Mercurial > octave
annotate scripts/special-matrix/gallery.m @ 28896:90fea9cc9caa
test: Add expected error message <Invalid call> to BIST tests for nargin.
* acosd.m, acot.m, acotd.m, acoth.m, acsc.m, acscd.m, acsch.m, asec.m,
asecd.m, asech.m, asind.m, atand.m, cosd.m, cot.m, cotd.m, coth.m, csc.m,
cscd.m, csch.m, sec.m, secd.m, sech.m, sind.m, tand.m, cart2pol.m, cart2sph.m,
celldisp.m, common_size.m, deal.m, del2.m, fliplr.m, integral2.m, interp1.m,
isequal.m, isequaln.m, nextpow2.m, pol2cart.m, quad2d.m, quadl.m, quadv.m,
randi.m, rat.m, repelem.m, rescale.m, shiftdim.m, sortrows.m, sph2cart.m,
xor.m, convhull.m, delaunay.m, delaunayn.m, griddata.m, griddatan.m,
inpolygon.m, voronoi.m, voronoin.m, listdlg.m, msgbox.m, questdlg.m,
rmappdata.m, setappdata.m, __gripe_missing_component__.m,
get_first_help_sentence.m, type.m, which.m, cmpermute.m, cmunique.m,
gray2ind.m, imfinfo.m, imshow.m, imwrite.m, ind2rgb.m, movie.m, rgb2ind.m,
importdata.m, bandwidth.m, condeig.m, gls.m, housh.m, linsolve.m, logm.m,
lscov.m, normest.m, normest1.m, ols.m, ordeig.m, planerot.m, qzhess.m, rref.m,
copyfile.m, delete.m, dos.m, fileparts.m, getfield.m, menu.m, mkdir.m,
movefile.m, orderfields.m, publish.m, setfield.m, substruct.m, unix.m,
unpack.m, decic.m, ode23.m, ode23s.m, ode45.m, fminsearch.m, lsqnonneg.m,
pqpnonneg.m, sqp.m, annotation.m, lighting.m, shading.m, area.m, compass.m,
contourc.m, feather.m, fplot.m, hist.m, isocaps.m, isocolors.m, isonormals.m,
isosurface.m, ostreamtube.m, pie.m, pie3.m, reducepatch.m, reducevolume.m,
rose.m, smooth3.m, stairs.m, stem.m, stem3.m, stream2.m, stream3.m,
streamline.m, streamribbon.m, streamtube.m, surfnorm.m, trimesh.m, trisurf.m,
colstyle.m, hgload.m, linkprop.m, meshgrid.m, ndgrid.m, padecoef.m, polyfit.m,
polyval.m, unmkpp.m, profexport.m, ismember.m, unique.m, movfun.m, movslice.m,
periodogram.m, sinc.m, spdiags.m, sprandsym.m, betaincinv.m, ellipke.m,
factor.m, gammainc.m, gammaincinv.m, isprime.m, lcm.m, gallery.m, hadamard.m,
bounds.m, corrcoef.m, discrete_rnd.m, empirical_rnd.m, histc.m, mode.m,
movmad.m, movmax.m, movmean.m, movmedian.m, movmin.m, movprod.m, movstd.m,
movsum.m, movvar.m, ranks.m, runlength.m, zscore.m, str2num.m, strchr.m,
strsplit.m, strtok.m, untabify.m, assert.m, demo.m, example.m, speed.m, test.m,
datenum.m, datevec.m, webread.m, webwrite.m:
Add expected error message <Invalid call> to BIST tests for nargin.
Remove redundant tests for nargin greater than the number of declared inputs
which are now handled by interpreter.
| author | Rik <rik@octave.org> |
|---|---|
| date | Sun, 11 Oct 2020 21:59:35 -0700 |
| parents | de5f2f9a64ff |
| children | 930cb0ae24fd |
| rev | line source |
|---|---|
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1 ######################################################################## |
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2 ## |
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3 ## Copyright (C) 1989-2020 The Octave Project Developers |
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4 ## |
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5 ## See the file COPYRIGHT.md in the top-level directory of this |
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6 ## distribution or <https://octave.org/copyright/>. |
| 16634 | 7 ## |
| 8 ## This file is part of Octave. | |
| 9 ## | |
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10 ## Octave is free software: you can redistribute it and/or modify it |
| 16634 | 11 ## under the terms of the GNU General Public License as published by |
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12 ## the Free Software Foundation, either version 3 of the License, or |
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13 ## (at your option) any later version. |
| 16634 | 14 ## |
| 15 ## Octave is distributed in the hope that it will be useful, but | |
| 16 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
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17 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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18 ## GNU General Public License for more details. |
| 16634 | 19 ## |
| 20 ## You should have received a copy of the GNU General Public License | |
| 21 ## along with Octave; see the file COPYING. If not, see | |
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22 ## <https://www.gnu.org/licenses/>. |
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23 ## |
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24 ######################################################################## |
| 16634 | 25 |
| 26 ## -*- texinfo -*- | |
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27 ## @deftypefn {} {} gallery (@var{name}) |
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28 ## @deftypefnx {} {} gallery (@var{name}, @var{args}) |
| 16634 | 29 ## Create interesting matrices for testing. |
| 30 ## | |
| 31 ## @end deftypefn | |
| 32 ## | |
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33 ## @deftypefn {} {@var{c} =} gallery ("cauchy", @var{x}) |
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34 ## @deftypefnx {} {@var{c} =} gallery ("cauchy", @var{x}, @var{y}) |
| 16634 | 35 ## Create a Cauchy matrix. |
| 36 ## | |
| 37 ## @end deftypefn | |
| 38 ## | |
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39 ## @deftypefn {} {@var{c} =} gallery ("chebspec", @var{n}) |
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40 ## @deftypefnx {} {@var{c} =} gallery ("chebspec", @var{n}, @var{k}) |
| 16634 | 41 ## Create a Chebyshev spectral differentiation matrix. |
| 42 ## | |
| 43 ## @end deftypefn | |
| 44 ## | |
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45 ## @deftypefn {} {@var{c} =} gallery ("chebvand", @var{p}) |
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46 ## @deftypefnx {} {@var{c} =} gallery ("chebvand", @var{m}, @var{p}) |
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47 ## Create a @nospell{Vandermonde}-like matrix for the Chebyshev polynomials. |
| 16634 | 48 ## |
| 49 ## @end deftypefn | |
| 50 ## | |
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51 ## @deftypefn {} {@var{a} =} gallery ("chow", @var{n}) |
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52 ## @deftypefnx {} {@var{a} =} gallery ("chow", @var{n}, @var{alpha}) |
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53 ## @deftypefnx {} {@var{a} =} gallery ("chow", @var{n}, @var{alpha}, @var{delta}) |
| 16634 | 54 ## Create a Chow matrix -- a singular Toeplitz lower Hessenberg matrix. |
| 55 ## | |
| 56 ## @end deftypefn | |
| 57 ## | |
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58 ## @deftypefn {} {@var{c} =} gallery ("circul", @var{v}) |
| 16634 | 59 ## Create a circulant matrix. |
| 60 ## | |
| 61 ## @end deftypefn | |
| 62 ## | |
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63 ## @deftypefn {} {@var{a} =} gallery ("clement", @var{n}) |
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64 ## @deftypefnx {} {@var{a} =} gallery ("clement", @var{n}, @var{k}) |
| 16634 | 65 ## Create a tridiagonal matrix with zero diagonal entries. |
| 66 ## | |
| 67 ## @end deftypefn | |
| 68 ## | |
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69 ## @deftypefn {} {@var{c} =} gallery ("compar", @var{a}) |
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70 ## @deftypefnx {} {@var{c} =} gallery ("compar", @var{a}, @var{k}) |
| 16634 | 71 ## Create a comparison matrix. |
| 72 ## | |
| 73 ## @end deftypefn | |
| 74 ## | |
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75 ## @deftypefn {} {@var{a} =} gallery ("condex", @var{n}) |
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76 ## @deftypefnx {} {@var{a} =} gallery ("condex", @var{n}, @var{k}) |
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77 ## @deftypefnx {} {@var{a} =} gallery ("condex", @var{n}, @var{k}, @var{theta}) |
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78 ## Create a @nospell{"counterexample"} matrix to a condition estimator. |
| 16634 | 79 ## |
| 80 ## @end deftypefn | |
| 81 ## | |
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82 ## @deftypefn {} {@var{a} =} gallery ("cycol", [@var{m} @var{n}]) |
