Mercurial > hg > octave-avbm
changeset 16636:2510fffc05e1
gallery: new function
| author | Carnë Draug <carandraug@octave.org> |
|---|---|
| date | Fri, 10 May 2013 01:02:58 +0100 |
| parents | 8d32a887754a |
| children | 25e418d23a4b |
| files | NEWS doc/interpreter/matrix.txi scripts/help/__unimplemented__.m scripts/special-matrix/gallery.m scripts/special-matrix/module.mk |
| diffstat | 5 files changed, 2676 insertions(+), 2 deletions(-) [+] |
line wrap: on
line diff
--- a/NEWS +++ b/NEWS @@ -180,7 +180,7 @@ betaincinv ellipj findfigs polyeig tetramesh cmpermute ellipke fminsearch rgbplot waterfall - cmunique erfcinv importdata shrinkfaces + cmunique erfcinv importdata shrinkfaces gallery colorcube erfi iscolormap splinefit dawson expint lines strjoin
--- a/doc/interpreter/matrix.txi +++ b/doc/interpreter/matrix.txi @@ -235,6 +235,8 @@ The following functions return famous matrix forms. +@DOCSTRING(gallery) + @DOCSTRING(hadamard) @DOCSTRING(hankel)
--- a/scripts/help/__unimplemented__.m +++ b/scripts/help/__unimplemented__.m @@ -203,7 +203,6 @@ "frame2im", "freqspace", "funm", - "gallery", "gammaincinv", "getframe", "getpixelposition",
new file mode 100644 --- /dev/null +++ b/scripts/special-matrix/gallery.m @@ -0,0 +1,2672 @@ +## Copyright (C) 1989-1995 Nicholas .J. Higham +## Copyright (C) 2013 Carnë Draug +## +## This file is part of Octave. +## +## Octave is free software; you can redistribute it and/or modify it +## under the terms of the GNU General Public License as published by +## the Free Software Foundation; either version 3 of the License, or (at +## your option) any later version. +## +## Octave is distributed in the hope that it will be useful, but +## WITHOUT ANY WARRANTY; without even the implied warranty of +## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU +## General Public License for more details. +## +## You should have received a copy of the GNU General Public License +## along with Octave; see the file COPYING. If not, see +## <http://www.gnu.org/licenses/>. + +## -*- texinfo -*- +## @deftypefn {Function File} {} gallery (@var{name}) +## @deftypefnx {Function File} {} gallery (@var{name}, @var{args}) +## Create interesting matrices for testing. +## +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{c} =} gallery ("cauchy", @var{x}) +## @deftypefnx {Function File} {@var{c} =} gallery ("cauchy", @var{x}, @var{y}) +## Create a Cauchy matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{c} =} gallery ("chebspec", @var{n}) +## @deftypefnx {Function File} {@var{c} =} gallery ("chebspec", @var{n}, @var{k}) +## Create a Chebyshev spectral differentiation matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{c} =} gallery ("chebvand", @var{p}) +## @deftypefnx {Function File} {@var{c} =} gallery ("chebvand", @var{m}, @var{p}) +## Create a Vandermonde-like matrix for the Chebyshev polynomials. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("chow", @var{n}) +## @deftypefnx {Function File} {@var{a} =} gallery ("chow", @var{n}, @var{alpha}) +## @deftypefnx {Function File} {@var{a} =} gallery ("chow", @var{n}, @var{alpha}, @var{delta}) +## Create a Chow matrix -- a singular Toeplitz lower Hessenberg matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{c} =} gallery ("circul", @var{v}) +## Create a circulant matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("clement", @var{n}) +## @deftypefnx {Function File} {@var{a} =} gallery ("clement", @var{n}, @var{k}) +## Create a tridiagonal matrix with zero diagonal entries. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{c} =} gallery ("compar", @var{a}) +## @deftypefnx {Function File} {@var{c} =} gallery ("compar", @var{a}, @var{k}) +## Create a comparison matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("condex", @var{n}) +## @deftypefnx {Function File} {@var{a} =} gallery ("condex", @var{n}, @var{k}) +## @deftypefnx {Function File} {@var{a} =} gallery ("condex", @var{n}, @var{k}, @var{theta}) +## Create a `counterexample' matrix to a condition estimator. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("cycol", [@var{m} @var{n}]) +## @deftypefnx {Function File} {@var{a} =} gallery ("cycol", @var{n}) +## @deftypefnx {Function File} {@var{a} =} gallery (@dots{}, @var{k}) +## Create a matrix whose columns repeat cyclically. +## +## @end deftypefn +## +## @deftypefn {Function File} {[@var{c},@var{d}, @var{e}] =} gallery ("dorr", @var{n}) +## @deftypefnx {Function File} {[@var{c},@var{d}, @var{e}] =} gallery ("dorr", @var{n}, @var{theta}) +## @deftypefnx {Function File} {@var{a} =} gallery ("dorr", @dots{}) +## Create a diagonally dominant, ill conditioned, tridiagonal matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("dramadah", @var{n}) +## @deftypefnx {Function File} {@var{a} =} gallery ("dramadah", @var{n}, @var{k}) +## Create a (0, 1) matrix whose inverse has large integer entries. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("fiedler", @var{c}) +## Create a symmetric Fiedler matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("forsythe", @var{n}) +## @deftypefnx {Function File} {@var{a} =} gallery ("forsythe", @var{n}, @var{alpha}) +## @deftypefnx {Function File} {@var{a} =} gallery ("forsythe", @var{n}, @var{alpha}, @var{lambda}) +## Create a Forsythe matrix (a perturbed Jordan block). +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{f} =} gallery ("frank", @var{n}) +## @deftypefnx {Function File} {@var{f} =} gallery ("frank", @var{n}, @var{k}) +## Create a Frank matrix (ill conditioned eigenvalues). +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{c} =} gallery ("gcdmat", @var{n}) +## Create a greatest common divisor matrix. +## +## @var{c} is an @var{n}-by-@var{n} matrix whose values correspond to the +## greatest common divisor of its coordinate values, i.e., @var{c}(i,j) +## correspond @code{gcd (i, j)}. +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("gearmat", @var{n}) +## @deftypefnx {Function File} {@var{a} =} gallery ("gearmat", @var{n}, @var{i}) +## @deftypefnx {Function File} {@var{a} =} gallery ("gearmat", @var{n}, @var{i}, @var{j}) +## Create a Gear matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{g} =} gallery ("grcar", @var{n}) +## @deftypefnx {Function File} {@var{g} =} gallery ("grcar", @var{n}, @var{k}) +## Create a Toeplitz matrix with sensitive eigenvalues. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("hanowa", @var{n}) +## @deftypefnx {Function File} {@var{a} =} gallery ("hanowa", @var{n}, @var{d}) +## Create a matrix whose eigenvalues lie on a vertical line in the complex plane. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{v} =} gallery ("house", @var{x}) +## @deftypefnx {Function File} {[@var{v}, @var{beta}] =} gallery ("house", @var{x}) +## Create a householder matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("invhess", @var{x}) +## @deftypefnx {Function File} {@var{a} =} gallery ("invhess", @var{x}, @var{y}) +## Create the inverse of an upper Hessenberg matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("invol", @var{n}) +## Create an involutory matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("ipjfact", @var{n}) +## @deftypefnx {Function File} {@var{a} =} gallery ("ipjfact", @var{n}, @var{k}) +## Create an Hankel matrix with factorial elements. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("jordbloc", @var{n}) +## @deftypefnx {Function File} {@var{a} =} gallery ("jordbloc", @var{n}, @var{lambda}) +## Create a Jordan block. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{u} =} gallery ("kahan", @var{n}) +## @deftypefnx {Function File} {@var{u} =} gallery ("kahan", @var{n}, @var{theta}) +## @deftypefnx {Function File} {@var{u} =} gallery ("kahan", @var{n}, @var{theta}, @var{pert}) +## Create a Kahan matrix (upper trapezoidal). +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("kms", @var{n}) +## @deftypefnx {Function File} {@var{a} =} gallery ("kms", @var{n}, @var{rho}) +## Create a Kac-Murdock-Szego Toeplitz matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{b} =} gallery ("krylov", @var{a}) +## @deftypefnx {Function File} {@var{b} =} gallery ("krylov", @var{a}, @var{x}) +## @deftypefnx {Function File} {@var{b} =} gallery ("krylov", @var{a}, @var{x}, @var{j}) +## Create a Krylov matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("lauchli", @var{n}) +## @deftypefnx {Function File} {@var{a} =} gallery ("lauchli", @var{n}, @var{mu}) +## Create a Lauchli matrix (rectangular). +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("lehmer", @var{n}) +## Create a Lehmer matrix (symmetric positive definite). +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{t} =} gallery ("lesp", @var{n}) +## Create a tridiagonal matrix with real, sensitive eigenvalues. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("lotkin", @var{n}) +## Create a Lotkin matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("minij", @var{n}) +## Create a symmetric positive definite matrix MIN(i,j). +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("moler", @var{n}) +## @deftypefnx {Function File} {@var{a} =} gallery ("moler", @var{n}, @var{alpha}) +## Create a Moler matrix (symmetric positive definite). +## +## @end deftypefn +## +## @deftypefn {Function File} {[@var{a}, @var{t}] =} gallery ("neumann", @var{n}) +## Create a singular matrix from the discrete Neumann problem (sparse). +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{q} =} gallery ("orthog", @var{n}) +## @deftypefnx {Function File} {@var{q} =} gallery ("orthog", @var{n}, @var{k}) +## Create orthogonal and nearly orthogonal matrices. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("parter", @var{n}) +## Create a Parter matrix (a Toeplitz matrix with singular values near pi). +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{p} =} gallery ("pei", @var{n}) +## @deftypefnx {Function File} {@var{p} =} gallery ("pei", @var{n}, @var{alpha}) +## Create a Pei matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("poisson", @var{n}) +## Create a block tridiagonal matrix from Poisson's equation (sparse). +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("prolate", @var{n}) +## @deftypefnx {Function File} {@var{a} =} gallery ("prolate", @var{n}, @var{w}) +## Create a prolate matrix (symmetric, ill-conditioned Toeplitz matrix). +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{h} =} gallery ("randhess", @var{x}) +## Create a random, orthogonal upper Hessenberg matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("rando", @var{n}) +## @deftypefnx {Function File} {@var{a} =} gallery ("rando", @var{n}, @var{k}) +## Create a random matrix with elements -1, 0 or 1. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("randsvd", @var{n}) +## @deftypefnx {Function File} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa}) +## @deftypefnx {Function File} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa}, @var{mode}) +## @deftypefnx {Function File} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa}, @var{mode}, @var{kl}) +## @deftypefnx {Function File} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa}, @var{mode}, @var{kl}, @var{ku}) +## Create a random matrix with pre-assigned singular values. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("redheff", @var{n}) +## Create a zero and ones matrix of Redheffer associated with the Riemann +## hypothesis. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("riemann", @var{n}) +## Create a matrix associated with the Riemann hypothesis. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("ris", @var{n}) +## Create a symmetric Hankel matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("smoke", @var{n}) +## @deftypefnx {Function File} {@var{a} =} gallery ("smoke", @var{n}, @var{k}) +## Create a complex matrix, with a `smoke ring' pseudospectrum. