)
+error ("gallery: NY must be a positive scalar for wathen matrix.");
+elseif (! isscalar (k))
+error ("gallery: K must be a scalar for wathen matrix.");
+endif
+e1 = [ 6  -6   2  -8
+-6  32  -6  20
+2  -6   6  -6
+-8  20  -6  32 ];
+e2 = [ 3  -8   2  -6
+-8  16  -8  20
+2  -8   3  -8
+-6  20  -8  16 ];
+e = [ e1  e2
+e2' e1] / 45;
+n = 3*nx*ny + 2*nx + 2*ny + 1;
+A = sparse (n, n);
+rho = 100 * rand (nx, ny);
+for j = 1:ny
+for i = 1:nx
+##
+##   For the element (I,J), determine the indices of the 8 nodes.
+##
+nn(1) = 3*j*nx + 2*i + 2*j + 1;
+nn(2) = nn(1) - 1;
+nn(3) = nn(2) - 1;
+nn(4) = (3*j - 1) * nx + 2*j + i - 1;
+nn(5) = 3 * (j-1) * nx + 2*i + 2*j - 3;
+nn(6) = nn(5) + 1;
+nn(7) = nn(6) + 1;
+nn(8) = nn(4) + 1;
+em = e * rho(i,j);
+for krow = 1:8
+for kcol = 1:8
+A(nn(krow),nn(kcol)) = A(nn(krow),nn(kcol)) + em(krow,kcol);
+endfor
+endfor
+endfor
+endfor
+## If requested, return A with diagonal scaling.
+if (k)
+A = diag (diag (A)) \ A;
+endif
+endfunction
+function [A, b] = wilk (n)
+## WILK   Various specific matrices devised/discussed by Wilkinson.
+##        [A, b] = WILK(N) is the matrix or system of order N.
+##        N = 3: upper triangular system Ux=b illustrating inaccurate solution.
+##        N = 4: lower triangular system Lx=b, ill-conditioned.
+##        N = 5: HILB(6)(1:5,2:6)*1.8144.  Symmetric positive definite.
+##        N = 21: W21+, tridiagonal.   Eigenvalue problem.
+##
+##        References:
+##        J.H. Wilkinson, Error analysis of direct methods of matrix inversion,
+##           J. Assoc. Comput. Mach., 8 (1961),  pp. 281-330.
+##        J.H. Wilkinson, Rounding Errors in Algebraic Processes, Notes on Applied
+##           Science No. 32, Her Majesty's Stationery Office, London, 1963.
+##        J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University
+##           Press, 1965.
+if (nargin != 1)
+error ("gallery: 1 argument is required for wilk matrix.");
+elseif (! isnumeric (n) || ! isscalar (n))
+error ("gallery: N must be a numeric scalar for wilk matrix.");
+endif
+if (n == 3)
+## Wilkinson (1961) p.323.
+A = [ 1e-10   0.9  -0.4
+0       0.9  -0.4
+0       0     1e-10 ];
+b = [ 0
+0
+1];
+elseif (n == 4)
+## Wilkinson (1963) p.105.
+A = [0.9143e-4  0          0          0
+0.8762     0.7156e-4  0          0
+0.7943     0.8143     0.9504e-4  0
+0.8017     0.6123     0.7165     0.7123e-4];
+b = [0.6524
+0.3127
+0.4186
+0.7853];
+elseif (n == 5)
+## Wilkinson (1965), p.234.
+A = hilb (6);
+A = A(1:5, 2:6) * 1.8144;
+elseif (n == 21)
+## Wilkinson (1965), p.308.
+E = diag (ones (n-1, 1), 1);
+m = (n-1)/2;
+A = diag (abs (-m:m)) + E + E';
+else
+error ("gallery: unknown N `%d' for wilk matrix.", n);
+endif
+endfunction
+## NOTE: bandred is part of the Test Matrix Toolbox and is used by randsvd()
+function A = bandred (A, kl, ku)
+## BANDRED  Band reduction by two-sided unitary transformations.
+##          B = BANDRED(A, KL, KU) is a matrix unitarily equivalent to A
+##          with lower bandwidth KL and upper bandwidth KU
+##          (i.e. B(i,j) = 0 if i > j+KL or j > i+KU).
+##          The reduction is performed using Householder transformations.
+##          If KU is omitted it defaults to KL.
+##
+##          Called by RANDSVD.
+##          This is a `standard' reduction.  Cf. reduction to bidiagonal form
+##          prior to computing the SVD.  This code is a little wasteful in that
+##          it computes certain elements which are immediately set to zero!
+##
+##          Reference:
+##          G.H. Golub and C.F. Van Loan, Matrix Computations, second edition,
+##          Johns Hopkins University Press, Baltimore, Maryland, 1989.
+##          Section 5.4.3.
+##  Check for special case where order of left/right transformations matters.
+##  Easiest approach is to work on the transpose, flipping back at the end.
+flip = false;
+if (ku == 0)
+flip = true;
+A = A';
+[ku, kl] = deal (kl, ku);
+endif
+[m, n] = size (A);
+for j = 1:min (min (m, n), max (m-kl-1, n-ku-1))
+if (j+kl+1 <= m)
+[v, beta] = house (A(j+kl:m,j));
+temp = A(j+kl:m,j:n);
+A(j+kl:m,j:n) = temp - beta*v*(v'*temp);
+A(j+kl+1:m,j) = zeros (m-j-kl, 1);
+endif
+if (j+ku+1 <= n)
+[v, beta] = house (A(j,j+ku:n)');
+temp = A(j:m,j+ku:n);
+A(j:m,j+ku:n) = temp - beta*(temp*v)*v';
+A(j,j+ku+1:n) = zeros (1, n-j-ku);
+endif
+endfor
+if (flip)
+A = A';
+endif
+endfunction

Mercurial > octave-antonio

comparison scripts/special-matrix/gallery.m @ 16634:2510fffc05e1