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83 ## @deftypefnx {} {@var{a} =} gallery ("cycol", @var{n}) |
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84 ## @deftypefnx {} {@var{a} =} gallery (@dots{}, @var{k}) |
| 16634 | 85 ## Create a matrix whose columns repeat cyclically. |
| 86 ## | |
| 87 ## @end deftypefn | |
| 88 ## | |
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89 ## @deftypefn {} {[@var{c}, @var{d}, @var{e}] =} gallery ("dorr", @var{n}) |
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90 ## @deftypefnx {} {[@var{c}, @var{d}, @var{e}] =} gallery ("dorr", @var{n}, @var{theta}) |
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91 ## @deftypefnx {} {@var{a} =} gallery ("dorr", @dots{}) |
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92 ## Create a diagonally dominant, ill-conditioned, tridiagonal matrix. |
| 16634 | 93 ## |
| 94 ## @end deftypefn | |
| 95 ## | |
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96 ## @deftypefn {} {@var{a} =} gallery ("dramadah", @var{n}) |
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97 ## @deftypefnx {} {@var{a} =} gallery ("dramadah", @var{n}, @var{k}) |
| 16634 | 98 ## Create a (0, 1) matrix whose inverse has large integer entries. |
| 99 ## | |
| 100 ## @end deftypefn | |
| 101 ## | |
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102 ## @deftypefn {} {@var{a} =} gallery ("fiedler", @var{c}) |
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103 ## Create a symmetric @nospell{Fiedler} matrix. |
| 16634 | 104 ## |
| 105 ## @end deftypefn | |
| 106 ## | |
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107 ## @deftypefn {} {@var{a} =} gallery ("forsythe", @var{n}) |
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108 ## @deftypefnx {} {@var{a} =} gallery ("forsythe", @var{n}, @var{alpha}) |
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109 ## @deftypefnx {} {@var{a} =} gallery ("forsythe", @var{n}, @var{alpha}, @var{lambda}) |
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110 ## Create a @nospell{Forsythe} matrix (a perturbed Jordan block). |
| 16634 | 111 ## |
| 112 ## @end deftypefn | |
| 113 ## | |
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114 ## @deftypefn {} {@var{f} =} gallery ("frank", @var{n}) |
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115 ## @deftypefnx {} {@var{f} =} gallery ("frank", @var{n}, @var{k}) |
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116 ## Create a Frank matrix (ill-conditioned eigenvalues). |
| 16634 | 117 ## |
| 118 ## @end deftypefn | |
| 119 ## | |
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120 ## @deftypefn {} {@var{c} =} gallery ("gcdmat", @var{n}) |
| 16634 | 121 ## Create a greatest common divisor matrix. |
| 122 ## | |
| 123 ## @var{c} is an @var{n}-by-@var{n} matrix whose values correspond to the | |
| 124 ## greatest common divisor of its coordinate values, i.e., @var{c}(i,j) | |
| 125 ## correspond @code{gcd (i, j)}. | |
| 126 ## @end deftypefn | |
| 127 ## | |
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128 ## @deftypefn {} {@var{a} =} gallery ("gearmat", @var{n}) |
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129 ## @deftypefnx {} {@var{a} =} gallery ("gearmat", @var{n}, @var{i}) |
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130 ## @deftypefnx {} {@var{a} =} gallery ("gearmat", @var{n}, @var{i}, @var{j}) |
| 16634 | 131 ## Create a Gear matrix. |
| 132 ## | |
| 133 ## @end deftypefn | |
| 134 ## | |
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135 ## @deftypefn {} {@var{g} =} gallery ("grcar", @var{n}) |
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136 ## @deftypefnx {} {@var{g} =} gallery ("grcar", @var{n}, @var{k}) |
| 16634 | 137 ## Create a Toeplitz matrix with sensitive eigenvalues. |
| 138 ## | |
| 139 ## @end deftypefn | |
| 140 ## | |
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141 ## @deftypefn {} {@var{a} =} gallery ("hanowa", @var{n}) |
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142 ## @deftypefnx {} {@var{a} =} gallery ("hanowa", @var{n}, @var{d}) |
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143 ## Create a matrix whose eigenvalues lie on a vertical line in the complex |
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144 ## plane. |
| 16634 | 145 ## |
| 146 ## @end deftypefn | |
| 147 ## | |
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148 ## @deftypefn {} {@var{v} =} gallery ("house", @var{x}) |
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149 ## @deftypefnx {} {[@var{v}, @var{beta}] =} gallery ("house", @var{x}) |
| 16634 | 150 ## Create a householder matrix. |
| 151 ## | |
| 152 ## @end deftypefn | |
| 153 ## | |
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154 ## @deftypefn {} {@var{a} =} gallery ("integerdata", @var{imax}, [@var{M} @var{N} @dots{}], @var{j}) |
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155 ## @deftypefnx {} {@var{a} =} gallery ("integerdata", @var{imax}, @var{M}, @var{N}, @dots{}, @var{j}) |
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156 ## @deftypefnx {} {@var{a} =} gallery ("integerdata", [@var{imin}, @var{imax}], [@var{M} @var{N} @dots{}], @var{j}) |
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157 ## @deftypefnx {} {@var{a} =} gallery ("integerdata", [@var{imin}, @var{imax}], @var{M}, @var{N}, @dots{}, @var{j}) |
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158 ## @deftypefnx {} {@var{a} =} gallery ("integerdata", @dots{}, "@var{class}") |
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159 ## Create a matrix with random integers in the range [1, @var{imax}]. |
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160 ## If @var{imin} is given then the integers are in the range |
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161 ## [@var{imin}, @var{imax}]. |
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162 ## |
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163 ## The second input is a matrix of dimensions describing the size of the |
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164 ## output. The dimensions can also be input as comma-separated arguments. |
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165 ## |
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166 ## The input @var{j} is an integer index in the range [0, 2^32-1]. The values |
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167 ## of the output matrix are always exactly the same (reproducibility) for a |
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168 ## given size input and @var{j} index. |
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169 ## |
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170 ## The final optional argument determines the class of the resulting matrix. |
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171 ## Possible values for @var{class}: @qcode{"uint8"}, @qcode{"uint16"}, |
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172 ## @qcode{"uint32"}, @qcode{"int8"}, @qcode{"int16"}, int32", @qcode{"single"}, |
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173 ## @qcode{"double"}. The default is @qcode{"double"}. |
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174 ## |
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175 ## @end deftypefn |
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176 ## |
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177 ## @deftypefn {} {@var{a} =} gallery ("invhess", @var{x}) |
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178 ## @deftypefnx {} {@var{a} =} gallery ("invhess", @var{x}, @var{y}) |
| 16634 | 179 ## Create the inverse of an upper Hessenberg matrix. |
| 180 ## | |
| 181 ## @end deftypefn | |
| 182 ## | |
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183 ## @deftypefn {} {@var{a} =} gallery ("invol", @var{n}) |
| 16634 | 184 ## Create an involutory matrix. |
| 185 ## | |
| 186 ## @end deftypefn | |
| 187 ## | |
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188 ## @deftypefn {} {@var{a} =} gallery ("ipjfact", @var{n}) |
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189 ## @deftypefnx {} {@var{a} =} gallery ("ipjfact", @var{n}, @var{k}) |
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190 ## Create a Hankel matrix with factorial elements. |