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{t} =} gallery ("toeppd", @var{n}) +## @deftypefnx {Function File} {@var{t} =} gallery ("toeppd", @var{n}, @var{m}) +## @deftypefnx {Function File} {@var{t} =} gallery ("toeppd", @var{n}, @var{m}, @var{w}) +## @deftypefnx {Function File} {@var{t} =} gallery ("toeppd", @var{n}, @var{m}, @var{w}, @var{theta}) +## Create a symmetric positive definite Toeplitz matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{p} =} gallery ("toeppen", @var{n}) +## @deftypefnx {Function File} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}) +## @deftypefnx {Function File} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b}) +## @deftypefnx {Function File} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b}, @var{c}) +## @deftypefnx {Function File} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b}, @var{c}, @var{d}) +## @deftypefnx {Function File} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b}, @var{c}, @var{d}, @var{e}) +## Create a pentadiagonal Toeplitz matrix (sparse). +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("tridiag", @var{x}, @var{y}, @var{z}) +## @deftypefnx {Function File} {@var{a} =} gallery ("tridiag", @var{n}) +## @deftypefnx {Function File} {@var{a} =} gallery ("tridiag", @var{n}, @var{c}, @var{d}, @var{e}) +## Create a tridiagonal matrix (sparse). +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{t} =} gallery ("triw", @var{n}) +## @deftypefnx {Function File} {@var{t} =} gallery ("triw", @var{n}, @var{alpha}) +## @deftypefnx {Function File} {@var{t} =} gallery ("triw", @var{n}, @var{alpha}, @var{k}) +## Create an upper triangular matrix discussed by Kahan, Golub and Wilkinson. +## +## @end deftypefn +## +## @deftypefn {Function File} {@var{a} =} gallery ("wathen", @var{nx}, @var{ny}) +## @deftypefnx {Function File} {@var{a} =} gallery ("wathen", @var{nx}, @var{ny}, @var{k}) +## Create the Wathen matrix. +## +## @end deftypefn +## +## @deftypefn {Function File} {[@var{a}, @var{b}] =} gallery ("wilk", @var{n}) +## Create various specific matrices devised/discussed by Wilkinson. +## +## @end deftypefn + +## Code for most of the individual matrices (except binomial, gcdmat, +## integerdata, leslie, normaldata, randcolu, randcorr, randjorth, sampling, +## uniformdata) by Nicholas .J. Higham <Nicholas.J.Higham@manchester.ac.uk> +## Adapted for Octave and into single gallery function by Carnë Draug + +function [varargout] = gallery (name, varargin) + + if (nargin < 1) + print_usage (); + elseif (! ischar (name)) + error ("gallery: NAME must be a string."); + endif + + ## NOTE: there isn't a lot of input check in the individual functions + ## that actually build the functions. This is by design. The original + ## code by Higham did not perform it and was propagated to Matlab, so + ## for compatibility, we also don't make it. For example, arguments + ## that behave as switches, and in theory accepting a value of 0 or 1, + ## will use a value of 0, for any value other than 1 (only check made + ## is if the value is equal to 1). It will often also accept string + ## values instead of numeric. Only input check added was where it + ## would be causing an error anyway. + + ## we will always want to return at least 1 output + n_out = nargout; + if (n_out == 0) + n_out = 1; + endif + + switch (tolower (name)) + case "binomial" + error ("gallery: matrix %s not implemented.", name); + case "cauchy" , [varargout{1:n_out}] = cauchy (varargin{:}); + case "chebspec" , [varargout{1:n_out}] = chebspec (varargin{:}); + case "chebvand" , [varargout{1:n_out}] = chebvand (varargin{:}); + case "chow" , [varargout{1:n_out}] = chow (varargin{:}); + case "circul" , [varargout{1:n_out}] = circul (varargin{:}); + case "clement" , [varargout{1:n_out}] = clement (varargin{:}); + case "compar" , [varargout{1:n_out}] = compar (varargin{:}); + case "condex" , [varargout{1:n_out}] = condex (varargin{:}); + case "cycol" , [varargout{1:n_out}] = cycol (varargin{:}); + case "dorr" , [varargout{1:n_out}] = dorr (varargin{:}); + case "dramadah" , [varargout{1:n_out}] = dramadah (varargin{:}); + case "fiedler" , [varargout{1:n_out}] = fiedler (varargin{:}); + case "forsythe" , [varargout{1:n_out}] = forsythe (varargin{:}); + case "frank" , [varargout{1:n_out}] = frank (varargin{:}); + case "gearmat" , [varargout{1:n_out}] = gearmat (varargin{:}); + case "gcdmat" , [varargout{1:n_out}] = gcdmat (varargin{:}); + case "grcar" , [varargout{1:n_out}] = grcar (varargin{:}); + case "hanowa" , [varargout{1:n_out}] = hanowa (varargin{:}); + case "house" , [varargout{1:n_out}] = house (varargin{:}); + case "integerdata" + error ("gallery: matrix %s not implemented.", name); + case "invhess" , [varargout{1:n_out}] = invhess (varargin{:}); + case "invol" , [varargout{1:n_out}] = invol (varargin{:}); + case "ipjfact" , [varargout{1:n_out}] = ipjfact (varargin{:}); + case "jordbloc" , [varargout{1:n_out}] = jordbloc (varargin{:}); + case "kahan" , [varargout{1:n_out}] = kahan (varargin{:}); + case "kms" , [varargout{1:n_out}] = kms (varargin{:}); + case "krylov" , [varargout{1:n_out}] = krylov (varargin{:}); + case "lauchli" , [varargout{1:n_out}] = lauchli (varargin{:}); + case "lehmer" , [varargout{1:n_out}] = lehmer (varargin{:}); + case "leslie" + error ("gallery: matrix %s not implemented.", name); + case "lesp" , [varargout{1:n_out}] = lesp (varargin{:}); + case "lotkin" , [varargout{1:n_out}] = lotkin (varargin{:}); + case "minij" , [varargout{1:n_out}] = minij (varargin{:}); + case "moler" , [varargout{1:n_out}] = moler (varargin{:}); + case "neumann" , [varargout{1:n_out}] = neumann (varargin{:}); + case "normaldata" + error ("gallery: matrix %s not implemented.", name); + case "orthog" , [varargout{1:n_out}] = orthog (varargin{:}); + case "parter" , [varargout{1:n_out}] = parter (varargin{:}); + case "pei" , [varargout{1:n_out}] = pei (varargin{:}); + case "poisson" , [varargout{1:n_out}] = poisson (varargin{:}); + case "prolate" , [varargout{1:n_out}] = prolate (varargin{:}); + case "randcolu" + error ("gallery: matrix %s not implemented.", name); + case "randcorr" + error ("gallery: matrix %s not implemented.", name); + case "randhess" , [varargout{1:n_out}] = randhess (varargin{:}); + case "randjorth" + error ("gallery: matrix %s not implemented.", name); + case "rando" , [varargout{1:n_out}] = rando (varargin{:}); + case "randsvd" , [varargout{1:n_out}] = randsvd (varargin{:}); + case "redheff" , [varargout{1:n_out}] = redheff (varargin{:}); + case "riemann" , [varargout{1:n_out}] = riemann (varargin{:}); + case "ris" , [varargout{1:n_out}] = ris (varargin{:}); + case "sampling" + error ("gallery: matrix %s not implemented.", name); + case "smoke" , [varargout{1:n_out}] = smoke (varargin{:}); + case "toeppd" , [varargout{1:n_out}] = toeppd (varargin{:}); + case "toeppen" , [varargout{1:n_out}] = toeppen (varargin{:}); + case "tridiag" , [varargout{1:n_out}] = tridiag (varargin{:}); + case "triw" , [varargout{1:n_out}] = triw (varargin{:}); + case "uniformdata" + error ("gallery: matrix %s not implemented.", name); + case "wathen" , [varargout{1:n_out}] = wathen (varargin{:}); + case "wilk" , [varargout{1:n_out}] = wilk (varargin{:}); + otherwise + error ("gallery: unknown matrix with NAME %s", name); + endswitch + +endfunction + +function C = cauchy (x, y) + ##CAUCHY Cauchy matrix. + ## C = CAUCHY(X, Y), where X, Y are N-vectors, is the N-by-N matrix + ## with C(i,j) = 1/(X(i)+Y(j)). By default, Y = X. + ## Special case: if X is a scalar CAUCHY(X) is the same as CAUCHY(1:X). + ## Explicit formulas are known for DET(C) (which is nonzero if X and Y + ## both have distinct elements) and the elements of INV(C). + ## C is totally positive if 0 < X(1) < ... < X(N) and + ## 0 < Y(1) < ... < Y(N). + ## + ## References: + ## N.J. Higham, Accuracy and Stability of Numerical Algorithms, + ## Society for Industrial and Applied Mathematics, Philadelphia, PA, + ## USA, 1996; sec. 26.1. + ## D.E. Knuth, The Art of Computer Programming, Volume 1, + ## Fundamental Algorithms, second edition, Addison-Wesley, Reading, + ## Massachusetts, 1973, p. 36. + ## E.E. Tyrtyshnikov, Cauchy-Toeplitz matrices and some applications, + ## Linear Algebra and Appl., 149 (1991), pp. 1-18. + ## O. Taussky and M. Marcus, Eigenvalues of finite matrices, in + ## Survey of Numerical Analysis, J. Todd, ed., McGraw-Hill, New York, + ## pp. 279-313, 1962. (States the totally positive property on p. 295.) + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 or 2 arguments are required for cauchy matrix."); + elseif (! isnumeric (x)) + error ("gallery: X must be numeric for cauchy matrix."); + elseif (nargin == 2 && ! isnumeric (y)) + error ("gallery: Y must be numeric for cauchy matrix."); + endif + + n = numel (x); + if (isscalar (x) && fix (x) == x) + n = x; + x = 1:n; + elseif (n > 1 && isvector (x)) + ## do nothing + else + error ("gallery: X be an integer or a vector for cauchy matrix."); + endif + + if (nargin == 1) + y = x; + endif + + ## Ensure x and y are column vectors + x = x(:); + y = y(:); + if (numel (x) != numel (y)) + error ("gallery: X and Y must be vectors of same length for cauchy matrix."); + endif + + C = x * ones (1, n) + ones (n, 1) * y.'; + C = ones (n) ./ C; +endfunction + +function C = chebspec (n, k = 0) + ## CHEBSPEC Chebyshev spectral differentiation matrix. + ## C = CHEBSPEC(N, K) is a Chebyshev spectral differentiation + ## matrix of order N. K = 0 (the default) or 1. + ## For K = 0 (`no boundary conditions'), C is nilpotent, with + ## C^N = 0 and it has the null vector ONES(N,1). + ## C is similar to a Jordan block of size N with eigenvalue zero. + ## For K = 1, C is nonsingular and well-conditioned, and its eigenvalues + ## have negative real parts. + ## For both K, the computed eigenvector matrix X from EIG is + ## ill-conditioned (MESH(REAL(X)) is interesting). + ## + ## References: + ## C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral + ## Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988; p. 69. + ## L.N. Trefethen and M.R. Trummer, An instability phenomenon in + ## spectral methods, SIAM J. Numer. Anal., 24 (1987), pp. 1008-1023. + ## D. Funaro, Computing the inverse of the Chebyshev collocation + ## derivative, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 1050-1057. + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 to 2 arguments are required for chebspec matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for chebspec matrix."); + elseif (! isnumeric (k) || ! isscalar (k)) + error ("gallery: K must be a scalar for chebspec matrix."); + endif + + ## k = 1 case obtained from k = 0 case with one bigger n. + switch (k) + case (0), # do nothing + case (1), n = n + 1; + otherwise + error ("gallery: unknown K `%d' for chebspec matrix.", k); + endswitch + + n = n-1; + C = zeros (n+1); + + one = ones (n+1, 1); + x = cos ((0:n)' * (pi/n)); + d = ones (n+1, 1); + d(1) = 2; + d(n+1) = 2; + + ## eye(size(C)) on next line avoids div by zero. + C = (d * (one./d)') ./ (x*one'-one*x' + eye (size (C))); + + ## Now fix diagonal and signs. + C(1,1) = (2*n^2+1)/6; + for i = 2:n+1 + if (rem (i, 2) == 0) + C(:,i) = -C(:,i); + C(i,:) = -C(i,:); + endif + if (i < n+1) + C(i,i) = -x(i)/(2*(1-x(i)^2)); + else + C(n+1,n+1) = -C(1,1); + endif + endfor + + if (k == 1) + C = C(2:n+1,2:n+1); + endif +endfunction + +function C = chebvand (m, p) + ## CHEBVAND Vandermonde-like matrix for the Chebyshev polynomials. + ## C = CHEBVAND(P), where P is a vector, produces the (primal) + ## Chebyshev Vandermonde matrix based on the points P, + ## i.e., C(i,j) = T_{i-1}(P(j)), where T_{i-1} is the Chebyshev + ## polynomial of degree i-1. + ## CHEBVAND(M,P) is a rectangular version of CHEBVAND(P) with M rows. + ## Special case: If P is a scalar then P equally spaced points on + ## [0,1] are used. + ## + ## Reference: + ## N.J. Higham, Stability analysis of algorithms for solving confluent + ## Vandermonde-like systems, SIAM J. Matrix Anal. Appl., 11 (1990), + ## pp. 23-41. + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 or 2 arguments are required for chebvand matrix."); + endif + + ## because the order of the arguments changes if nargin is 1 or 2 ... + + if (nargin == 1) + p = m; + endif + + n = numel (p); + if (! isnumeric (p)) + error ("gallery: P must be numeric for chebvand matrix."); + elseif (isscalar (p) && fix (p) == p) + n = p; + p = linspace (0, 1, n); + elseif (n > 1 && isvector (p)) + ## do nothing + endif + p = p(:).'; # Ensure p is a row vector. + + if (nargin == 1) + m = n; + elseif (! isnumeric (m) || ! isscalar (m)) + error ("gallery: M must be a scalar for chebvand matrix."); + endif + + C = ones (m, n); + if (m != 1) + C(2,:) = p; + ## Use Chebyshev polynomial recurrence. + for i = 3:m + C(i,:) = 2.