Mercurial > octave-antonio
view scripts/special-matrix/gallery.m @ 20162:2645f9ef8c88 stable
doc: Update more docstrings to have one sentence summary as first line.
Reviewed specfun, special-matrix, testfun, and time script directories.
* scripts/specfun/expint.m, scripts/specfun/isprime.m,
scripts/specfun/legendre.m, scripts/specfun/primes.m,
scripts/specfun/reallog.m, scripts/specfun/realsqrt.m,
scripts/special-matrix/gallery.m, scripts/special-matrix/hadamard.m,
scripts/special-matrix/hankel.m, scripts/special-matrix/hilb.m,
scripts/special-matrix/invhilb.m, scripts/special-matrix/magic.m,
scripts/special-matrix/pascal.m, scripts/special-matrix/rosser.m,
scripts/special-matrix/toeplitz.m, scripts/special-matrix/vander.m,
scripts/special-matrix/wilkinson.m, scripts/testfun/assert.m,
scripts/testfun/demo.m, scripts/testfun/example.m, scripts/testfun/fail.m,
scripts/testfun/rundemos.m, scripts/testfun/runtests.m,
scripts/testfun/speed.m, scripts/time/asctime.m, scripts/time/calendar.m,
scripts/time/clock.m, scripts/time/ctime.m, scripts/time/datenum.m,
scripts/time/datestr.m, scripts/time/datevec.m, scripts/time/etime.m,
scripts/time/is_leap_year.m, scripts/time/now.m, scripts/time/weekday.m:
Update more docstrings to have one sentence summary as first line.
| author | Rik <rik@octave.org> |
|---|---|
| date | Sun, 03 May 2015 17:00:11 -0700 |
| parents | 941e782d0429 |
| children |
line wrap: on
line source
## Copyright (C) 1989-1995 Nicholas .J. Higham ## Copyright (C) 2013-2015 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 @nospell{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 @nospell{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 ("integerdata", @var{imax}, [@var{M} @var{N} @dots{}], @var{j}) ## @deftypefnx {Function File} {@var{a} =} gallery ("integerdata", @var{imax}, @var{M}, @var{N}, @dots{}, @var{j}) ## @deftypefnx {Function File} {@var{a} =} gallery ("integerdata", [@var{imin}, @var{imax}], [@var{M} @var{N} @dots{}], @var{j}) ## @deftypefnx {Function File} {@var{a} =} gallery ("integerdata", [@var{imin}, @var{imax}], @var{M}, @var{N}, @dots{}, @var{j}) ## @deftypefnx {Function File} {@var{a} =} gallery ("integerdata", @dots{}, "@var{class}") ## Create a matrix with random integers in the range [1, @var{imax}]. ## If @var{imin} is given then the integers are in the range ## [@var{imin}, @var{imax}]. ## ## The second input is a matrix of dimensions describing the size of the output. ## The dimensions can also be input as comma-separated arguments. ## ## The input @var{j} is an integer index in the range [0, 2^32-1]. The values ## of the output matrix are always exactly the same (reproducibility) for a ## given size input and @var{j} index. ## ## The final optional argument determines the class of the resulting matrix. ## Possible values for @var{class}: @qcode{"uint8"}, @qcode{"uint16"}, ## @qcode{"uint32"}, @qcode{"int8"}, @qcode{"int16"}, int32", @qcode{"single"}, ## @qcode{"double"}. The default is @qcode{"double"}. ## ## @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 a 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 @nospell{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 @nospell{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 @nospell{Lauchli} matrix (rectangular). ## ## @end deftypefn ## ## @deftypefn {Function File} {@var{a} =} gallery ("lehmer", @var{n}) ## Create a @nospell{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 @nospell{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 @nospell{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{a} =} gallery ("normaldata", [@var{M} @var{N} @dots{}], @var{j}) ## @deftypefnx {Function File} {@var{a} =} gallery ("normaldata", @var{M}, @var{N}, @dots{}, @var{j}) ## @deftypefnx {Function File} {@var{a} =} gallery ("normaldata", @dots{}, "@var{class}") ## Create a matrix with random samples from the standard normal distribution ## (mean = 0, std = 1). ## ## The first input is a matrix of dimensions describing the size of the output. ## The dimensions can also be input as comma-separated arguments. ## ## The input @var{j} is an integer index in the range [0, 2^32-1]. The values ## of the output matrix are always exactly the same (reproducibility) for a ## given size input and @var{j} index. ## ## The final optional argument determines the class of the resulting matrix. ## Possible values for @var{class}: @qcode{"single"}, @qcode{"double"}. ## The default is @qcode{"double"}. ## ## @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 @nospell{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 @nospell{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 ## @nospell{Kahan, Golub, and Wilkinson}. ## ## @end deftypefn ## ## @deftypefn {Function File} {@var{a} =} gallery ("uniformdata", [@var{M} @var{N} @dots{}], @var{j}) ## @deftypefnx {Function File} {@var{a} =} gallery ("uniformdata", @var{M}, @var{N}, @dots{}, @var{j}) ## @deftypefnx {Function File} {@var{a} =} gallery ("uniformdata", @dots{}, "@var{class}") ## Create a matrix with random samples from the standard uniform distribution ## (range [0,1]). ## ## The first input is a matrix of dimensions describing the size of the output. ## The dimensions can also be input as comma-separated arguments. ## ## The input @var{j} is an integer index in the range [0, 2^32-1]. The values ## of the output matrix are always exactly the same (reproducibility) for a ## given size input and @var{j} index. ## ## The final optional argument determines the class of the resulting matrix. ## Possible values for @var{class}: @qcode{"single"}, @qcode{"double"}. ## The default is @qcode{"double"}. ## ## @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 @nospell{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", [varargout{1:n_out}] = integerdata (varargin{:}); 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" , [varargout{1:n_out}] = normaldata (varargin{:}); 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" , [varargout{1:n_out}] = uniformdata (varargin{:}); 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 (v); if (isscalar (v) && fix (v) == v) n = v; v = 1:n; elseif (n > 1 && isvector (v)) ## 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 2-D 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. ## 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 nonzero scalar, and abs (I) <= N for gearmat matrix."); elseif (! isnumeric (j) || ! isscalar (j) || i == 0 || abs (j) <= n) error ("gallery: J must be a nonzero 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 = integerdata (varargin) if (nargin < 3) error ("gallery: At least 3 arguments required for integerdata matrix."); endif if (isnumeric (varargin{end})) jidx = varargin{end}; svec = [varargin{:}]; varargin(end) = []; elseif (ischar (varargin{end})) if (nargin < 4) error (["gallery: CLASS argument requires 4 inputs " ... "for integerdata matrix."]); endif jidx = varargin{end-1}; svec = [varargin{1:end-1}]; varargin(end-1) = []; else error (["gallery: J must be an integer in the range [0, 2^32-1] " ... "for integerdata matrix"]); endif if (! (isnumeric (jidx) && isscalar (jidx) && jidx == fix (jidx) && jidx >= 0 && jidx <= 0xFFFFFFFF)) error (["gallery: J must be an integer in the range [0, 2^32-1] " ... "for integerdata matrix"]); endif ## Save and restore random state. Initialization done so that reproducible ## data is available from gallery depending on the jidx and size vector. randstate = rand ("state"); unwind_protect rand ("state", svec); A = randi (varargin{:}); unwind_protect_cleanup rand ("state", randstate); end_unwind_protect 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(:); ## FIXME: 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 2-D 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 = bsxfun (@min, 1:n, (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 A = normaldata (varargin) if (nargin < 2) error ("gallery: At least 2 arguments required for normaldata matrix."); endif if (isnumeric (varargin{end})) jidx = varargin{end}; svec = [varargin{:}]; varargin(end) = []; elseif (ischar (varargin{end})) if (nargin < 3) error (["gallery: CLASS argument requires 3 inputs " ... "for normaldata matrix."]); endif jidx = varargin{end-1}; svec = [varargin{1:end-1}]; varargin(end-1) = []; else error (["gallery: J must be an integer in the range [0, 2^32-1] " ... "for normaldata matrix"]); endif if (! (isnumeric (jidx) && isscalar (jidx) && jidx == fix (jidx) && jidx >= 0 && jidx <= 0xFFFFFFFF)) error (["gallery: J must be an integer in the range [0, 2^32-1] " ... "for normaldata matrix"]); endif ## Save and restore random state. Initialization done so that reproducible ## data is available from gallery depending on the jidx and size vector. randstate = randn ("state"); unwind_protect randn ("state", svec); A = randn (varargin{:}); unwind_protect_cleanup randn ("state", randstate); end_unwind_protect 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); endfor 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; endif ## 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); endif 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; endif 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 ("isnumeric", {a b c d e})) || any (cellfun ("numel", {a b c d e}) != 1)) 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 = n(end) - 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 || k < 0) error ("gallery: K must be a numeric integer >= 0 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 = uniformdata (varargin) if (nargin < 2) error ("gallery: At least 2 arguments required for uniformdata matrix."); endif if (isnumeric (varargin{end})) jidx = varargin{end}; svec = [varargin{:}]; varargin(end) = []; elseif (ischar (varargin{end})) if (nargin < 3) error (["gallery: CLASS argument requires 3 inputs " ... "for uniformdata matrix."]); endif jidx = varargin{end-1}; svec = [varargin{1:end-1}]; varargin(end-1) = []; else error (["gallery: J must be an integer in the range [0, 2^32-1] " ... "for uniformdata matrix"]); endif if (! (isnumeric (jidx) && isscalar (jidx) && jidx == fix (jidx) && jidx >= 0 && jidx <= 0xFFFFFFFF)) error (["gallery: J must be an integer in the range [0, 2^32-1] " ... "for uniformdata matrix"]); endif ## Save and restore random state. Initialization done so that reproducible ## data is available from gallery depending on the jidx and size vector. randstate = rand ("state"); unwind_protect rand ("state", svec); A = rand (varargin{:}); unwind_protect_cleanup rand ("state", randstate); end_unwind_protect 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 ## BIST testing for just a few functions to verify that the main gallery ## dispatch function works. %assert (gallery ("clement", 3), [0 1 0; 2 0 2; 0 1 0]) %assert (gallery ("invhess", 2), [1 -1; 1 2]) ## Test input validation of main dispatch function only %!error gallery () %!error <NAME must be a string> gallery (123) %!error <matrix binomial not implemented> gallery ("binomial") %!error <unknown matrix with NAME foobar> gallery ("foobar") %!assert (gallery ("minij", 4), [1 1 1 1; 1 2 2 2; 1 2 3 3; 1 2 3 4]) %!assert (gallery ("minij", 1), 1) %!assert (gallery ("minij", 0), []) %!assert (gallery ("minij", -1), [])