| 16634 | 191 ## |
| 192 ## @end deftypefn | |
| 193 ## | |
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194 ## @deftypefn {} {@var{a} =} gallery ("jordbloc", @var{n}) |
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195 ## @deftypefnx {} {@var{a} =} gallery ("jordbloc", @var{n}, @var{lambda}) |
| 16634 | 196 ## Create a Jordan block. |
| 197 ## | |
| 198 ## @end deftypefn | |
| 199 ## | |
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200 ## @deftypefn {} {@var{u} =} gallery ("kahan", @var{n}) |
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201 ## @deftypefnx {} {@var{u} =} gallery ("kahan", @var{n}, @var{theta}) |
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202 ## @deftypefnx {} {@var{u} =} gallery ("kahan", @var{n}, @var{theta}, @var{pert}) |
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203 ## Create a @nospell{Kahan} matrix (upper trapezoidal). |
| 16634 | 204 ## |
| 205 ## @end deftypefn | |
| 206 ## | |
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207 ## @deftypefn {} {@var{a} =} gallery ("kms", @var{n}) |
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208 ## @deftypefnx {} {@var{a} =} gallery ("kms", @var{n}, @var{rho}) |
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209 ## Create a @nospell{Kac-Murdock-Szego} Toeplitz matrix. |
| 16634 | 210 ## |
| 211 ## @end deftypefn | |
| 212 ## | |
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213 ## @deftypefn {} {@var{b} =} gallery ("krylov", @var{a}) |
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214 ## @deftypefnx {} {@var{b} =} gallery ("krylov", @var{a}, @var{x}) |
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215 ## @deftypefnx {} {@var{b} =} gallery ("krylov", @var{a}, @var{x}, @var{j}) |
| 16634 | 216 ## Create a Krylov matrix. |
| 217 ## | |
| 218 ## @end deftypefn | |
| 219 ## | |
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220 ## @deftypefn {} {@var{a} =} gallery ("lauchli", @var{n}) |
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221 ## @deftypefnx {} {@var{a} =} gallery ("lauchli", @var{n}, @var{mu}) |
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222 ## Create a @nospell{Lauchli} matrix (rectangular). |
| 16634 | 223 ## |
| 224 ## @end deftypefn | |
| 225 ## | |
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226 ## @deftypefn {} {@var{a} =} gallery ("lehmer", @var{n}) |
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227 ## Create a @nospell{Lehmer} matrix (symmetric positive definite). |
| 16634 | 228 ## |
| 229 ## @end deftypefn | |
| 230 ## | |
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231 ## @deftypefn {} {@var{t} =} gallery ("lesp", @var{n}) |
| 16634 | 232 ## Create a tridiagonal matrix with real, sensitive eigenvalues. |
| 233 ## | |
| 234 ## @end deftypefn | |
| 235 ## | |
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236 ## @deftypefn {} {@var{a} =} gallery ("lotkin", @var{n}) |
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237 ## Create a @nospell{Lotkin} matrix. |
| 16634 | 238 ## |
| 239 ## @end deftypefn | |
| 240 ## | |
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241 ## @deftypefn {} {@var{a} =} gallery ("minij", @var{n}) |
| 16634 | 242 ## Create a symmetric positive definite matrix MIN(i,j). |
| 243 ## | |
| 244 ## @end deftypefn | |
| 245 ## | |
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246 ## @deftypefn {} {@var{a} =} gallery ("moler", @var{n}) |
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247 ## @deftypefnx {} {@var{a} =} gallery ("moler", @var{n}, @var{alpha}) |
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248 ## Create a @nospell{Moler} matrix (symmetric positive definite). |
| 16634 | 249 ## |
| 250 ## @end deftypefn | |
| 251 ## | |
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252 ## @deftypefn {} {[@var{a}, @var{t}] =} gallery ("neumann", @var{n}) |
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253 ## Create a singular matrix from the discrete @nospell{Neumann} problem |
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254 ## (sparse). |
| 16634 | 255 ## |
| 256 ## @end deftypefn | |
| 257 ## | |
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258 ## @deftypefn {} {@var{a} =} gallery ("normaldata", [@var{M} @var{N} @dots{}], @var{j}) |
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259 ## @deftypefnx {} {@var{a} =} gallery ("normaldata", @var{M}, @var{N}, @dots{}, @var{j}) |
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260 ## @deftypefnx {} {@var{a} =} gallery ("normaldata", @dots{}, "@var{class}") |
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261 ## Create a matrix with random samples from the standard normal distribution |
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262 ## (mean = 0, std = 1). |
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263 ## |
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264 ## The first input is a matrix of dimensions describing the size of the output. |
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265 ## The dimensions can also be input as comma-separated arguments. |
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266 ## |
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267 ## The input @var{j} is an integer index in the range [0, 2^32-1]. The values |
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268 ## of the output matrix are always exactly the same (reproducibility) for a |
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269 ## given size input and @var{j} index. |
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270 ## |
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271 ## The final optional argument determines the class of the resulting matrix. |
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272 ## Possible values for @var{class}: @qcode{"single"}, @qcode{"double"}. |
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273 ## The default is @qcode{"double"}. |
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274 ## |
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275 ## @end deftypefn |
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276 ## |
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277 ## @deftypefn {} {@var{q} =} gallery ("orthog", @var{n}) |
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278 ## @deftypefnx {} {@var{q} =} gallery ("orthog", @var{n}, @var{k}) |
| 16634 | 279 ## Create orthogonal and nearly orthogonal matrices. |
| 280 ## | |
| 281 ## @end deftypefn | |
| 282 ## | |
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283 ## @deftypefn {} {@var{a} =} gallery ("parter", @var{n}) |
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284 ## Create a @nospell{Parter} matrix (a Toeplitz matrix with singular values |
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285 ## near pi). |
| 16634 | 286 ## |
| 287 ## @end deftypefn | |
| 288 ## | |
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289 ## @deftypefn {} {@var{p} =} gallery ("pei", @var{n}) |
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290 ## @deftypefnx {} {@var{p} =} gallery ("pei", @var{n}, @var{alpha}) |
| 16634 | 291 ## Create a Pei matrix. |
| 292 ## | |
| 293 ## @end deftypefn | |
| 294 ## | |
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295 ## @deftypefn {} {@var{a} =} gallery ("poisson", @var{n}) |
| 16634 | 296 ## Create a block tridiagonal matrix from Poisson's equation (sparse). |
| 297 ## | |
| 298 ## @end deftypefn | |
| 299 ## | |
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300 ## @deftypefn {} {@var{a} =} gallery ("prolate", @var{n}) |
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301 ## @deftypefnx {} {@var{a} =} gallery ("prolate", @var{n}, @var{w}) |
| 16634 | 302 ## Create a prolate matrix (symmetric, ill-conditioned Toeplitz matrix). |
| 303 ## | |
| 304 ## @end deftypefn | |
| 305 ## | |
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306 ## @deftypefn {} {@var{h} =} gallery ("randhess", @var{x}) |
| 16634 | 307 ## Create a random, orthogonal upper Hessenberg matrix. |
| 308 ## | |
| 309 ## @end deftypefn | |
| 310 ## | |
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311 ## @deftypefn {} {@var{a} =} gallery ("rando", @var{n}) |