*p.*C(i-1,:) - C(i-2,:); + endfor + endif +endfunction + +function A = chow (n, alpha = 1, delta = 0) + ## CHOW Chow matrix - a singular Toeplitz lower Hessenberg matrix. + ## A = CHOW(N, ALPHA, DELTA) is a Toeplitz lower Hessenberg matrix + ## A = H(ALPHA) + DELTA*EYE, where H(i,j) = ALPHA^(i-j+1). + ## H(ALPHA) has p = FLOOR(N/2) zero eigenvalues, the rest being + ## 4*ALPHA*COS( k*PI/(N+2) )^2, k=1:N-p. + ## Defaults: ALPHA = 1, DELTA = 0. + ## + ## References: + ## T.S. Chow, A class of Hessenberg matrices with known + ## eigenvalues and inverses, SIAM Review, 11 (1969), pp. 391-395. + ## G. Fairweather, On the eigenvalues and eigenvectors of a class of + ## Hessenberg matrices, SIAM Review, 13 (1971), pp. 220-221. + + if (nargin < 1 || nargin > 3) + error ("gallery: 1 to 3 arguments are required for chow matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for chow matrix."); + elseif (! isnumeric (alpha) || ! isscalar (alpha)) + error ("gallery: ALPHA must be a scalar for chow matrix."); + elseif (! isnumeric (delta) || ! isscalar (delta)) + error ("gallery: DELTA must be a scalar for chow matrix."); + endif + + A = toeplitz (alpha.^(1:n), [alpha 1 zeros(1, n-2)]) + delta * eye (n); +endfunction + +function C = circul (v) + ## CIRCUL Circulant matrix. + ## C = CIRCUL(V) is the circulant matrix whose first row is V. + ## (A circulant matrix has the property that each row is obtained + ## from the previous one by cyclically permuting the entries one step + ## forward; it is a special Toeplitz matrix in which the diagonals + ## `wrap round'.) + ## Special case: if V is a scalar then C = CIRCUL(1:V). + ## The eigensystem of C (N-by-N) is known explicitly. If t is an Nth + ## root of unity, then the inner product of V with W = [1 t t^2 ... t^N] + ## is an eigenvalue of C, and W(N:-1:1) is an eigenvector of C. + ## + ## Reference: + ## P.J. Davis, Circulant Matrices, John Wiley, 1977. + + if (nargin != 1) + error ("gallery: 1 argument is required for circul matrix."); + elseif (! isnumeric (v)) + error ("gallery: V must be numeric for circul matrix."); + endif + + n = numel (x); + if (isscalar (x) && fix (x) == x) + n = v; + v = 1:n; + elseif (n > 1 && isvector (x)) + ## do nothing + else + error ("gallery: X must be a scalar or a vector for circul matrix."); + endif + + v = v(:).'; # Make sure v is a row vector + C = toeplitz ([v(1) v(n:-1:2)], v); +endfunction + +function A = clement (n, k = 0) + ## CLEMENT Clement matrix - tridiagonal with zero diagonal entries. + ## CLEMENT(N, K) is a tridiagonal matrix with zero diagonal entries + ## and known eigenvalues. It is singular if N is odd. About 64 + ## percent of the entries of the inverse are zero. The eigenvalues + ## are plus and minus the numbers N-1, N-3, N-5, ..., (1 or 0). + ## For K = 0 (the default) the matrix is unsymmetric, while for + ## K = 1 it is symmetric. + ## CLEMENT(N, 1) is diagonally similar to CLEMENT(N). + ## + ## Similar properties hold for TRIDIAG(X,Y,Z) where Y = ZEROS(N,1). + ## The eigenvalues still come in plus/minus pairs but they are not + ## known explicitly. + ## + ## References: + ## P.A. Clement, A class of triple-diagonal matrices for test + ## purposes, SIAM Review, 1 (1959), pp. 50-52. + ## A. Edelman and E. Kostlan, The road from Kac's matrix to Kac's + ## random polynomials. In John~G. Lewis, editor, Proceedings of + ## the Fifth SIAM Conference on Applied Linear Algebra Society + ## for Industrial and Applied Mathematics, Philadelphia, 1994, + ## pp. 503-507. + ## O. Taussky and J. Todd, Another look at a matrix of Mark Kac, + ## Linear Algebra and Appl., 150 (1991), pp. 341-360. + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 or 2 arguments are required for clement matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for clement matrix."); + elseif (! isnumeric (k) || ! isscalar (k)) + error ("gallery: K must be a numeric scalar for clement matrix."); + endif + + n = n-1; + x = n:-1:1; + z = 1:n; + + if (k == 0) + A = diag (x, -1) + diag (z, 1); + elseif (k == 1) + y = sqrt (x.*z); + A = diag (y, -1) + diag (y, 1); + else + error ("gallery: K must have a value of 0 or 1 for clement matrix."); + endif +endfunction + +function C = compar (A, k = 0) + ## COMP Comparison matrices. + ## COMP(A) is DIAG(B) - TRIL(B,-1) - TRIU(B,1), where B = ABS(A). + ## COMP(A, 1) is A with each diagonal element replaced by its + ## absolute value, and each off-diagonal element replaced by minus + ## the absolute value of the largest element in absolute value in + ## its row. However, if A is triangular COMP(A, 1) is too. + ## COMP(A, 0) is the same as COMP(A). + ## COMP(A) is often denoted by M(A) in the literature. + ## + ## Reference (e.g.): + ## N.J. Higham, A survey of condition number estimation for + ## triangular matrices, SIAM Review, 29 (1987), pp. 575-596. + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 or 2 arguments are required for compar matrix."); + elseif (! isnumeric (A) || ndims (A) != 2) + error ("gallery: A must be a 2D matrix for compar matrix."); + elseif (! isnumeric (k) || ! isscalar (k)) + error ("gallery: K must be a numeric scalar for compar matrix."); + endif + + [m, n] = size (A); + p = min (m, n); + + if (k == 0) + ## This code uses less temporary storage than + ## the `high level' definition above. + C = -abs (A); + for j = 1:p + C(j,j) = abs (A(j,j)); + endfor + + elseif (k == 1) + C = A'; + for j = 1:p + C(k,k) = 0; + endfor + mx = max (abs (C)); + C = -mx'*ones (1, n); + for j = 1:p + C(j,j) = abs (A(j,j)); + endfor + if (all (A == tril (A))), C = tril (C); endif + if (all (A == triu (A))), C = triu (C); endif + + else + error ("gallery: K must have a value of 0 or 1 for compar matrix."); + endif + +endfunction + +function A = condex (n, k = 4, theta = 100) + ## CONDEX `Counterexamples' to matrix condition number estimators. + ## CONDEX(N, K, THETA) is a `counterexample' matrix to a condition + ## estimator. It has order N and scalar parameter THETA (default 100). + ## If N is not equal to the `natural' size of the matrix then + ## the matrix is padded out with an identity matrix to order N. + ## The matrix, its natural size, and the estimator to which it applies + ## are specified by K (default K = 4) as follows: + ## K = 1: 4-by-4, LINPACK (RCOND) + ## K = 2: 3-by-3, LINPACK (RCOND) + ## K = 3: arbitrary, LINPACK (RCOND) (independent of THETA) + ## K = 4: N >= 4, SONEST (Higham 1988) + ## (Note that in practice the K = 4 matrix is not usually a + ## counterexample because of the rounding errors in forming it.) + ## + ## References: + ## A.K. Cline and R.K. Rew, A set of counter-examples to three + ## condition number estimators, SIAM J. Sci. Stat. Comput., + ## 4 (1983), pp. 602-611. + ## N.J. Higham, FORTRAN codes for estimating the one-norm of a real or + ## complex matrix, with applications to condition estimation + ## (Algorithm 674), ACM Trans. Math. Soft., 14 (1988), pp. 381-396. + + if (nargin < 1 || nargin > 3) + error ("gallery: 1 to 3 arguments are required for condex matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for condex matrix."); + elseif (! isnumeric (k) || ! isscalar (k)) + error ("gallery: K must be a numeric scalar for condex matrix."); + elseif (! isnumeric (theta) || ! isscalar (theta)) + error ("gallery: THETA must be a numeric scalar for condex matrix."); + endif + + if (k == 1) # Cline and Rew (1983), Example B. + A = [1 -1 -2*theta 0 + 0 1 theta -theta + 0 1 1+theta -(theta+1) + 0 0 0 theta]; + + elseif (k == 2) # Cline and Rew (1983), Example C. + A = [1 1-2/theta^2 -2 + 0 1/theta -1/theta + 0 0 1]; + + elseif (k == 3) # Cline and Rew (1983), Example D. + A = gallery ("triw", n, -1)'; + A(n,n) = -1; + + elseif (k == 4) # Higham (1988), p. 390. + x = ones (n, 3); # First col is e + x(2:n,2) = zeros (n-1, 1); # Second col is e(1) + + ## Third col is special vector b in SONEST + x(:, 3) = (-1).^[0:n-1]' .* ( 1 + [0:n-1]'/(n-1) ); + + Q = orth (x); # Q*Q' is now the orthogonal projector onto span(e(1),e,b)). + P = eye (n) - Q*Q'; + A = eye (n) + theta*P; + + else + error ("gallery: unknown estimator K `%d' for condex matrix.", k); + endif + + ## Pad out with identity as necessary. + m = columns (A); + if (m < n) + for i = n:-1:m+1 + A(i,i) = 1; + endfor + endif +endfunction + +function A = cycol (n, k) + ## CYCOL Matrix whose columns repeat cyclically. + ## A = CYCOL([M N], K) is an M-by-N matrix of the form A = B(1:M,1:N) + ## where B = [C C C...] and C = RANDN(M, K). Thus A's columns repeat + ## cyclically, and A has rank at most K. K need not divide N. + ## K defaults to ROUND(N/4). + ## CYCOL(N, K), where N is a scalar, is the same as CYCOL([N N], K). + ## + ## This type of matrix can lead to underflow problems for Gaussian + ## elimination: see NA Digest Volume 89, Issue 3 (January 22, 1989). + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 or 2 arguments are required for cycol matrix."); + elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n) + error ("gallery: N must be a 1 or 2 element integer for cycol matrix."); + elseif (! isnumeric (k) || ! isscalar (k)) + error ("gallery: K must be a scalar for cycol matrix."); + endif + + ## Parameter n specifies dimension: m-by-n + m = n(1); + n = n(end); + + if (nargin < 2) + k = max (round (n/4), 1); + endif + + A = randn (m, k); + for i = 2:ceil (n/k) + A = [A A(:,1:k)]; + endfor + A = A(:,1:n); +endfunction + +function [c, d, e] = dorr (n, theta = 0.01) + ## DORR Dorr matrix - diagonally dominant, ill conditioned, tridiagonal. + ## [C, D, E] = DORR(N, THETA) returns the vectors defining a row diagonally + ## dominant, tridiagonal M-matrix that is ill conditioned for small + ## values of the parameter THETA >= 0. + ## If only one output parameter is supplied then + ## C = FULL(TRIDIAG(C,D,E)), i.e., the matrix iself is returned. + ## The columns of INV(C) vary greatly in norm. THETA defaults to 0.01. + ## The amount of diagonal dominance is given by (ignoring rounding errors): + ## COMP(C)*ONES(N,1) = THETA*(N+1)^2 * [1 0 0 ... 0 1]'. + ## + ## Reference: + ## F.W. Dorr, An example of ill-conditioning in the numerical + ## solution of singular perturbation problems, Math. Comp., 25 (1971), + ## pp. 271-283. + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 or 2 arguments are required for dorr matrix."); + elseif (! isscalar (n) || ! isnumeric (n) || fix (n) != n) + error ("gallery: N must be an integer for dorr matrix."); + elseif (! isscalar (theta) || ! isnumeric (theta)) + error ("gallery: THETA must be a numeric scalar for dorr matrix."); + endif + + c = zeros (n, 1); + e = c; + d = c; + ## All length n for convenience. Make c, e of length n-1 later. + + h = 1/(n+1); + m = floor ((n+1)/2); + term = theta/h^2; + + i = (1:m)'; + c(i) = -term * ones (m, 1); + e(i) = c(i) - (0.5-i*h)/h; + d(i) = -(c(i) + e(i)); + + i = (m+1:n)'; + e(i) = -term * ones (n-m, 1); + c(i) = e(i) + (0.5-i*h)/h; + d(i) = -(c(i) + e(i)); + + c = c(2:n); + e = e(1:n-1); + + if (nargout <= 1) + c = tridiag (c, d, e); + endif +endfunction + +function A = dramadah (n, k = 1) + ## DRAMADAH A (0,1) matrix whose inverse has large integer entries. + ## An anti-Hadamard matrix A is a matrix with elements 0 or 1 for + ## which MU(A) := NORM(INV(A),'FRO') is maximal. + ## A = DRAMADAH(N, K) is an N-by-N (0,1) matrix for which MU(A) is + ## relatively large, although not necessarily maximal. + ## Available types (the default is K = 1): + ## K = 1: A is Toeplitz, with ABS(DET(A)) = 1, and MU(A) > c(1.75)^N, + ## where c is a constant. + ## K = 2: A is upper triangular and Toeplitz. + ## The inverses of both types have integer entries. + ## + ## Another interesting (0,1) matrix: + ## K = 3: A has maximal determinant among (0,1) lower Hessenberg + ## matrices: det(A) = the n'th Fibonacci number. A is Toeplitz. + ## The eigenvalues have an interesting distribution in the complex + ## plane. + ## + ## References: + ## R.L. Graham and N.J.A. Sloane, Anti-Hadamard matrices, + ## Linear Algebra and Appl., 62 (1984), pp. 113-137. + ## L. Ching, The maximum determinant of an nxn lower Hessenberg + ## (0,1) matrix, Linear Algebra and Appl., 183 (1993), pp. 147-153. + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 to 2 arguments are required for dramadah matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for dramadah matrix."); + elseif (! isnumeric (k) || ! isscalar (k)) + error ("gallery: K must be a numeric scalar for dramadah matrix."); + endif + + switch (k) + case (1) # Toeplitz + c = ones (n, 1); + for i = 2:4:n + m = min (1, n-i); + c(i:i+m) = zeros (m+1, 1); + endfor + r = zeros (n, 1); + r(1:4) = [1 1 0 1]; + if (n < 4) + r = r(1:n); + endif + A = toeplitz (c, r); + + case (2) # Upper triangular and Toeplitz + c = zeros (n, 1); + c(1) = 1; + r = ones (n, 1); + for i= 3:2:n + r(i) = 0; + endfor + A = toeplitz (c, r); + + case (3) # Lower Hessenberg + c = ones (n, 1); + for i= 2:2:n + c(i) = 0; + endfor + A = toeplitz (c, [1 1 zeros(1,n-2)]); + + otherwise + error ("gallery: unknown K `%d' for dramadah matrix.", k); + endswitch +endfunction + +function A = fiedler (c) + ## FIEDLER Fiedler matrix - symmetric. + ## A = FIEDLER(C), where C is an n-vector, is the n-by-n symmetric + ## matrix with elements ABS(C(i)-C(j)). + ## Special case: if C is a scalar, then A = FIEDLER(1:C) + ## (i.e. A(i,j) = ABS(i-j)). + ## Properties: + ## FIEDLER(N) has a dominant positive eigenvalue and all the other + ## eigenvalues are negative (Szego, 1936). + ## Explicit formulas for INV(A) and DET(A) are given by Todd (1977) + ## and attributed to Fiedler. These indicate that INV(A) is + ## tridiagonal except for nonzero (1,n) and (n,1) elements. + ## [I think these formulas are valid only if the elements of + ## C are in increasing or decreasing order---NJH.] + ## + ## References: + ## G. Szego, Solution to problem 3705, Amer. Math. Monthly, + ## 43 (1936), pp. 246-259. + ## J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, + ## Birkhauser, Basel, and Academic Press, New York, 1977, p. 159. + + if (nargin != 1) + error ("gallery: 1 argument is required for fiedler matrix."); + elseif (! isnumeric (c)) + error ("gallery: C must be numeric for fiedler matrix."); + endif + + n = numel (c); + if (isscalar (c) && fix (c) == c) + n = c; + c = 1:n; + elseif (n > 1 && isvector (c)) + ## do nothing + else + error ("gallery: C must be an integer or a vector for fiedler matrix."); + endif + c = c(:).'; # Ensure c is a row vector. + + A = ones (n, 1) * c; + A = abs (A - A.'); # NB. array transpose. +endfunction + +function A = forsythe (n, alpha = sqrt (eps), lambda = 0) + ## FORSYTHE Forsythe matrix - a perturbed Jordan block. + ## FORSYTHE(N, ALPHA, LAMBDA) is the N-by-N matrix equal to + ## JORDBLOC(N, LAMBDA) except it has an ALPHA in the (N,1) position. + ## It has the characteristic polynomial + ## DET(A-t*EYE) = (LAMBDA-t)^N - (-1)^N ALPHA. + ## ALPHA defaults to SQRT(EPS) and LAMBDA to 0. + + if (nargin < 1 || nargin > 3) + error ("gallery: 1 to 3 arguments are required for forsythe matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for forsythe matrix."); + elseif (! isnumeric (alpha) || ! isscalar (alpha)) + error ("gallery: ALPHA must be a numeric scalar for forsythe matrix."); + elseif (! isnumeric (lambda) || ! isscalar (lambda)) + error ("gallery: LAMBDA must be a numeric scalar for forsythe matrix."); + endif + + A = jordbloc (n, lambda); + A(n,1) = alpha; +endfunction + +function F = frank (n, k = 0) + ## FRANK Frank matrix---ill conditioned eigenvalues. + ## F = FRANK(N, K) is the Frank matrix of order N. It is upper + ## Hessenberg with determinant 1. K = 0 is the default; if K = 1 the + ## elements are reflected about the anti-diagonal (1,N)--(N,1). + ## F has all positive eigenvalues and they occur in reciprocal pairs + ## (so that 1 is an eigenvalue if N is odd). + ## The eigenvalues of F may be obtained in terms of the zeros of the + ## Hermite polynomials. + ## The FLOOR(N/2) smallest eigenvalues of F are ill conditioned, + ## the more so for bigger N. + ## + ## DET(FRANK(N)') comes out far from 1 for large N---see Frank (1958) + ## and Wilkinson (1960) for discussions. + ## + ## This version incorporates improvements suggested by W. Kahan. + ## + ## References: + ## W.L. Frank, Computing eigenvalues of complex matrices by determinant + ## evaluation and by methods of Danilewski and Wielandt, J. Soc. + ## Indust. Appl. Math., 6 (1958), pp. 378-392 (see pp. 385, 388). + ## G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the + ## computation of the Jordan canonical form, SIAM Review, 18 (1976), + ## pp. 578-619 (Section 13). + ## H. Rutishauser, On test matrices, Programmation en Mathematiques + ## Numeriques, Editions Centre Nat. Recherche Sci., Paris, 165, + ## 1966, pp. 349-365. Section 9. + ## J.H. Wilkinson, Error analysis of floating-point computation, + ## Numer. Math., 2 (1960), pp. 319-340 (Section 8). + ## J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University + ## Press, 1965 (pp. 92-93). + ## The next two references give details of the eigensystem, as does + ## Rutishauser (see above). + ## P.J. Eberlein, A note on the matrices denoted by B_n, SIAM J. Appl. + ## Math., 20 (1971), pp. 87-92. + ## J.M. Varah, A generalization of the Frank matrix, SIAM J. Sci. Stat. + ## Comput., 7 (1986), pp. 835-839. + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 to 2 arguments are required for frank matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for frank matrix."); + elseif (! isnumeric (k) || ! isscalar (k)) + error ("gallery: K must be a numeric scalar for frank matrix."); + endif + + p = n:-1:1; + F = triu (p(ones (n, 1), :) - diag (ones (n-1, 1), -1), -1); + + switch (k) + case (0), # do nothing + case (1), F = F(p,p)'; + otherwise + error ("gallery: K must have a value of 0 or 1 for frank matrix."); + endswitch +endfunction + +function c = gcdmat (n) + if (nargin != 1) + error ("gallery: 1 argument is required for gcdmat matrix."); + elseif (! isscalar (n) || ! isnumeric (n) || fix (n) != n) + error ("gallery: N must be an integer for gcdmat matrix."); + endif + c = gcd (repmat ((1:n)', [1 n]), repmat (1:n, [n 1])); +endfunction + +function A = gearmat (n, i = n, j = -n) + ## NOTE: this function was named gearm in the original Test Matrix Toolbox + ## GEARMAT Gear matrix. + ## A = GEARMAT(N,I,J) is the N-by-N matrix with ones on the sub- and + ## super-diagonals, SIGN(I) in the (1,ABS(I)) position, SIGN(J) + ## in the (N,N+1-ABS(J)) position, and zeros everywhere else. + ## Defaults: I = N, j = -N. + ## All eigenvalues are of the form 2*COS(a) and the eigenvectors + ## are of the form [SIN(w+a), SIN(w+2a), ..., SIN(w+Na)]. + ## The values of a and w are given in the reference below. + ## A can have double and triple eigenvalues and can be defective. + ## GEARMAT(N) is singular. + ## + ## (GEAR is a Simulink function, hence GEARMAT for Gear matrix.) + ## Reference: + ## C.W. Gear, A simple set of test matrices for eigenvalue programs, + ## Math. Comp., 23 (1969), pp. 119-125. + + if (nargin < 1 || nargin > 3) + error ("gallery: 1 to 3 arguments are required for gearmat matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for gearmat matrix."); + elseif (! isnumeric (i) || ! isscalar (i) || i == 0 || abs (i) <= n) + error ("gallery: I must be a non-zero scalar, and abs (I) <= N for gearmat matrix."); + elseif (! isnumeric (j) || ! isscalar (j) || i == 0 || abs (j) <= n) + error ("gallery: J must be a non-zero scalar, and abs (J) <= N for gearmat matrix."); + endif + + A = diag (ones (n-1, 1), -1) + diag (ones (n-1, 1), 1); + A(1, abs (i)) = sign (i); + A(n, n+1 - abs (j)) = sign (j); +endfunction + +function G = grcar (n, k = 3) + ## GRCAR Grcar matrix - a Toeplitz matrix with sensitive eigenvalues. + ## GRCAR(N, K) is an N-by-N matrix with -1s on the + ## subdiagonal, 1s on the diagonal, and K superdiagonals of 1s. + ## The default is K = 3. The eigenvalues of this matrix form an + ## interesting pattern in the complex plane (try PS(GRCAR(32))). + ## + ## References: + ## J.F. Grcar, Operator coefficient methods for linear equations, + ## Report SAND89-8691, Sandia National Laboratories, Albuquerque, + ## New Mexico, 1989 (Appendix 2). + ## N.M. Nachtigal, L. Reichel and L.N. Trefethen, A hybrid GMRES + ## algorithm for nonsymmetric linear systems, SIAM J. Matrix Anal. + ## Appl., 13 (1992), pp. 796-825. + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 to 2 arguments are required for grcar matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for grcar matrix."); + elseif (! isnumeric (k) || ! isscalar (k)) + error ("gallery: K must be a numeric scalar for grcar matrix."); + endif + + G = tril (triu (ones (n)), k) - diag (ones (n-1, 1), -1); +endfunction + +function A = hanowa (n, d = -1) + ## HANOWA A matrix whose eigenvalues lie on a vertical line in the complex plane. + ## HANOWA(N, d) is the N-by-N block 2x2 matrix (thus N = 2M must be even) + ## [d*EYE(M) -DIAG(1:M) + ## DIAG(1:M) d*EYE(M)] + ## It has complex eigenvalues lambda(k) = d +/- k*i (1 <= k <= M). + ## Parameter d defaults to -1. + ## + ## Reference: + ## E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary + ## Differential Equations I: Nonstiff Problems, Springer-Verlag, + ## Berlin, 1987. (pp. 86-87) + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 to 2 arguments are required for hanowa matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for hanowa matrix."); + elseif (rem (n, 2) != 0) + error ("gallery: N must be even for hanowa matrix."); + elseif (! isnumeric (lambda) || ! isscalar (lambda)) + error ("gallery: D must be a numeric scalar for hanowa matrix."); + endif + + m = n/2; + A = [ d*eye(m) -diag(1:m) + diag(1:m) d*eye(m) ]; +endfunction + +function [v, beta] = house (x) + ## HOUSE Householder matrix. + ## If [v, beta] = HOUSE(x) then H = EYE - beta*v*v' is a Householder + ## matrix such that Hx = -sign(x(1))*norm(x)*e_1. + ## NB: If x = 0 then v = 0, beta = 1 is returned. + ## x can be real or complex. + ## sign(x) := exp(i*arg(x)) ( = x./abs(x) when x ~= 0). + ## + ## Theory: (textbook references Golub & Van Loan 1989, 38-43; + ## Stewart 1973, 231-234, 262; Wilkinson 1965, 48-50). + ## Hx = y: (I - beta*v*v')x = -s*e_1. + ## Must have |s| = norm(x), v = x+s*e_1, and + ## x'y = x'Hx =(x'Hx)' real => arg(s) = arg(x(1)). + ## So take s = sign(x(1))*norm(x) (which avoids cancellation). + ## v'v = (x(1)+s)^2 + x(2)^2 + ... + x(n)^2 + ## = 2*norm(x)*(norm(x) + |x(1)|). + ## + ## References: + ## G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, + ## Johns Hopkins University Press, Baltimore, Maryland, 1989. + ## G.W. Stewart, Introduction to Matrix Computations, Academic Press, + ## New York, 1973, + ## J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University + ## Press, 1965. + + if (nargin != 1) + error ("gallery: 1 argument is required for house matrix."); + elseif (! isnumeric (x) || ! isvector (x) || numel (x) <= 1) + error ("gallery: X must be a vector for house matrix."); + endif + + ## must be a column vector + x = x(:); + + s = norm (x) * (sign (x(1)) + (x(1) == 0)); # Modification for sign (0) == 1. + v = x; + if (s == 0) + ## Quit if x is the zero vector. + beta = 1; + else + v(1) = v(1) + s; + beta = 1/(s'*v(1)); # NB the conjugated s. + ## beta = 1/(abs (s) * (abs (s) +abs(x(1)) would guarantee beta real. + ## But beta as above can be non-real (due to rounding) only when x is complex. + endif +endfunction + +function A = invhess (x, y) + ## INVHESS Inverse of an upper Hessenberg matrix. + ## INVHESS(X, Y), where X is an N-vector and Y an N-1 vector, + ## is the matrix whose lower triangle agrees with that of + ## ONES(N,1)*X' and whose strict upper triangle agrees with + ## that of [1 Y]*ONES(1,N). + ## The matrix is nonsingular if X(1) ~= 0 and X(i+1) ~= Y(i) + ## for all i, and its inverse is an upper Hessenberg matrix. + ## If Y is omitted it defaults to -X(1:N-1). + ## Special case: if X is a scalar INVHESS(X) is the same as + ## INVHESS(1:X). + ## + ## References: + ## F.N. Valvi and V.S. Geroyannis, Analytic inverses and + ## determinants for a class of matrices, IMA Journal of Numerical + ## Analysis, 7 (1987), pp. 123-128. + ## W.