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312 ## @deftypefnx {} {@var{a} =} gallery ("rando", @var{n}, @var{k}) |
| 16634 | 313 ## Create a random matrix with elements -1, 0 or 1. |
| 314 ## | |
| 315 ## @end deftypefn | |
| 316 ## | |
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317 ## @deftypefn {} {@var{a} =} gallery ("randsvd", @var{n}) |
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318 ## @deftypefnx {} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa}) |
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319 ## @deftypefnx {} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa}, @var{mode}) |
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320 ## @deftypefnx {} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa}, @var{mode}, @var{kl}) |
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321 ## @deftypefnx {} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa}, @var{mode}, @var{kl}, @var{ku}) |
| 16634 | 322 ## Create a random matrix with pre-assigned singular values. |
| 323 ## | |
| 324 ## @end deftypefn | |
| 325 ## | |
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326 ## @deftypefn {} {@var{a} =} gallery ("redheff", @var{n}) |
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327 ## Create a zero and ones matrix of @nospell{Redheffer} associated with the |
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328 ## Riemann hypothesis. |
| 16634 | 329 ## |
| 330 ## @end deftypefn | |
| 331 ## | |
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332 ## @deftypefn {} {@var{a} =} gallery ("riemann", @var{n}) |
| 16634 | 333 ## Create a matrix associated with the Riemann hypothesis. |
| 334 ## | |
| 335 ## @end deftypefn | |
| 336 ## | |
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337 ## @deftypefn {} {@var{a} =} gallery ("ris", @var{n}) |
| 16634 | 338 ## Create a symmetric Hankel matrix. |
| 339 ## | |
| 340 ## @end deftypefn | |
| 341 ## | |
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342 ## @deftypefn {} {@var{a} =} gallery ("smoke", @var{n}) |
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343 ## @deftypefnx {} {@var{a} =} gallery ("smoke", @var{n}, @var{k}) |
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344 ## Create a complex matrix, with a @nospell{"smoke ring"} pseudospectrum. |
| 16634 | 345 ## |
| 346 ## @end deftypefn | |
| 347 ## | |
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348 ## @deftypefn {} {@var{t} =} gallery ("toeppd", @var{n}) |
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349 ## @deftypefnx {} {@var{t} =} gallery ("toeppd", @var{n}, @var{m}) |
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350 ## @deftypefnx {} {@var{t} =} gallery ("toeppd", @var{n}, @var{m}, @var{w}) |
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351 ## @deftypefnx {} {@var{t} =} gallery ("toeppd", @var{n}, @var{m}, @var{w}, @var{theta}) |
| 16634 | 352 ## Create a symmetric positive definite Toeplitz matrix. |
| 353 ## | |
| 354 ## @end deftypefn | |
| 355 ## | |
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356 ## @deftypefn {} {@var{p} =} gallery ("toeppen", @var{n}) |
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357 ## @deftypefnx {} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}) |
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358 ## @deftypefnx {} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b}) |
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359 ## @deftypefnx {} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b}, @var{c}) |
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360 ## @deftypefnx {} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b}, @var{c}, @var{d}) |
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361 ## @deftypefnx {} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b}, @var{c}, @var{d}, @var{e}) |
| 16634 | 362 ## Create a pentadiagonal Toeplitz matrix (sparse). |
| 363 ## | |
| 364 ## @end deftypefn | |
| 365 ## | |
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366 ## @deftypefn {} {@var{a} =} gallery ("tridiag", @var{x}, @var{y}, @var{z}) |
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367 ## @deftypefnx {} {@var{a} =} gallery ("tridiag", @var{n}) |
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368 ## @deftypefnx {} {@var{a} =} gallery ("tridiag", @var{n}, @var{c}, @var{d}, @var{e}) |
| 16634 | 369 ## Create a tridiagonal matrix (sparse). |
| 370 ## | |
| 371 ## @end deftypefn | |
| 372 ## | |
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373 ## @deftypefn {} {@var{t} =} gallery ("triw", @var{n}) |
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374 ## @deftypefnx {} {@var{t} =} gallery ("triw", @var{n}, @var{alpha}) |
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375 ## @deftypefnx {} {@var{t} =} gallery ("triw", @var{n}, @var{alpha}, @var{k}) |
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376 ## Create an upper triangular matrix discussed by |
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377 ## @nospell{Kahan, Golub, and Wilkinson}. |
| 16634 | 378 ## |
| 379 ## @end deftypefn | |
| 380 ## | |
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381 ## @deftypefn {} {@var{a} =} gallery ("uniformdata", [@var{M} @var{N} @dots{}], @var{j}) |
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382 ## @deftypefnx {} {@var{a} =} gallery ("uniformdata", @var{M}, @var{N}, @dots{}, @var{j}) |
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383 ## @deftypefnx {} {@var{a} =} gallery ("uniformdata", @dots{}, "@var{class}") |
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384 ## Create a matrix with random samples from the standard uniform distribution |
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385 ## (range [0,1]). |
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386 ## |
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387 ## The first input is a matrix of dimensions describing the size of the output. |
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388 ## The dimensions can also be input as comma-separated arguments. |
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389 ## |
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390 ## The input @var{j} is an integer index in the range [0, 2^32-1]. The values |
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391 ## of the output matrix are always exactly the same (reproducibility) for a |
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392 ## given size input and @var{j} index. |
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393 ## |
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394 ## The final optional argument determines the class of the resulting matrix. |
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395 ## Possible values for @var{class}: @qcode{"single"}, @qcode{"double"}. |
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396 ## The default is @qcode{"double"}. |
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397 ## |
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398 ## @end deftypefn |
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399 ## |
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400 ## @deftypefn {} {@var{a} =} gallery ("wathen", @var{nx}, @var{ny}) |
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401 ## @deftypefnx {} {@var{a} =} gallery ("wathen", @var{nx}, @var{ny}, @var{k}) |
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402 ## Create the @nospell{Wathen} matrix. |
| 16634 | 403 ## |
| 404 ## @end deftypefn | |
| 405 ## | |
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406 ## @deftypefn {} {[@var{a}, @var{b}] =} gallery ("wilk", @var{n}) |
| 16634 | 407 ## Create various specific matrices devised/discussed by Wilkinson. |
| 408 ## | |
| 409 ## @end deftypefn | |
| 410 | |
| 411 ## Code for most of the individual matrices (except binomial, gcdmat, | |
| 412 ## integerdata, leslie, normaldata, randcolu, randcorr, randjorth, sampling, | |
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413 ## uniformdata) by Nicholas J. Higham <Nicholas.J.Higham@manchester.ac.uk> |
| 16634 | 414 ## Adapted for Octave and into single gallery function by Carnë Draug |
| 415 | |