-L. Cao and W.J. Stewart, A note on inverses of Hessenberg-like + ## matrices, Linear Algebra and Appl., 76 (1986), pp. 233-240. + ## Y. Ikebe, On inverses of Hessenberg matrices, Linear Algebra and + ## Appl., 24 (1979), pp. 93-97. + ## P. Rozsa, On the inverse of band matrices, Integral Equations and + ## Operator Theory, 10 (1987), pp. 82-95. + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 to 2 arguments are required for invhess matrix."); + elseif (! isnumeric (x)) + error ("gallery: X must be numeric for invhess matrix."); + endif + + if (isscalar (x) && fix (x) == x) + n = x; + x = 1:n; + elseif (! isscalar (x) && isvector (x)) + n = numel (n); + else + error ("gallery: X must be an integer scalar, or a vector for invhess matrix."); + endif + + if (nargin < 2) + y = -x(1:end-1); + elseif (! isvector (y) || numel (y) != numel (x) -1) + error ("gallery: Y must be a vector of length -1 than X for invhess matrix."); + endif + + x = x(:); + y = y(:); + + ## On next line, z = x'; A = z(ones(n,1),:) would be more efficient. + A = ones (n, 1) * x'; + for j = 2:n + A(1:j-1,j) = y(1:j-1); + endfor +endfunction + +function A = invol (n) + ## INVOL An involutory matrix. + ## A = INVOL(N) is an N-by-N involutory (A*A = EYE(N)) and + ## ill-conditioned matrix. + ## It is a diagonally scaled version of HILB(N). + ## NB: B = (EYE(N)-A)/2 and B = (EYE(N)+A)/2 are idempotent (B*B = B). + ## + ## Reference: + ## A.S. Householder and J.A. Carpenter, The singular values + ## of involutory and of idempotent matrices, Numer. Math. 5 (1963), + ## pp. 234-237. + + if (nargin != 1) + error ("gallery: 1 argument is required for invol matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for invol matrix."); + endif + + A = hilb (n); + + d = -n; + A(:, 1) = d * A(:, 1); + + for i = 1:n-1 + d = -(n+i)*(n-i)*d/(i*i); + A(i+1,:) = d * A(i+1,:); + endfor +endfunction + +function [A, detA] = ipjfact (n, k = 0) + ## IPJFACT A Hankel matrix with factorial elements. + ## A = IPJFACT(N, K) is the matrix with + ## A(i,j) = (i+j)! (K = 0, default) + ## A(i,j) = 1/(i+j)! (K = 1) + ## Both are Hankel matrices. + ## The determinant and inverse are known explicitly. + ## If a second output argument is present, d = DET(A) is returned: + ## [A, d] = IPJFACT(N, K); + ## + ## Suggested by P. R. Graves-Morris. + ## + ## Reference: + ## M.J.C. Gover, The explicit inverse of factorial Hankel matrices, + ## Dept. of Mathematics, University of Bradford, 1993. + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 to 2 arguments are required for ipjfact matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for ipjfact matrix."); + elseif (! isnumeric (k) || ! isscalar (k)) + error ("gallery: K must be a numeric scalar for ipjfact matrix."); + endif + + c = cumprod (2:n+1); + d = cumprod (n+1:2*n) * c(n-1); + + A = hankel (c, d); + + switch (k) + case (0), # do nothing + case (1), A = ones (n) ./ A; + otherwise + error ("gallery: K must have a value of 0 or 1 for ipjfact matrix."); + endswitch + + if (nargout == 2) + d = 1; + + if (k == 0) + for i = 1:n-1 + d = d * prod (1:i+1) * prod (1:n-i); + endfor + d = d * prod (1:n+1); + + elseif (k == 1) + for i = 0:n-1 + d = d * prod (1:i) / prod (1:n+1+i); + endfor + if (rem (n*(n-1)/2, 2)) + d = -d; + endif + + else + error ("gallery: K must have a value of 0 or 1 for ipjfact matrix."); + endif + + detA = d; + endif +endfunction + +function J = jordbloc (n, lambda = 1) + ## JORDBLOC Jordan block. + ## JORDBLOC(N, LAMBDA) is the N-by-N Jordan block with eigenvalue + ## LAMBDA. LAMBDA = 1 is the default. + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 to 2 arguments are required for jordbloc matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for jordbloc matrix."); + elseif (! isnumeric (lambda) || ! isscalar (lambda)) + error ("gallery: LAMBDA must be a numeric scalar for jordbloc matrix."); + endif + + J = lambda * eye (n) + diag (ones (n-1, 1), 1); +endfunction + +function U = kahan (n, theta = 1.2, pert = 25) + ## KAHAN Kahan matrix - upper trapezoidal. + ## KAHAN(N, THETA) is an upper trapezoidal matrix + ## that has some interesting properties regarding estimation of + ## condition and rank. + ## The matrix is N-by-N unless N is a 2-vector, in which case it + ## is N(1)-by-N(2). + ## The parameter THETA defaults to 1.2. + ## The useful range of THETA is 0 < THETA < PI. + ## + ## To ensure that the QR factorization with column pivoting does not + ## interchange columns in the presence of rounding errors, the diagonal + ## is perturbed by PERT*EPS*diag( [N:-1:1] ). + ## The default is PERT = 25, which ensures no interchanges for KAHAN(N) + ## up to at least N = 90 in IEEE arithmetic. + ## KAHAN(N, THETA, PERT) uses the given value of PERT. + ## + ## The inverse of KAHAN(N, THETA) is known explicitly: see + ## Higham (1987, p. 588), for example. + ## The diagonal perturbation was suggested by Christian Bischof. + ## + ## References: + ## W. Kahan, Numerical linear algebra, Canadian Math. Bulletin, + ## 9 (1966), pp. 757-801. + ## N.J. Higham, A survey of condition number estimation for + ## triangular matrices, SIAM Review, 29 (1987), pp. 575-596. + + if (nargin < 1 || nargin > 3) + error ("gallery: 1 to 3 arguments are required for kahan matrix."); + elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n) + error ("gallery: N must be a 1 or 2 element integer for kahan matrix."); + elseif (! isnumeric (theta) || ! isscalar (theta)) + error ("gallery: THETA must be a numeric scalar for kahan matrix."); + elseif (! isnumeric (pert) || ! isscalar (pert)) + error ("gallery: PERT must be a numeric scalar for kahan matrix."); + endif + + ## Parameter n specifies dimension: r-by-n + r = n(1); + n = n(end); + + s = sin (theta); + c = cos (theta); + + U = eye (n) - c * triu (ones (n), 1); + U = diag (s.^[0:n-1]) * U + pert*eps* diag ([n:-1:1]); + if (r > n) + U(r,n) = 0; # Extend to an r-by-n matrix + else + U = U(1:r,:); # Reduce to an r-by-n matrix + endif +endfunction + +function A = kms (n, rho = 0.5) + ## KMS Kac-Murdock-Szego Toeplitz matrix. + ## A = KMS(N, RHO) is the N-by-N Kac-Murdock-Szego Toeplitz matrix with + ## A(i,j) = RHO^(ABS((i-j))) (for real RHO). + ## If RHO is complex, then the same formula holds except that elements + ## below the diagonal are conjugated. + ## RHO defaults to 0.5. + ## Properties: + ## A has an LDL' factorization with + ## L = INV(TRIW(N,-RHO,1)'), + ## D(i,i) = (1-ABS(RHO)^2)*EYE(N) except D(1,1) = 1. + ## A is positive definite if and only if 0 < ABS(RHO) < 1. + ## INV(A) is tridiagonal. + ## + ## Reference: + ## W.F. Trench, Numerical solution of the eigenvalue problem + ## for Hermitian Toeplitz matrices, SIAM J. Matrix Analysis and Appl., + ## 10 (1989), pp. 135-146 (and see the references therein). + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 to 2 arguments are required for lauchli matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error("gallery: N must be an integer for lauchli matrix.") + elseif (! isscalar (mu)) + error("gallery: MU must be a scalar for lauchli matrix.") + endif + + A = (1:n)'*ones(1,n); + A = abs(A - A'); + A = rho .^ A; + if imag(rho) + A = conj(tril(A,-1)) + triu(A); + endif +endfunction + +function B = krylov (A, x, j) + ## KRYLOV Krylov matrix. + ## KRYLOV(A, x, j) is the Krylov matrix + ## [x, Ax, A^2x, ..., A^(j-1)x], + ## where A is an n-by-n matrix and x is an n-vector. + ## Defaults: x = ONES(n,1), j = n. + ## KRYLOV(n) is the same as KRYLOV(RANDN(n)). + ## + ## Reference: + ## G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, + ## Johns Hopkins University Press, Baltimore, Maryland, 1989, p. 369. + + if (nargin < 1 || nargin > 3) + error ("gallery: 1 to 3 arguments are required for krylov matrix."); + elseif (! isnumeric (A) || ! issquare (A) || ndims (A) != 2) + error ("gallery: A must be a square 2D matrix for krylov matrix."); + endif + + n = length (A); + if (isscalar (A)) + n = A; + A = randn (n); + endif + + if (nargin < 2) + x = ones (n, 1); + elseif (! isvector (x) || numel (x) != n) + error ("gallery: X must be a vector of length equal to A for krylov matrix."); + endif + + if (nargin < 3) + j = n; + elseif (! isnumeric (j) || ! isscalar (j) || fix (j) != j) + error ("gallery: J must be an integer for krylov matrix."); + endif + + B = ones (n, j); + B(:,1) = x(:); + for i = 2:j + B(:,i) = A*B(:,i-1); + endfor +endfunction + +function A = lauchli (n, mu = sqrt (eps)) + ## LAUCHLI Lauchli matrix - rectangular. + ## LAUCHLI(N, MU) is the (N+1)-by-N matrix [ONES(1,N); MU*EYE(N))]. + ## It is a well-known example in least squares and other problems + ## that indicates the dangers of forming A'*A. + ## MU defaults to SQRT(EPS). + ## + ## Reference: + ## P. Lauchli, Jordan-Elimination und Ausgleichung nach + ## kleinsten Quadraten, Numer. Math, 3 (1961), pp. 226-240. + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 to 2 arguments are required for lauchli matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for lauchli matrix."); + elseif (! isscalar (mu)) + error ("gallery: MU must be a scalar for lauchli matrix."); + endif + + A = [ones(1, n) + mu*eye(n) ]; +endfunction + +function A = lehmer (n) + ## LEHMER Lehmer matrix - symmetric positive definite. + ## A = LEHMER(N) is the symmetric positive definite N-by-N matrix with + ## A(i,j) = i/j for j >= i. + ## A is totally nonnegative. INV(A) is tridiagonal, and explicit + ## formulas are known for its entries. + ## N <= COND(A) <= 4*N*N. + ## + ## References: + ## M. Newman and J. Todd, The evaluation of matrix inversion + ## programs, J. Soc. Indust. Appl. Math., 6 (1958), pp. 466-476. + ## Solutions to problem E710 (proposed by D.H. Lehmer): The inverse + ## of a matrix, Amer. Math. Monthly, 53 (1946), pp. 534-535. + ## J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, + ## Birkhauser, Basel, and Academic Press, New York, 1977, p. 154. + + if (nargin != 1) + error ("gallery: 1 argument is required for lehmer matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for lehmer matrix."); + endif + + A = ones (n, 1) * (1:n); + A = A./A'; + A = tril (A) + tril (A, -1)'; +endfunction + +function T = lesp (n) + ## LESP A tridiagonal matrix with real, sensitive eigenvalues. + ## LESP(N) is an N-by-N matrix whose eigenvalues are real and smoothly + ## distributed in the interval approximately [-2*N-3.5, -4.5]. + ## The sensitivities of the eigenvalues increase exponentially as + ## the eigenvalues grow more negative. + ## The matrix is similar to the symmetric tridiagonal matrix with + ## the same diagonal entries and with off-diagonal entries 