| 416 function [varargout] = gallery (name, varargin) | |
| 417 | |
| 418 if (nargin < 1) | |
| 419 print_usage (); | |
| 420 elseif (! ischar (name)) | |
| 421 error ("gallery: NAME must be a string."); | |
| 422 endif | |
| 423 | |
| 424 ## NOTE: there isn't a lot of input check in the individual functions | |
| 425 ## that actually build the functions. This is by design. The original | |
| 426 ## code by Higham did not perform it and was propagated to Matlab, so | |
| 427 ## for compatibility, we also don't make it. For example, arguments | |
| 428 ## that behave as switches, and in theory accepting a value of 0 or 1, | |
| 429 ## will use a value of 0, for any value other than 1 (only check made | |
| 430 ## is if the value is equal to 1). It will often also accept string | |
| 431 ## values instead of numeric. Only input check added was where it | |
| 432 ## would be causing an error anyway. | |
| 433 | |
| 434 ## we will always want to return at least 1 output | |
| 435 n_out = nargout; | |
| 436 if (n_out == 0) | |
| 437 n_out = 1; | |
| 438 endif | |
| 439 | |
| 440 switch (tolower (name)) | |
| 441 case "binomial" | |
| 442 error ("gallery: matrix %s not implemented.", name); | |
| 443 case "cauchy" , [varargout{1:n_out}] = cauchy (varargin{:}); | |
| 444 case "chebspec" , [varargout{1:n_out}] = chebspec (varargin{:}); | |
| 445 case "chebvand" , [varargout{1:n_out}] = chebvand (varargin{:}); | |
| 446 case "chow" , [varargout{1:n_out}] = chow (varargin{:}); | |
| 447 case "circul" , [varargout{1:n_out}] = circul (varargin{:}); | |
| 448 case "clement" , [varargout{1:n_out}] = clement (varargin{:}); | |
| 449 case "compar" , [varargout{1:n_out}] = compar (varargin{:}); | |
| 450 case "condex" , [varargout{1:n_out}] = condex (varargin{:}); | |
| 451 case "cycol" , [varargout{1:n_out}] = cycol (varargin{:}); | |
| 452 case "dorr" , [varargout{1:n_out}] = dorr (varargin{:}); | |
| 453 case "dramadah" , [varargout{1:n_out}] = dramadah (varargin{:}); | |
| 454 case "fiedler" , [varargout{1:n_out}] = fiedler (varargin{:}); | |
| 455 case "forsythe" , [varargout{1:n_out}] = forsythe (varargin{:}); | |
| 456 case "frank" , [varargout{1:n_out}] = frank (varargin{:}); | |
| 457 case "gearmat" , [varargout{1:n_out}] = gearmat (varargin{:}); | |
| 458 case "gcdmat" , [varargout{1:n_out}] = gcdmat (varargin{:}); | |
| 459 case "grcar" , [varargout{1:n_out}] = grcar (varargin{:}); | |
| 460 case "hanowa" , [varargout{1:n_out}] = hanowa (varargin{:}); | |
| 461 case "house" , [varargout{1:n_out}] = house (varargin{:}); | |
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462 case "integerdata", [varargout{1:n_out}] = integerdata (varargin{:}); |
| 16634 | 463 case "invhess" , [varargout{1:n_out}] = invhess (varargin{:}); |
| 464 case "invol" , [varargout{1:n_out}] = invol (varargin{:}); | |
| 465 case "ipjfact" , [varargout{1:n_out}] = ipjfact (varargin{:}); | |
| 466 case "jordbloc" , [varargout{1:n_out}] = jordbloc (varargin{:}); | |
| 467 case "kahan" , [varargout{1:n_out}] = kahan (varargin{:}); | |
| 468 case "kms" , [varargout{1:n_out}] = kms (varargin{:}); | |
| 469 case "krylov" , [varargout{1:n_out}] = krylov (varargin{:}); | |
| 470 case "lauchli" , [varargout{1:n_out}] = lauchli (varargin{:}); | |
| 471 case "lehmer" , [varargout{1:n_out}] = lehmer (varargin{:}); | |
| 472 case "leslie" | |
| 473 error ("gallery: matrix %s not implemented.", name); | |
| 474 case "lesp" , [varargout{1:n_out}] = lesp (varargin{:}); | |
| 475 case "lotkin" , [varargout{1:n_out}] = lotkin (varargin{:}); | |
| 476 case "minij" , [varargout{1:n_out}] = minij (varargin{:}); | |
| 477 case "moler" , [varargout{1:n_out}] = moler (varargin{:}); | |
| 478 case "neumann" , [varargout{1:n_out}] = neumann (varargin{:}); | |
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479 case "normaldata" , [varargout{1:n_out}] = normaldata (varargin{:}); |
| 16634 | 480 case "orthog" , [varargout{1:n_out}] = orthog (varargin{:}); |
| 481 case "parter" , [varargout{1:n_out}] = parter (varargin{:}); | |
| 482 case "pei" , [varargout{1:n_out}] = pei (varargin{:}); | |
| 483 case "poisson" , [varargout{1:n_out}] = poisson (varargin{:}); | |
| 484 case "prolate" , [varargout{1:n_out}] = prolate (varargin{:}); | |
| 485 case "randcolu" | |
| 486 error ("gallery: matrix %s not implemented.", name); | |
| 487 case "randcorr" | |
| 488 error ("gallery: matrix %s not implemented.", name); | |
| 489 case "randhess" , [varargout{1:n_out}] = randhess (varargin{:}); | |
| 490 case "randjorth" | |
| 491 error ("gallery: matrix %s not implemented.", name); | |
| 492 case "rando" , [varargout{1:n_out}] = rando (varargin{:}); | |
| 493 case "randsvd" , [varargout{1:n_out}] = randsvd (varargin{:}); | |
| 494 case "redheff" , [varargout{1:n_out}] = redheff (varargin{:}); | |
| 495 case "riemann" , [varargout{1:n_out}] = riemann (varargin{:}); | |
| 496 case "ris" , [varargout{1:n_out}] = ris (varargin{:}); | |
| 497 case "sampling" | |
| 498 error ("gallery: matrix %s not implemented.", name); | |
| 499 case "smoke" , [varargout{1:n_out}] = smoke (varargin{:}); | |
| 500 case "toeppd" , [varargout{1:n_out}] = toeppd (varargin{:}); | |
| 501 case "toeppen" , [varargout{1:n_out}] = toeppen (varargin{:}); | |
| 502 case "tridiag" , [varargout{1:n_out}] = tridiag (varargin{:}); | |
| 503 case "triw" , [varargout{1:n_out}] = triw (varargin{:}); | |
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504 case "uniformdata" , [varargout{1:n_out}] = uniformdata (varargin{:}); |
| 16634 | 505 case "wathen" , [varargout{1:n_out}] = wathen (varargin{:}); |
| 506 case "wilk" , [varargout{1:n_out}] = wilk (varargin{:}); | |
| 507 otherwise | |
| 508 error ("gallery: unknown matrix with NAME %s", name); | |
| 509 endswitch | |
| 510 | |
| 511 endfunction | |
| 512 | |
| 513 function C = cauchy (x, y) | |
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514 ## CAUCHY Cauchy matrix. |
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515 ## C = CAUCHY(X, Y), where X, Y are N-vectors, is the N-by-N matrix |
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516 ## with C(i,j) = 1/(X(i)+Y(j)). By default, Y = X. |
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517 ## Special case: if X is a scalar CAUCHY(X) is the same as CAUCHY(1:X). |
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518 ## Explicit formulas are known for DET(C) (which is nonzero if X and Y |
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519 ## both have distinct elements) and the elements of INV(C). |
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520 ## C is totally positive if 0 < X(1) < ... < X(N) and |
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521 ## 0 < Y(1) < ... < Y(N). |
| 16634 | 522 ## |
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523 ## References: |
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524 ## N.J. Higham, Accuracy and Stability of Numerical Algorithms, |
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525 ## Society for Industrial and Applied Mathematics, Philadelphia, PA, |
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526 ## USA, 1996; sec. 26.1. |
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527 ## D.E. Knuth, The Art of Computer Programming, Volume 1, |
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528 ## Fundamental Algorithms, second edition, Addison-Wesley, Reading, |
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529 ## Massachusetts, 1973, p. 36. |
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530 ## E.E. Tyrtyshnikov, Cauchy-Toeplitz matrices and some applications, |
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531 ## Linear Algebra and Appl., 149 (1991), pp. 1-18. |
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532 ## O. Taussky and M. Marcus, Eigenvalues of finite matrices, in |
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533 ## Survey of Numerical Analysis, J. Todd, ed., McGraw-Hill, New York, |
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534 ## pp. 279-313, 1962. (States the totally positive property on p. 295.) |
| 16634 | 535 |
| 536 if (nargin < 1 || nargin > 2) | |
| 537 error ("gallery: 1 or 2 arguments are required for cauchy matrix."); | |
| 538 elseif (! isnumeric (x)) | |