1, + ## via a similarity transformation with D = diag(1!,2!,...,N!). + ## + ## References: + ## H.W.J. Lenferink and M.N. Spijker, On the use of stability regions in + ## the numerical analysis of initial value problems, + ## Math. Comp., 57 (1991), pp. 221-237. + ## L.N. Trefethen, Pseudospectra of matrices, in Numerical Analysis 1991, + ## Proceedings of the 14th Dundee Conference, + ## D.F. Griffiths and G.A. Watson, eds, Pitman Research Notes in + ## Mathematics, volume 260, Longman Scientific and Technical, Essex, + ## UK, 1992, pp. 234-266. + + if (nargin != 1) + error ("gallery: 1 argument is required for lesp matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for lesp matrix."); + endif + + x = 2:n; + T = full (tridiag (ones (size (x)) ./x, -(2*[x n+1]+1), x)); +endfunction + +function A = lotkin (n) + ## LOTKIN Lotkin matrix. + ## A = LOTKIN(N) is the Hilbert matrix with its first row altered to + ## all ones. A is unsymmetric, ill-conditioned, and has many negative + ## eigenvalues of small magnitude. + ## The inverse has integer entries and is known explicitly. + ## + ## Reference: + ## M. Lotkin, A set of test matrices, MTAC, 9 (1955), pp. 153-161. + + if (nargin != 1) + error ("gallery: 1 argument is required for lotkin matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for lotkin matrix."); + endif + + A = hilb (n); + A(1,:) = ones (1, n); +endfunction + +function A = minij (n) + ## MINIJ Symmetric positive definite matrix MIN(i,j). + ## A = MINIJ(N) is the N-by-N symmetric positive definite matrix with + ## A(i,j) = MIN(i,j). + ## Properties, variations: + ## INV(A) is tridiagonal: it is minus the second difference matrix + ## except its (N,N) element is 1. + ## 2*A-ONES(N) (Givens' matrix) has tridiagonal inverse and + ## eigenvalues .5*sec^2([2r-1)PI/4N], r=1:N. + ## (N+1)*ONES(N)-A also has a tridiagonal inverse. + ## + ## References: + ## J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, + ## Birkhauser, Basel, and Academic Press, New York, 1977, p. 158. + ## D.E. Rutherford, Some continuant determinants arising in physics and + ## chemistry---II, Proc. Royal Soc. Edin., 63, A (1952), pp. 232-241. + ## (For the eigenvalues of Givens' matrix.) + + if (nargin != 1) + error ("gallery: 1 argument is required for minij matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for minij matrix."); + endif + + A = min (ones (n, 1) * (1:n), (1:n)' * ones (1, n)); +endfunction + +function A = moler (n, alpha = -1) + ## MOLER Moler matrix - symmetric positive definite. + ## A = MOLER(N, ALPHA) is the symmetric positive definite N-by-N matrix + ## U'*U where U = TRIW(N, ALPHA). + ## For ALPHA = -1 (the default) A(i,j) = MIN(i,j)-2, A(i,i) = i. + ## A has one small eigenvalue. + ## + ## Nash (1990) attributes the ALPHA = -1 matrix to Moler. + ## + ## Reference: + ## J.C. Nash, Compact Numerical Methods for Computers: Linear + ## Algebra and Function Minimisation, second edition, Adam Hilger, + ## Bristol, 1990 (Appendix 1). + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 to 2 arguments are required for moler matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for moler matrix."); + elseif (! isscalar (alpha)) + error ("gallery: ALPHA must be a scalar for moler matrix."); + endif + + A = triw (n, alpha)' * triw (n, alpha); +endfunction + +function [A, T] = neumann (n) + ## NEUMANN Singular matrix from the discrete Neumann problem (sparse). + ## NEUMANN(N) is the singular, row diagonally dominant matrix resulting + ## from discretizing the Neumann problem with the usual five point + ## operator on a regular mesh. + ## It has a one-dimensional null space with null vector ONES(N,1). + ## The dimension N should be a perfect square, or else a 2-vector, + ## in which case the dimension of the matrix is N(1)*N(2). + ## + ## Reference: + ## R.J. Plemmons, Regular splittings and the discrete Neumann + ## problem, Numer. Math., 25 (1976), pp. 153-161. + + if (nargin != 1) + error ("gallery: 1 argument is required for neumann matrix."); + elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n) + error ("gallery: N must be a 1 or 2 element integer for neumann matrix."); + endif + + if (isscalar (n)) + m = sqrt (n); + if (m^2 != n) + error ("gallery: N must be a perfect square for neumann matrix."); + endif + n(1) = m; + n(2) = m; + endif + + T = tridiag (n(1), -1, 2, -1); + T(1,2) = -2; + T(n(1),n(1)-1) = -2; + + A = kron (T, eye (n(2))) + kron (eye (n(2)), T); +endfunction + +function Q = orthog (n, k = 1) + ## ORTHOG Orthogonal and nearly orthogonal matrices. + ## Q = ORTHOG(N, K) selects the K'th type of matrix of order N. + ## K > 0 for exactly orthogonal matrices, K < 0 for diagonal scalings of + ## orthogonal matrices. + ## Available types: (K = 1 is the default) + ## K = 1: Q(i,j) = SQRT(2/(n+1)) * SIN( i*j*PI/(n+1) ) + ## Symmetric eigenvector matrix for second difference matrix. + ## K = 2: Q(i,j) = 2/SQRT(2*n+1)) * SIN( 2*i*j*PI/(2*n+1) ) + ## Symmetric. + ## K = 3: Q(r,s) = EXP(2*PI*i*(r-1)*(s-1)/n) / SQRT(n) (i=SQRT(-1)) + ## Unitary, the Fourier matrix. Q^4 is the identity. + ## This is essentially the same matrix as FFT(EYE(N))/SQRT(N)! + ## K = 4: Helmert matrix: a permutation of a lower Hessenberg matrix, + ## whose first row is ONES(1:N)/SQRT(N). + ## K = 5: Q(i,j) = SIN( 2*PI*(i-1)*(j-1)/n ) + COS( 2*PI*(i-1)*(j-1)/n ). + ## Symmetric matrix arising in the Hartley transform. + ## K = -1: Q(i,j) = COS( (i-1)*(j-1)*PI/(n-1) ) + ## Chebyshev Vandermonde-like matrix, based on extrema of T(n-1). + ## K = -2: Q(i,j) = COS( (i-1)*(j-1/2)*PI/n) ) + ## Chebyshev Vandermonde-like matrix, based on zeros of T(n). + ## + ## References: + ## N.J. Higham and D.J. Higham, Large growth factors in Gaussian + ## elimination with pivoting, SIAM J. Matrix Analysis and Appl., + ## 10 (1989), pp. 155-164. + ## P. Morton, On the eigenvectors of Schur's matrix, J. Number Theory, + ## 12 (1980), pp. 122-127. (Re. ORTHOG(N, 3)) + ## H.O. Lancaster, The Helmert Matrices, Amer. Math. Monthly, 72 (1965), + ## pp. 4-12. + ## D. Bini and P. Favati, On a matrix algebra related to the discrete + ## Hartley transform, SIAM J. Matrix Anal. Appl., 14 (1993), + ## pp. 500-507. + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 to 2 arguments are required for orthog matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for orthog matrix."); + elseif (! isnumeric (k) || ! isscalar (k)) + error ("gallery: K must be a numeric scalar for orthog matrix."); + endif + + switch (k) + case (1) + ## E'vectors second difference matrix + m = (1:n)'*(1:n) * (pi/(n+1)); + Q = sin (m) * sqrt (2/(n+1)); + + case (2) + m = (1:n)'*(1:n) * (2*pi/(2*n+1)); + Q = sin (m) * (2/ sqrt (2*n+1)); + + case (3) + ## Vandermonde based on roots of unity + m = 0:n-1; + Q = exp (m'*m*2*pi* sqrt (-1) / n) / sqrt (n); + + case (4) + ## Helmert matrix + Q = tril (ones (n)); + Q(1,2:n) = ones (1, n-1); + for i = 2:n + Q(i,i) = -(i-1); + end + Q = diag (sqrt ([n 1:n-1] .* [1:n])) \ Q; + + case (5) + ## Hartley matrix + m = (0:n-1)'*(0:n-1) * (2*pi/n); + Q = (cos (m) + sin (m)) / sqrt (n); + + case (-1) + ## extrema of T(n-1) + m = (0:n-1)'*(0:n-1) * (pi/(n-1)); + Q = cos (m); + + case (-2) + ## zeros of T(n) + m = (0:n-1)'*(.5:n-.5) * (pi/n); + Q = cos (m); + + otherwise + error ("gallery: unknown K `%d' for orthog matrix.", k); + endswitch +endfunction + +function A = parter (n) + ## PARTER Parter matrix - a Toeplitz matrix with singular values near PI. + ## PARTER(N) is the matrix with (i,j) element 1/(i-j+0.5). + ## It is a Cauchy matrix and a Toeplitz matrix. + ## + ## At the Second SIAM Conference on Linear Algebra, Raleigh, N.C., + ## 1985, Cleve Moler noted that most of the singular values of + ## PARTER(N) are very close to PI. An explanation of the phenomenon + ## was given by Parter; see also the paper by Tyrtyshnikov. + ## + ## References: + ## The MathWorks Newsletter, Volume 1, Issue 1, March 1986, page 2. + ## S.V. Parter, On the distribution of the singular values of Toeplitz + ## matrices, Linear Algebra and Appl., 80 (1986), pp. 115-130. + ## E.E. Tyrtyshnikov, Cauchy-Toeplitz matrices and some applications, + ## Linear Algebra and Appl., 149 (1991), pp. 1-18. + + if (nargin != 1) + error ("gallery: 1 argument is required for parter matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for parter matrix."); + endif + + A = cauchy ((1:n) + 0.5, -(1:n)); +endfunction + +function P = pei (n, alpha = 1) + ## PEI Pei matrix. + ## PEI(N, ALPHA), where ALPHA is a scalar, is the symmetric matrix + ## ALPHA*EYE(N) + ONES(N). + ## If ALPHA is omitted then ALPHA = 1 is used. + ## The matrix is singular for ALPHA = 0, -N. + ## + ## Reference: + ## M.L. Pei, A test matrix for inversion procedures, + ## Comm. ACM, 5 (1962), p. 508. + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 to 2 arguments are required for pei matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for pei matrix."); + elseif (! isnumeric (w) || ! isscalar (w)) + error ("gallery: ALPHA must be a scalar for pei matrix."); + endif + + P = alpha * eye (n) + ones (n); +endfunction + +function A = poisson (n) + ## POISSON Block tridiagonal matrix from Poisson's equation (sparse). + ## POISSON(N) is the block tridiagonal matrix of order N^2 + ## resulting from discretizing Poisson's equation with the + ## 5-point operator on an N-by-N mesh. + ## + ## Reference: + ## G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, + ## Johns Hopkins University Press, Baltimore, Maryland, 1989 + ## (Section 4.5.4). + + if (nargin != 1) + error ("gallery: 1 argument is required for poisson matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for poisson matrix."); + endif + + S = tridiag (n, -1, 2, -1); + I = speye (n); + A = kron (I, S) + kron (S, I); +endfunction + +function A = prolate (n, w = 0.25) + ## PROLATE Prolate matrix - symmetric, ill-conditioned Toeplitz matrix. + ## A = PROLATE(N, W) is the N-by-N prolate matrix with parameter W. + ## It is a symmetric Toeplitz matrix. + ## If 0 < W < 0.5 then + ## - A is positive definite + ## - the eigenvalues of A are distinct, lie in (0, 1), and + ## tend to cluster around 0 and 1. + ## W defaults to 0.25. + ## + ## Reference: + ## J.M. Varah. The Prolate matrix. Linear Algebra and Appl., + ## 187:269--278, 1993. + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 to 2 arguments are required for prolate matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for prolate matrix."); + elseif (! isnumeric (w) || ! isscalar (w)) + error ("gallery: W must be a scalar for prolate matrix."); + endif + + a = zeros (n, 1); + a(1) = 2*w; + a(2:n) = sin (2*pi*w*(1:n-1)) ./ (pi*(1:n-1)); + + A = toeplitz(a); +endfunction + +function H = randhess (x) + ## NOTE: this function was named ohess in the original Test Matrix Toolbox + ## RANDHESS Random, orthogonal upper Hessenberg matrix. + ## H = RANDHESS(N) is an N-by-N real, random, orthogonal + ## upper Hessenberg matrix. + ## Alternatively, H = RANDHESS(X), where X is an arbitrary real + ## N-vector (N > 1) constructs H non-randomly using the elements + ## of X as parameters. + ## In both cases H is constructed via a product of N-1 Givens rotations. + ## + ## Note: See Gragg (1986) for how to represent an N-by-N (complex) + ## unitary Hessenberg matrix with positive subdiagonal elements in terms + ## of 2N-1 real parameters (the Schur parametrization). + ## This M-file handles the real case only and is intended simply as a + ## convenient way to generate random or non-random orthogonal Hessenberg + ## matrices. + ## + ## Reference: + ## W.B. Gragg, The QR algorithm for unitary Hessenberg matrices, + ## J. Comp. Appl. Math., 16 (1986), pp. 1-8. + + if (nargin != 1) + error ("gallery: 1 argument is required for randhess matrix."); + elseif (! isnumeric (x) || ! isreal (x)) + error ("gallery: N or X must be numeric real values for randhess matrix."); + endif + + if (isscalar (x)) + n = x; + x = rand (n-1, 1) * 2*pi; + H = eye (n); + H(n,n) = sign (randn); + elseif (isvector (x)) + n = numel (x); + H = eye (n); + H(n,n) = sign (x(n)) + (x(n) == 0); # Second term ensures H(n,n) nonzero. + else + error ("gallery: N or X must be a scalar or a vector for randhess matrix."); + endif + + for i = n:-1:2 + ## Apply Givens rotation through angle x(i-1). + theta = x(i-1); + c = cos (theta); + s = sin (theta); + H([i-1 i], :) = [ c*H(i-1,:)+s*H(i,:) + -s*H(i-1,:)+c*H(i,:) ]; + endfor +endfunction + +function A = rando (n, k = 1) + ## RANDO Random matrix with elements -1, 0 or 1. + ## A = RANDO(N, K) is a random N-by-N matrix with elements from + ## one of the following discrete distributions (default K = 1): + ## K = 1: A(i,j) = 0 or 1 with equal probability, + ## K = 2: A(i,j) = -1 or 1 with equal probability, + ## K = 3: A(i,j) = -1, 0 or 1 with equal probability. + ## N may be a 2-vector, in which case the matrix is N(1)-by-N(2). + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 to 2 arguments are required for rando matrix."); + elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n) + error ("gallery: N must be an integer for rando matrix."); + elseif (! isnumeric (k) || ! isscalar (k)) + error ("gallery: K must be a numeric scalar for smoke matrix."); + endif + + ## Parameter n specifies dimension: m-by-n. + m = n(1); + n = n(end); + + switch (k) + case (1), A = floor ( rand(m, n) + 0.5); # {0, 1} + case (2), A = 2*floor ( rand(m, n) + 0.5) -1; # {-1, 1} + case (3), A = round (3*rand(m, n) - 1.5); # {-1, 0, 1} + otherwise + error ("gallery: unknown K `%d' for smoke matrix.", k); + endswitch + +endfunction + +function A = randsvd (n, kappa = sqrt (1/eps), mode = 3, kl = n-1, ku = kl) + ## RANDSVD Random matrix with pre-assigned singular values. + ## RANDSVD(N, KAPPA, MODE, KL, KU) is a (banded) random matrix of order N + ## with COND(A) = KAPPA and singular values from the distribution MODE. + ## N may be a 2-vector, in which case the matrix is N(1)-by-N(2). + ## Available types: + ## MODE = 1: one large singular value, + ## MODE = 2: one small singular value, + ## MODE = 3: geometrically distributed singular values, + ## MODE = 4: arithmetically distributed singular values, + ## MODE = 5: random singular values with unif. dist. logarithm. + ## If omitted, MODE defaults to 3, and KAPPA defaults to SQRT(1/EPS). + ## If MODE < 0 then the effect is as for ABS(MODE) except that in the + ## original matrix of singular values the order of the diagonal entries + ## is reversed: small to large instead of large to small. + ## KL and KU are the lower and upper bandwidths respectively; if they + ## are omitted a full matrix is produced. + ## If only KL is present, KU defaults to KL. + ## Special case: if KAPPA < 0 then a random full symmetric positive + ## definite matrix is produced with COND(A) = -KAPPA and + ## eigenvalues distributed according to MODE. + ## KL and KU, if present, are ignored. + ## + ## Reference: + ## N.J. Higham, Accuracy and Stability of Numerical Algorithms, + ## Society for Industrial and Applied Mathematics, Philadelphia, PA, + ## USA, 1996; sec. 26.3. + ## + ## This routine is similar to the more comprehensive Fortran routine xLATMS + ## in the following reference: + ## J.W. Demmel and A. McKenney, A test matrix generation suite, + ## LAPACK Working Note #9, Courant Institute of Mathematical Sciences, + ## New York, 1989. + + if (nargin < 1 || nargin > 5) + error ("gallery: 1 to 5 arguments are required for randsvd matrix."); + elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n) + error ("gallery: N must be a 1 or 2 element integer vector for randsvd matrix."); + elseif (! isnumeric (kappa) || ! isscalar (kappa)) + error ("gallery: KAPPA must be a numeric scalar for randsvd matrix."); + elseif (abs (kappa) < 1) + error ("gallery: KAPPA must larger than or equal to 1 for randsvd matrix."); + elseif (! isnumeric (mode) || ! isscalar (mode)) + error ("gallery: MODE must be a numeric scalar for randsvd matrix."); + elseif (! isnumeric (kl) || ! isscalar (kl)) + error ("gallery: KL must be a numeric scalar for randsvd matrix."); + elseif (! isnumeric (ku) || ! isscalar (ku)) + error ("gallery: KU must be a numeric scalar for randsvd matrix."); + endif + + posdef = 0; + if (kappa < 0) + posdef = 1; + kappa = -kappa; + endif + + ## Parameter n specifies dimension: m-by-n. + m = n(1); + n = n(end); + p = min ([m n]); + + ## If A will be a vector + if (p == 1) + A = randn (m, n); + A = A / norm (A); + return + end + + ## Set up vector sigma of singular values. + switch (abs (mode)) + case (1) + sigma = ones (p, 1) ./ kappa; + sigma(1) = 1; + case (2) + sigma = ones (p, 1); + sigma(p) = 1 / kappa; + case (3) + factor = kappa^(-1/(p-1)); + sigma = factor.^[0:p-1]; + case (4) + sigma = ones (p, 1) - (0:p-1)'/(p-1)*(1-1/kappa); + case (5) + ## In this case cond (A) <= kappa. + rand ("uniform"); + sigma = exp (-rand (p, 1) * log (kappa)); + otherwise + error ("gallery: unknown MODE `%d' for randsvd matrix.", mode); + endswitch + + ## Convert to diagonal matrix of singular values. + if (mode < 0) + sigma = sigma (p:-1:1); + end + sigma = diag (sigma); + + if (posdef) + ## handle case where KAPPA was negative + Q = qmult (p); + A = Q' * sigma * Q; + A = (A + A') / 2; # Ensure matrix is symmetric. + return + endif + + if (m != n) + ## Expand to m-by-n diagonal matrix + sigma(m, n) = 0; + end + + if (kl == 0 && ku == 0) + ## Diagonal matrix requested - nothing more to do. + A = sigma; + else + ## A = U*sigma*V, where U, V are random orthogonal matrices from the + ## Haar distribution. + A = qmult (sigma'); + A = qmult (A'); + + if (kl < n-1 || ku < n-1) + ## Bandwidth reduction + A = bandred (A, kl, ku); + endif + endif +endfunction + +function A = redheff (n) + ## REDHEFF A (0,1) matrix of Redheffer associated with the Riemann hypothesis. + ## A = REDHEFF(N) is an N-by-N matrix of 0s and 1s defined by + ## A(i,j) = 1 if j = 1 or if i divides j, + ## A(i,j) = 0 otherwise. + ## It has N - FLOOR(LOG2(N)) - 1 eigenvalues equal to 1, + ## a real eigenvalue (the spectral radius) approximately SQRT(N), + ## a negative eigenvalue approximately -SQRT(N), + ## and the remaining eigenvalues are provably ``small''. + ## Barrett and Jarvis (1992) conjecture that + ## ``the small eigenvalues all lie inside the unit circle + ## ABS(Z) = 1'', + ## and a proof of this conjecture, together with a proof that some + ## eigenvalue tends to zero as N tends to infinity, would yield + ## a new proof of the prime number theorem. + ## The Riemann hypothesis is true if and only if + ## DET(A) = O( N^(1/2+epsilon) ) for every epsilon > 0 + ## (`!' denotes factorial). + ## See also RIEMANN. + ## + ## Reference: + ## W.W. Barrett and T.J. Jarvis, + ## Spectral Properties of a Matrix of Redheffer, + ## Linear Algebra and Appl., 162 (1992), pp. 673-683. + + if (nargin != 1) + error ("gallery: 1 argument is required for redheff matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for redheff matrix."); + endif + + i = (1:n)' * ones (1, n); + A = ! rem (i', i); + A(:,1) = ones (n, 1); +endfunction + +function A = riemann (n) + ## RIEMANN A matrix associated with the Riemann hypothesis. + ## A = RIEMANN(N) is an N-by-N matrix for which the + ## Riemann hypothesis is true if and only if + ## DET(A) = O( N! N^(-1/2+epsilon) ) for every epsilon > 0 + ## (`!' denotes factorial). + ## A = B(2:N+1, 2:N+1), where + ## B(i,j) = i-1 if i divides j and -1 otherwise. + ## Properties include, with M = N+1: + ## Each eigenvalue E(i) satisfies ABS(E(i)) <= M - 1/M. + ## i <= E(i) <= i+1 with at most M-SQRT(M) exceptions. + ## All integers in the interval (M/3, M/2] are eigenvalues. + ## + ## See also REDHEFF. + ## + ## Reference: + ## F. Roesler, Riemann's hypothesis as an eigenvalue problem, + ## Linear Algebra and Appl., 81 (1986), pp. 153-198. + + if (nargin != 1) + error ("gallery: 1 argument is required for riemann matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for riemann matrix."); + endif + + n = n+1; + i = (2:n)' * ones (1, n-1); + j = i'; + A = i .* (! rem (j, i)) - ones (n-1); +endfunction + +function A = ris (n) + ## NOTE: this function was named dingdong in the original Test Matrix Toolbox + ## RIS Dingdong matrix - a symmetric Hankel matrix. + ## A = RIS(N) is the symmetric N-by-N Hankel matrix with + ## A(i,j) = 0.5/(N-i-j+1.5). + ## The eigenvalues of A cluster around PI/2 and -PI/2. + ## + ## Invented by F.N. Ris. + ## + ## Reference: + ## J.C. Nash, Compact Numerical Methods for Computers: Linear + ## Algebra and Function Minimisation, second edition, Adam Hilger, + ## Bristol, 1990 (Appendix 1). + + if (nargin != 1) + error ("gallery: 1 argument is required for ris matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for ris matrix."); + endif + + p = -2*(1:n) + (n+1.5); + A = cauchy (p); +endfunction + +function A = smoke (n, k = 0) + ## SMOKE Smoke matrix - complex, with a `smoke ring' pseudospectrum. + ## SMOKE(N) is an N-by-N matrix with 1s on the + ## superdiagonal, 1 in the (N,1) position, and powers of + ## roots of unity along the diagonal. + ## SMOKE(N, 1) is the same except for a zero (N,1) element. + ## The eigenvalues of SMOKE(N, 1) are the N'th roots of unity; + ## those of SMOKE(N) are the N'th roots of unity times 2^(1/N). + ## + ## Try PS(SMOKE(32)). For SMOKE(N, 1) the pseudospectrum looks + ## like a sausage folded back on itself. + ## GERSH(SMOKE(N, 1)) is interesting. + ## + ## Reference: + ## L. Reichel and L.N. Trefethen, Eigenvalues and pseudo-eigenvalues of + ## Toeplitz matrices, Linear Algebra and Appl., 162-164:153-185, 1992. + + if (nargin < 1 || nargin > 2) + error ("gallery: 1 to 2 arguments are required for smoke matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be an integer for smoke matrix."); + elseif (! isnumeric (n) || ! isscalar (n)) + error ("gallery: K must be a numeric scalar for smoke matrix."); + endif + + w = exp(2*pi*i/n); + A = diag( [w.^(1:n-1) 1] ) + diag(ones(n-1,1),1); + + switch (k) + case (0), A(n,1) = 1; + case (1), # do nothing + otherwise, + error ("gallery: K must have a value of 0 or 1 for smoke matrix."); + endswitch +endfunction + +function T = toeppd (n, m = n, w = rand (m,1), theta = rand (m,1)) + ## NOTE: this function was named pdtoep in the original Test Matrix Toolbox + ## TOEPPD Symmetric positive definite Toeplitz matrix. + ## TOEPPD(N, M, W, THETA) is an N-by-N symmetric positive (semi-) + ## definite (SPD) Toeplitz matrix, comprised of the sum of M rank 2 + ## (or, for certain THETA, rank 1) SPD Toeplitz matrices. + ## Specifically, + ## T = W(1)*T(THETA(1)) + ... + W(M)*T(THETA(M)), + ## where T(THETA(k)) has (i,j) element COS(2*PI*THETA(k)*(i-j)). + ## Defaults: M = N, W = RAND(M,1), THETA = RAND(M,1). + ## + ## Reference: + ## G. Cybenko and C.F. Van Loan, Computing the minimum eigenvalue of + ## a symmetric positive definite Toeplitz matrix, SIAM J. Sci. Stat. + ## Comput., 7 (1986), pp. 123-131. + + if (nargin < 1 || nargin > 4) + error ("gallery: 1 to 4 arguments are required for toeppd matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be a numeric integer for toeppd matrix."); + elseif (! isnumeric (m) || ! isscalar (m) || fix (m) != m) + error ("gallery: M must be a numeric integer for toeppd matrix."); + elseif (numel (w) != m || numel (theta) != m) + error ("gallery: W and THETA must be vectors of length M for toeppd matrix."); + endif + + T = zeros (n); + E = 2*pi * ((1:n)' * ones (1, n) - ones (n, 1) * (1:n)); + + for i = 1:m + T = T + w(i) * cos (theta(i)*E); + endfor +endfunction + +function P = toeppen (n, a = 1, b = -10, c = 0, d = 10, e = 1) + ## NOTE: this function was named pentoep in the original Test Matrix Toolbox + ## TOEPPEN Pentadiagonal Toeplitz matrix (sparse). + ## P = TOEPPEN(N, A, B, C, D, E) is the N-by-N pentadiagonal + ## Toeplitz matrix with diagonals composed of the numbers + ## A =: P(3,1), B =: P(2,1), C =: P(1,1), D =: P(1,2), E =: P(1,3). + ## Default: (A,B,C,D,E) = (1,-10,0,10,1) (a matrix of Rutishauser). + ## This matrix has eigenvalues lying approximately on + ## the line segment 2*cos(2*t) + 20*i*sin(t). + ## + ## Interesting plots are + ## PS(FULL(TOEPPEN(32,0,1,0,0,1/4))) - `triangle' + ## PS(FULL(TOEPPEN(32,0,1/2,0,0,1))) - `propeller' + ## PS(FULL(TOEPPEN(32,0,1/2,1,1,1))) - `fish' + ## + ## References: + ## R.M. Beam and R.F. Warming, The asymptotic spectra of + ## banded Toeplitz and quasi-Toeplitz matrices, SIAM J. Sci. + ## Comput. 