| 539 error ("gallery: X must be numeric for cauchy matrix."); | |
| 540 elseif (nargin == 2 && ! isnumeric (y)) | |
| 541 error ("gallery: Y must be numeric for cauchy matrix."); | |
| 542 endif | |
| 543 | |
| 544 n = numel (x); | |
| 545 if (isscalar (x) && fix (x) == x) | |
| 546 n = x; | |
| 547 x = 1:n; | |
| 548 elseif (n > 1 && isvector (x)) | |
| 549 ## do nothing | |
| 550 else | |
| 551 error ("gallery: X be an integer or a vector for cauchy matrix."); | |
| 552 endif | |
| 553 | |
| 554 if (nargin == 1) | |
| 555 y = x; | |
| 556 endif | |
| 557 | |
| 558 ## Ensure x and y are column vectors | |
| 559 x = x(:); | |
| 560 y = y(:); | |
| 561 if (numel (x) != numel (y)) | |
| 562 error ("gallery: X and Y must be vectors of same length for cauchy matrix."); | |
| 563 endif | |
| 564 | |
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565 C = 1 ./ (x .+ y.'); |
| 16634 | 566 endfunction |
| 567 | |
| 568 function C = chebspec (n, k = 0) | |
| 569 ## CHEBSPEC Chebyshev spectral differentiation matrix. | |
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570 ## C = CHEBSPEC(N, K) is a Chebyshev spectral differentiation |
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571 ## matrix of order N. K = 0 (the default) or 1. |
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572 ## For K = 0 ('no boundary conditions'), C is nilpotent, with |
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573 ## C^N = 0 and it has the null vector ONES(N,1). |
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574 ## C is similar to a Jordan block of size N with eigenvalue zero. |
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575 ## For K = 1, C is nonsingular and well-conditioned, and its eigenvalues |
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576 ## have negative real parts. |
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577 ## For both K, the computed eigenvector matrix X from EIG is |
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578 ## ill-conditioned (MESH(REAL(X)) is interesting). |
| 16634 | 579 ## |
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580 ## References: |
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581 ## C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral |
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582 ## Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988; p. 69. |
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583 ## L.N. Trefethen and M.R. Trummer, An instability phenomenon in |
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584 ## spectral methods, SIAM J. Numer. Anal., 24 (1987), pp. 1008-1023. |
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585 ## D. Funaro, Computing the inverse of the Chebyshev collocation |
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586 ## derivative, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 1050-1057. |
| 16634 | 587 |
| 588 if (nargin < 1 || nargin > 2) | |
| 589 error ("gallery: 1 to 2 arguments are required for chebspec matrix."); | |
| 590 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 591 error ("gallery: N must be an integer for chebspec matrix."); | |
| 592 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 593 error ("gallery: K must be a scalar for chebspec matrix."); | |
| 594 endif | |
| 595 | |
| 596 ## k = 1 case obtained from k = 0 case with one bigger n. | |
| 597 switch (k) | |
| 598 case (0), # do nothing | |
| 599 case (1), n = n + 1; | |
| 600 otherwise | |
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601 error ("gallery: K should be either 0 or 1 for chebspec matrix."); |
| 16634 | 602 endswitch |
| 603 | |
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604 n -= 1; |
| 16634 | 605 C = zeros (n+1); |
| 606 | |
| 607 one = ones (n+1, 1); | |
| 608 x = cos ((0:n)' * (pi/n)); | |
| 609 d = ones (n+1, 1); | |
| 610 d(1) = 2; | |
| 611 d(n+1) = 2; | |
| 612 | |
| 613 ## eye(size(C)) on next line avoids div by zero. | |
| 614 C = (d * (one./d)') ./ (x*one'-one*x' + eye (size (C))); | |
| 615 | |
| 616 ## Now fix diagonal and signs. | |
| 617 C(1,1) = (2*n^2+1)/6; | |
| 618 for i = 2:n+1 | |
| 619 if (rem (i, 2) == 0) | |
| 620 C(:,i) = -C(:,i); | |
| 621 C(i,:) = -C(i,:); | |
| 622 endif | |
| 623 if (i < n+1) | |
| 624 C(i,i) = -x(i)/(2*(1-x(i)^2)); | |
| 625 else | |
| 626 C(n+1,n+1) = -C(1,1); | |
| 627 endif | |
| 628 endfor | |
| 629 | |
| 630 if (k == 1) | |
| 631 C = C(2:n+1,2:n+1); | |
| 632 endif | |
| 633 endfunction | |
| 634 | |
| 635 function C = chebvand (m, p) | |
| 636 ## CHEBVAND Vandermonde-like matrix for the Chebyshev polynomials. | |
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637 ## C = CHEBVAND(P), where P is a vector, produces the (primal) |
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638 ## Chebyshev Vandermonde matrix based on the points P, |
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639 ## i.e., C(i,j) = T_{i-1}(P(j)), where T_{i-1} is the Chebyshev |
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640 ## polynomial of degree i-1. |
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641 ## CHEBVAND(M,P) is a rectangular version of CHEBVAND(P) with M rows. |
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642 ## Special case: If P is a scalar then P equally spaced points on |
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643 ## [0,1] are used. |
| 16634 | 644 ## |
|
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645 ## Reference: |
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646 ## N.J. Higham, Stability analysis of algorithms for solving confluent |
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647 ## Vandermonde-like systems, SIAM J. Matrix Anal. Appl., 11 (1990), |
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648 ## pp. 23-41. |
| 16634 | 649 |
| 650 if (nargin < 1 || nargin > 2) | |
| 651 error ("gallery: 1 or 2 arguments are required for chebvand matrix."); | |
| 652 endif | |
| 653 | |
| 654 ## because the order of the arguments changes if nargin is 1 or 2 ... | |
| 655 | |
| 656 if (nargin == 1) | |
| 657 p = m; | |
| 658 endif | |
| 659 | |
| 660 n = numel (p); | |
| 661 if (! isnumeric (p)) | |
| 662 error ("gallery: P must be numeric for chebvand matrix."); | |
| 663 elseif (isscalar (p) && fix (p) == p) | |
| 664 n = p; | |
| 665 p = linspace (0, 1, n); | |
| 666 elseif (n > 1 && isvector (p)) | |
| 667 ## do nothing | |
| 668 endif | |
| 669 p = p(:).'; # Ensure p is a row vector. | |
| 670 | |
| 671 if (nargin == 1) | |
| 672 m = n; | |
| 673 elseif (! isnumeric (m) || ! isscalar (m)) | |
| 674 error ("gallery: M must be a scalar for chebvand matrix."); | |
| 675 endif | |
| 676 | |
| 677 C = ones (m, n); | |
| 678 if (m != 1) | |
| 679 C(2,:) = p; | |
| 680 ## Use Chebyshev polynomial recurrence. | |
| 681 for i = 3:m | |
| 682 C(i,:) = 2.*p.*C(i-1,:) - C(i-2,:); | |
| 683 endfor | |
| 684 endif | |
| 685 endfunction | |
| 686 | |
| 687 function A = chow (n, alpha = 1, delta = 0) | |
| 688 ## CHOW Chow matrix - a singular Toeplitz lower Hessenberg matrix. | |
|
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689 ## A = CHOW(N, ALPHA, DELTA) is a Toeplitz lower Hessenberg matrix |
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690 ## A = H(ALPHA) + DELTA*EYE, where H(i,j) = ALPHA^(i-j+1). |
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691 ## H(ALPHA) has p = FLOOR(N/2) zero eigenvalues, the rest being |
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|
692 ## 4*ALPHA*COS( k*PI/(N+2) )^2, k=1:N-p. |
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693 ## Defaults: ALPHA = 1, DELTA = 0. |
| 16634 | 694 ## |
|
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695 ## References: |
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696 ## T.S. Chow, A class of Hessenberg matrices with known |
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697 ## eigenvalues and inverses, SIAM Review, 11 (1969), pp. 391-395. |
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698 ## G. Fairweather, On the eigenvalues and eigenvectors of a class of |
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|
699 ## Hessenberg matrices, SIAM Review, 13 (1971), pp. 220-221. |
| 16634 | 700 |