14 (4), 1993, pp. 971-1006. + ## H. Rutishauser, On test matrices, Programmation en Mathematiques + ## Numeriques, Editions Centre Nat. Recherche Sci., Paris, 165, + ## 1966, pp. 349-365. + + if (nargin < 1 || nargin > 6) + error ("gallery: 1 to 6 arguments are required for toeppen matrix."); + elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) + error ("gallery: N must be a numeric integer for toeppen matrix."); + elseif (any (cellfun (@(x) ! isnumeric (x) || ! isscalar (x), {a b c d e}))) + error ("gallery: A, B, C, D and E must be numeric scalars for toeppen matrix."); + endif + + P = spdiags ([a*ones(n,1) b*ones(n,1) c*ones(n,1) d*ones(n,1) e*ones(n,1)], + -2:2, n, n); +endfunction + +function T = tridiag (n, x = -1, y = 2, z = -1) + ## TRIDIAG Tridiagonal matrix (sparse). + ## TRIDIAG(X, Y, Z) is the tridiagonal matrix with subdiagonal X, + ## diagonal Y, and superdiagonal Z. + ## X and Z must be vectors of dimension one less than Y. + ## Alternatively TRIDIAG(N, C, D, E), where C, D, and E are all + ## scalars, yields the Toeplitz tridiagonal matrix of order N + ## with subdiagonal elements C, diagonal elements D, and superdiagonal + ## elements E. This matrix has eigenvalues (Todd 1977) + ## D + 2*SQRT(C*E)*COS(k*PI/(N+1)), k=1:N. + ## TRIDIAG(N) is the same as TRIDIAG(N,-1,2,-1), which is + ## a symmetric positive definite M-matrix (the negative of the + ## second difference matrix). + ## + ## References: + ## J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, + ## Birkhauser, Basel, and Academic Press, New York, 1977, p. 155. + ## D.E. Rutherford, Some continuant determinants arising in physics and + ## chemistry---II, Proc. Royal Soc. Edin., 63, A (1952), pp. 232-241. + + if (nargin != 1 && nargin != 3 && nargin != 4) + error ("gallery: 1, 3, or 4 arguments are required for tridiag matrix."); + elseif (nargin == 3) + z = y; + y = x; + x = n; + endif + + ## Force column vectors + x = x(:); + y = y(:); + z = z(:); + + if (isscalar (x) && isscalar (y) && isscalar (z)) + x *= ones (n-1, 1); + z *= ones (n-1, 1); + y *= ones (n, 1); + elseif (numel (y) != numel (x) + 1) + error ("gallery: X must have one element less than Y for tridiag matrix."); + elseif (numel (y) != numel (z) + 1) + error ("gallery: Z must have one element less than Y for tridiag matrix."); + endif + + ## T = diag (x, -1) + diag (y) + diag (z, 1); # For non-sparse matrix. + n = numel (y); + T = spdiags ([[x;0] y [0;z]], -1:1, n, n); +endfunction + +function t = triw (n, alpha = -1, k = -1) + ## TRIW Upper triangular matrix discussed by Wilkinson and others. + ## TRIW(N, ALPHA, K) is the upper triangular matrix with ones on + ## the diagonal and ALPHAs on the first K >= 0 superdiagonals. + ## N may be a 2-vector, in which case the matrix is N(1)-by-N(2) and + ## upper trapezoidal. + ## Defaults: ALPHA = -1, + ## K = N - 1 (full upper triangle). + ## TRIW(N) is a matrix discussed by Kahan, Golub and Wilkinson. + ## + ## Ostrowski (1954) shows that + ## COND(TRIW(N,2)) = COT(PI/(4*N))^2, + ## and for large ABS(ALPHA), + ## COND(TRIW(N,ALPHA)) is approximately ABS(ALPHA)^N*SIN(PI/(4*N-2)). + ## + ## Adding -2^(2-N) to the (N,1) element makes TRIW(N) singular, + ## as does adding -2^(1-N) to all elements in the first column. + ## + ## References: + ## G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the + ## computation of the Jordan canonical form, SIAM Review, + ## 18(4), 1976, pp. 578-619. + ## W. Kahan, Numerical linear algebra, Canadian Math. Bulletin, + ## 9 (1966), pp. 757-801. + ## A.M. Ostrowski, On the spectrum of a one-parametric family of + ## matrices, J. Reine Angew. Math., 193 (3/4), 1954, pp. 143-160. + ## J.H. Wilkinson, Singular-value decomposition---basic aspects, + ## in D.A.H. Jacobs, ed., Numerical Software---Needs and Availability, + ## Academic Press, London, 1978, pp. 109-135. + + if (nargin < 1 || nargin > 3) + error ("gallery: 1 to 3 arguments are required for triw matrix."); + elseif (! isnumeric (n) || all (numel (n) != [1 2])) + error ("gallery: N must be a 1 or 2 elements vector for triw matrix."); + elseif (! isscalar (alpha)) + error ("gallery: ALPHA must be a scalar for triw matrix."); + elseif (! isscalar (k) || ! isnumeric (k) || fix (k) != k) + error ("gallery: K must be a numeric integer for triw matrix."); + endif + + m = n(1); # Parameter n specifies dimension: m-by-n. + n = n(end); + + t = tril (eye (m, n) + alpha * triu (ones (m, n), 1), k); +endfunction + +function A = wathen (nx, ny, k = 0) + ## # WATHEN returns the Wathen matrix. + ## + ## Discussion: + ## + ## The Wathen matrix is a finite element matrix which is sparse. + ## + ## The entries of the matrix depend in part on a physical quantity + ## related to density. That density is here assigned random values between + ## 0 and 100. + ## + ## A = WATHEN ( NX, NY ) is a sparse random N-by-N finite element matrix + ## where N = 3*NX*NY + 2*NX + 2*NY + 1. + ## + ## A is the consistent mass matrix for a regular NX-by-NY + ## grid of 8-node (serendipity) elements in 2 space dimensions. + ## + ## Here is an illustration for NX = 3, NX = 2: + ## + ## 23-24-25-26-27-28-29 + ## | | | | + ## 19 20 21 22 + ## | | | | + ## 12-13-14-15-16-17-18 + ## | | | | + ## 8 9 10 11 + ## | | | | + ## 1--2--3--4--5--6--7 + ## + ## For this example, the total number of nodes is, as expected, + ## + ## N = 3 * 3 * 2 + 2 * 2 + 2 * 3 + 1 = 29. + ## + ## A is symmetric positive definite for any (positive) values of + ## the density, RHO(NX,NY), which is chosen randomly in this routine. + ## + ## In particular, if D = DIAG(DIAG(A)), then + ## 0.25 <= EIG(INV(D)*A) <= 4.5 + ## for any positive integers NX and NY and any densities RHO(NX,NY). + ## + ## A = WATHEN ( NX, NY, 1 ) returns the diagonally scaled matrix. + ## + ## Modified: + ## + ## 17 September 2007 + ## + ## Author: + ## + ## Nicholas Higham + ## + ## Reference: + ## + ## Nicholas Higham, + ## Algorithm 694: A Collection of Test Matrices in MATLAB, + ## ACM Transactions on Mathematical Software, + ## Volume 17, Number 3, September 1991, pages 289-305. + ## + ## Andrew Wathen, + ## Realistic eigenvalue bounds for the Galerkin mass matrix, + ## IMA Journal of Numerical Analysis, + ## Volume 7, 1987, pages 449-457. + ## + ## Parameters: + ## + ## Input, integer NX, NY, the number of elements in the X and Y directions + ## of the finite element grid. NX and NY must each be at least 1. + ## + ## Optional input, integer K, is used to request that the diagonally scaled + ## version of the matrix be returned. This happens if K is specified with + ## the value 1. + ## + ## Output, sparse real A(N,N), the matrix. The dimension N is determined by + ## NX and NY, as described above. A is stored in the MATLAB sparse matrix + ## format. + + if (nargin < 2 || nargin > 3) + error ("gallery: 2 or 3 arguments are required for wathen matrix."); + elseif (! isnumeric (nx) || ! isscalar (nx) || nx < 1) + error ("gallery: NX must be a positive scalar for wathen matrix."); + elseif (! isnumeric (ny) || ! isscalar (ny) || ny < 1) + error ("gallery: NY must be a positive scalar for wathen matrix."); + elseif (! isscalar (k)) + error ("gallery: K must be a scalar for wathen matrix."); + endif + + e1 = [ 6 -6 2 -8 + -6 32 -6 20 + 2 -6 6 -6 + -8 20 -6 32 ]; + + e2 = [ 3 -8 2 -6 + -8 16 -8 20 + 2 -8 3 -8 + -6 20 -8 16 ]; + + e = [ e1 e2 + e2' e1] / 45; + + n = 3*nx*ny + 2*nx + 2*ny + 1; + + A = sparse (n, n); + + rho = 100 * rand (nx, ny); + + for j = 1:ny + for i = 1:nx + ## + ## For the element (I,J), determine the indices of the 8 nodes. + ## + nn(1) = 3*j*nx + 2*i + 2*j + 1; + nn(2) = nn(1) - 1; + nn(3) = nn(2) - 1; + nn(4) = (3*j - 1) * nx + 2*j + i - 1; + nn(5) = 3 * (j-1) * nx + 2*i + 2*j - 3; + nn(6) = nn(5) + 1; + nn(7) = nn(6) + 1; + nn(8) = nn(4) + 1; + + em = e * rho(i,j); + + for krow = 1:8 + for kcol = 1:8 + A(nn(krow),nn(kcol)) = A(nn(krow),nn(kcol)) + em(krow,kcol); + endfor + endfor + + endfor + endfor + + ## If requested, return A with diagonal scaling. + if (k) + A = diag (diag (A)) \ A; + endif +endfunction + +function [A, b] = wilk (n) + ## WILK Various specific matrices devised/discussed by Wilkinson. + ## [A, b] = WILK(N) is the matrix or system of order N. + ## N = 3: upper triangular system Ux=b illustrating inaccurate solution. + ## N = 4: lower triangular system Lx=b, ill-conditioned. + ## N = 5: HILB(6)(1:5,2:6)*1.8144. Symmetric positive definite. + ## N = 21: W21+, tridiagonal. Eigenvalue problem. + ## + ## References: + ## J.H. Wilkinson, Error analysis of direct methods of matrix inversion, + ## J. Assoc. Comput. Mach., 8 (1961), pp. 281-330. + ## J.H. Wilkinson, Rounding Errors in Algebraic Processes, Notes on Applied + ## Science No. 32, Her Majesty's Stationery Office, London, 1963. + ## J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University + ## Press, 1965. + + if (nargin != 1) + error ("gallery: 1 argument is required for wilk matrix."); + elseif (! isnumeric (n) || ! isscalar (n)) + error ("gallery: N must be a numeric scalar for wilk matrix."); + endif + + if (n == 3) + ## Wilkinson (1961) p.323. + A = [ 1e-10 0.9 -0.4 + 0 0.9 -0.4 + 0 0 1e-10 ]; + + b = [ 0 + 0 + 1]; + + elseif (n == 4) + ## Wilkinson (1963) p.105. + A = [0.9143e-4 0 0 0 + 0.8762 0.7156e-4 0 0 + 0.7943 0.8143 0.9504e-4 0 + 0.8017 0.6123 0.7165 0.7123e-4]; + + b = [0.6524 + 0.3127 + 0.4186 + 0.7853]; + + elseif (n == 5) + ## Wilkinson (1965), p.234. + A = hilb (6); + A = A(1:5, 2:6) * 1.8144; + + elseif (n == 21) + ## Wilkinson (1965), p.308. + E = diag (ones (n-1, 1), 1); + m = (n-1)/2; + A = diag (abs (-m:m)) + E + E'; + + else + error ("gallery: unknown N `%d' for wilk matrix.", n); + endif +endfunction + +## NOTE: bandred is part of the Test Matrix Toolbox and is used by randsvd() +function A = bandred (A, kl, ku) + ## BANDRED Band reduction by two-sided unitary transformations. + ## B = BANDRED(A, KL, KU) is a matrix unitarily equivalent to A + ## with lower bandwidth KL and upper bandwidth KU + ## (i.e. B(i,j) = 0 if i > j+KL or j > i+KU). + ## The reduction is performed using Householder transformations. + ## If KU is omitted it defaults to KL. + ## + ## Called by RANDSVD. + ## This is a `standard' reduction. Cf. reduction to bidiagonal form + ## prior to computing the SVD. This code is a little wasteful in that + ## it computes certain elements which are immediately set to zero! + ## + ## Reference: + ## G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, + ## Johns Hopkins University Press, Baltimore, Maryland, 1989. + ## Section 5.4.3. + + ## Check for special case where order of left/right transformations matters. + ## Easiest approach is to work on the transpose, flipping back at the end. + flip = false; + if (ku == 0) + flip = true; + A = A'; + [ku, kl] = deal (kl, ku); + endif + + [m, n] = size (A); + + for j = 1:min (min (m, n), max (m-kl-1, n-ku-1)) + if (j+kl+1 <= m) + [v, beta] = house (A(j+kl:m,j)); + temp = A(j+kl:m,j:n); + A(j+kl:m,j:n) = temp - beta*v*(v'*temp); + A(j+kl+1:m,j) = zeros (m-j-kl, 1); + endif + + if (j+ku+1 <= n) + [v, beta] = house (A(j,j+ku:n)'); + temp = A(j:m,j+ku:n); + A(j:m,j+ku:n) = temp - beta*(temp*v)*v'; + A(j,j+ku+1:n) = zeros (1, n-j-ku); + endif + endfor + + if (flip) + A = A'; + endif +endfunction