| 701 if (nargin < 1 || nargin > 3) | |
| 702 error ("gallery: 1 to 3 arguments are required for chow matrix."); | |
| 703 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 704 error ("gallery: N must be an integer for chow matrix."); | |
| 705 elseif (! isnumeric (alpha) || ! isscalar (alpha)) | |
| 706 error ("gallery: ALPHA must be a scalar for chow matrix."); | |
| 707 elseif (! isnumeric (delta) || ! isscalar (delta)) | |
| 708 error ("gallery: DELTA must be a scalar for chow matrix."); | |
| 709 endif | |
| 710 | |
| 711 A = toeplitz (alpha.^(1:n), [alpha 1 zeros(1, n-2)]) + delta * eye (n); | |
| 712 endfunction | |
| 713 | |
| 714 function C = circul (v) | |
| 715 ## CIRCUL Circulant matrix. | |
|
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716 ## C = CIRCUL(V) is the circulant matrix whose first row is V. |
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717 ## (A circulant matrix has the property that each row is obtained |
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718 ## from the previous one by cyclically permuting the entries one step |
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719 ## forward; it is a special Toeplitz matrix in which the diagonals |
|
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|
720 ## 'wrap round'.) |
|
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721 ## Special case: if V is a scalar then C = CIRCUL(1:V). |
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722 ## The eigensystem of C (N-by-N) is known explicitly. If t is an Nth |
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723 ## root of unity, then the inner product of V with W = [1 t t^2 ... t^N] |
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724 ## is an eigenvalue of C, and W(N:-1:1) is an eigenvector of C. |
| 16634 | 725 ## |
|
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726 ## Reference: |
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|
727 ## P.J. Davis, Circulant Matrices, John Wiley, 1977. |
| 16634 | 728 |
| 729 if (nargin != 1) | |
| 730 error ("gallery: 1 argument is required for circul matrix."); | |
| 731 elseif (! isnumeric (v)) | |
| 732 error ("gallery: V must be numeric for circul matrix."); | |
| 733 endif | |
| 734 | |
|
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735 n = numel (v); |
|
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|
736 if (isscalar (v) && fix (v) == v) |
| 16634 | 737 n = v; |
| 738 v = 1:n; | |
|
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|
739 elseif (n > 1 && isvector (v)) |
| 16634 | 740 ## do nothing |
| 741 else | |
| 742 error ("gallery: X must be a scalar or a vector for circul matrix."); | |
| 743 endif | |
| 744 | |
| 745 v = v(:).'; # Make sure v is a row vector | |
| 746 C = toeplitz ([v(1) v(n:-1:2)], v); | |
| 747 endfunction | |
| 748 | |
| 749 function A = clement (n, k = 0) | |
| 750 ## CLEMENT Clement matrix - tridiagonal with zero diagonal entries. | |
|
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751 ## CLEMENT(N, K) is a tridiagonal matrix with zero diagonal entries |
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752 ## and known eigenvalues. It is singular if N is odd. About 64 |
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753 ## percent of the entries of the inverse are zero. The eigenvalues |
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754 ## are plus and minus the numbers N-1, N-3, N-5, ..., (1 or 0). |
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755 ## For K = 0 (the default) the matrix is unsymmetric, while for |
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756 ## K = 1 it is symmetric. |
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757 ## CLEMENT(N, 1) is diagonally similar to CLEMENT(N). |
| 16634 | 758 ## |
|
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759 ## Similar properties hold for TRIDIAG(X,Y,Z) where Y = ZEROS(N,1). |
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760 ## The eigenvalues still come in plus/minus pairs but they are not |
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761 ## known explicitly. |
| 16634 | 762 ## |
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763 ## References: |
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764 ## P.A. Clement, A class of triple-diagonal matrices for test |
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765 ## purposes, SIAM Review, 1 (1959), pp. 50-52. |
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766 ## A. Edelman and E. Kostlan, The road from Kac's matrix to Kac's |
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767 ## random polynomials. In John~G. Lewis, editor, Proceedings of |
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768 ## the Fifth SIAM Conference on Applied Linear Algebra Society |
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769 ## for Industrial and Applied Mathematics, Philadelphia, 1994, |
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770 ## pp. 503-507. |
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771 ## O. Taussky and J. Todd, Another look at a matrix of Mark Kac, |
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772 ## Linear Algebra and Appl., 150 (1991), pp. 341-360. |
| 16634 | 773 |
| 774 if (nargin < 1 || nargin > 2) | |
| 775 error ("gallery: 1 or 2 arguments are required for clement matrix."); | |
| 776 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 777 error ("gallery: N must be an integer for clement matrix."); | |
| 778 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 779 error ("gallery: K must be a numeric scalar for clement matrix."); | |
| 780 endif | |
| 781 | |
|
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782 n -= 1; |
| 16634 | 783 x = n:-1:1; |
| 784 z = 1:n; | |
| 785 | |
| 786 if (k == 0) | |
| 787 A = diag (x, -1) + diag (z, 1); | |
| 788 elseif (k == 1) | |
| 789 y = sqrt (x.*z); | |
| 790 A = diag (y, -1) + diag (y, 1); | |
| 791 else | |
| 792 error ("gallery: K must have a value of 0 or 1 for clement matrix."); | |
| 793 endif | |
| 794 endfunction | |
| 795 | |
| 796 function C = compar (A, k = 0) | |
| 797 ## COMP Comparison matrices. | |
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798 ## COMP(A) is DIAG(B) - TRIL(B,-1) - TRIU(B,1), where B = ABS(A). |
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799 ## COMP(A, 1) is A with each diagonal element replaced by its |
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800 ## absolute value, and each off-diagonal element replaced by minus |
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801 ## the absolute value of the largest element in absolute value in |
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802 ## its row. However, if A is triangular COMP(A, 1) is too. |
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803 ## COMP(A, 0) is the same as COMP(A). |
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804 ## COMP(A) is often denoted by M(A) in the literature. |
| 16634 | 805 ## |
|
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806 ## Reference (e.g.): |
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|
807 ## N.J. Higham, A survey of condition number estimation for |
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|
808 ## triangular matrices, SIAM Review, 29 (1987), pp. 575-596. |
| 16634 | 809 |
| 810 if (nargin < 1 || nargin > 2) | |
| 811 error ("gallery: 1 or 2 arguments are required for compar matrix."); | |
| 812 elseif (! isnumeric (A) || ndims (A) != 2) | |
| 17199 | 813 error ("gallery: A must be a 2-D matrix for compar matrix."); |
| 16634 | 814 elseif (! isnumeric (k) || ! isscalar (k)) |
| 815 error ("gallery: K must be a numeric scalar for compar matrix."); | |
| 816 endif | |
| 817 | |
| 818 [m, n] = size (A); | |
| 819 p = min (m, n); | |
| 820 | |
| 821 if (k == 0) | |
| 822 ## This code uses less temporary storage than | |
|
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|
823 ## the 'high level' definition above. |
| 16634 | 824 C = -abs (A); |
| 825 for j = 1:p | |
| 826 C(j,j) = abs (A(j,j)); | |
| 827 endfor | |
| 828 | |
| 829 elseif (k == 1) | |
| 830 C = A'; | |
| 831 for j = 1:p | |
| 832 C(k,k) = 0; | |
| 833 endfor | |
| 834 mx = max (abs (C)); | |
| 835 C = -mx'*ones (1, n); | |
| 836 for j = 1:p | |
| 837 C(j,j) = abs (A(j,j)); | |
| 838 endfor | |
| 839 if (all (A == tril (A))), C = tril (C); endif | |
| 840 if (all (A == triu (A))), C = triu (C); endif | |
| 841 | |
| 842 else | |
| 843 error ("gallery: K must have a value of 0 or 1 for compar matrix."); | |
| 844 endif | |
| 845 | |
| 846 endfunction | |
| 847 | |
| 848 function A = condex (n, k = 4, theta = 100) | |
|
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849 ## CONDEX 'Counterexamples' to matrix condition number estimators. |
|
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|
850 ## CONDEX(N, K, THETA) is a 'counterexample' matrix to a condition |
|
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851 ## estimator. It has order N and scalar parameter THETA (default 100). |
|
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|
852 ## If N is not equal to the 'natural' size of the matrix then |
|
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|
853 ## the matrix is padded out with an identity matrix to order N. |
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|
854 ## The matrix, its natural size, and the estimator to which it applies |
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855 ## are specified by K (default K = 4) as follows: |
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856 ## K = 1: 4-by-4, LINPACK (RCOND) |
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857 ## K = 2: 3-by-3, LINPACK (RCOND) |
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858 ## K = 3: arbitrary, LINPACK (RCOND) (independent of THETA) |
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859 ## K = 4: N >= 4, SONEST (Higham 1988) |
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|
860 ## (Note that in practice the K = 4 matrix is not usually a |
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|
861 ## counterexample because of the rounding errors in forming it.) |
| 16634 | 862 ## |
|
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863 ## References: |
|
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864 ## A.K. Cline and R.K. Rew, A set of counter-examples to three |
|
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|
865 ## condition number estimators, SIAM J. Sci. Stat. Comput., |
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866 ## 4 (1983), pp. 602-611. |
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|
867 ## N.J. Higham, FORTRAN codes for estimating the one-norm of a real or |
|
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868 ## complex matrix, with applications to condition estimation |
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|
869 ## (Algorithm 674), ACM Trans. Math. Soft., 14 (1988), pp. 381-396. |
| 16634 | 870 |
| 871 if (nargin < 1 || nargin > 3) | |
| 872 error ("gallery: 1 to 3 arguments are required for condex matrix."); | |
| 873 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 874 error ("gallery: N must be an integer for condex matrix."); | |
| 875 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 876 error ("gallery: K must be a numeric scalar for condex matrix."); | |
| 877 elseif (! isnumeric (theta) || ! isscalar (theta)) | |
| 878 error ("gallery: THETA must be a numeric scalar for condex matrix."); | |
| 879 endif | |
| 880 | |
| 881 if (k == 1) # Cline and Rew (1983), Example B. | |
| 882 A = [1 -1 -2*theta 0 | |
| 883 0 1 theta -theta | |
| 884 0 1 1+theta -(theta+1) | |
| 885 0 0 0 theta]; | |
| 886 | |
| 887 elseif (k == 2) # Cline and Rew (1983), Example C. | |
| 888 A = [1 1-2/theta^2 -2 | |
| 889 0 1/theta -1/theta | |
| 890 0 0 1]; | |
| 891 | |
| 892 elseif (k == 3) # Cline and Rew (1983), Example D. | |
| 893 A = gallery ("triw", n, -1)'; | |
| 894 A(n,n) = -1; | |
| 895 | |
| 896 elseif (k == 4) # Higham (1988), p. 390. | |
| 897 x = ones (n, 3); # First col is e | |
| 898 x(2:n,2) = zeros (n-1, 1); # Second col is e(1) | |
| 899 | |
| 900 ## Third col is special vector b in SONEST | |
| 901 x(:, 3) = (-1).^[0:n-1]' .* ( 1 + [0:n-1]'/(n-1) ); | |
| 902 | |
| 903 Q = orth (x); # Q*Q' is now the orthogonal projector onto span(e(1),e,b)). | |
| 904 P = eye (n) - Q*Q'; | |
| 905 A = eye (n) + theta*P; | |
| 906 | |
| 907 else | |
|
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|
908 error ("gallery: unknown estimator K '%d' for condex matrix.", k); |
| 16634 | 909 endif |
| 910 | |
| 911 ## Pad out with identity as necessary. | |
| 912 m = columns (A); | |
| 913 if (m < n) | |
| 914 for i = n:-1:m+1 | |
| 915 A(i,i) = 1; | |
| 916 endfor | |
| 917 endif | |
| 918 endfunction | |
| 919 | |
|
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|
920 function A = cycol (n, k = max (round (n(end)/4), 1)) |
| 16634 | 921 ## CYCOL Matrix whose columns repeat cyclically. |
|
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|
922 ## A = CYCOL([M N], K) is an M-by-N matrix of the form A = B(1:M,1:N) |
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923 ## where B = [C C C...] and C = RANDN(M, K). Thus A's columns repeat |
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924 ## cyclically, and A has rank at most K. K need not divide N. |
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925 ## K defaults to ROUND(N/4). |
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926 ## CYCOL(N, K), where N is a scalar, is the same as CYCOL([N N], K). |
| 16634 | 927 ## |
|
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928 ## This type of matrix can lead to underflow problems for Gaussian |
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|
929 ## elimination: see NA Digest Volume 89, Issue 3 (January 22, 1989). |
| 16634 | 930 |
| 931 if (nargin < 1 || nargin > 2) | |
| 932 error ("gallery: 1 or 2 arguments are required for cycol matrix."); | |
| 933 elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n) | |
| 934 error ("gallery: N must be a 1 or 2 element integer for cycol matrix."); | |
| 935 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 936 error ("gallery: K must be a scalar for cycol matrix."); | |
| 937 endif | |
| 938 | |
| 939 ## Parameter n specifies dimension: m-by-n | |
| 940 m = n(1); | |
| 941 n = n(end); | |
| 942 | |
| 943 A = randn (m, k); | |
| 944 for i = 2:ceil (n/k) | |
| 945 A = [A A(:,1:k)]; | |
| 946 endfor | |
| 947 A = A(:,1:n); | |
| 948 endfunction | |
| 949 | |
| 950 function [c, d, e] = dorr (n, theta = 0.01) | |
| 951 ## DORR Dorr matrix - diagonally dominant, ill conditioned, tridiagonal. | |
|
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952 ## [C, D, E] = DORR(N, THETA) returns the vectors defining a row diagonally |
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953 ## dominant, tridiagonal M-matrix that is ill conditioned for small |
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954 ## values of the parameter THETA >= 0. |
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955 ## If only one output parameter is supplied then |
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956 ## C = FULL(TRIDIAG(C,D,E)), i.e., the matrix iself is returned. |
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957 ## The columns of INV(C) vary greatly in norm. THETA defaults to 0.01. |
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958 ## The amount of diagonal dominance is given by (ignoring rounding errors): |
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959 ## COMP(C)*ONES(N,1) = THETA*(N+1)^2 * [1 0 0 ... 0 1]'. |
| 16634 | 960 ## |
|
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961 ## Reference: |
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962 ## F.W. Dorr, An example of ill-conditioning in the numerical |
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|
963 ## solution of singular perturbation problems, Math. Comp., 25 (1971), |
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|
964 ## pp. 271-283. |
| 16634 | 965 |
| 966 if (nargin < 1 || nargin > 2) | |
| 967 error ("gallery: 1 or 2 arguments are required for dorr matrix."); | |
| 968 elseif (! isscalar (n) || ! isnumeric (n) || fix (n) != n) | |
| 969 error ("gallery: N must be an integer for dorr matrix."); | |
| 970 elseif (! isscalar (theta) || ! isnumeric (theta)) | |
| 971 error ("gallery: THETA must be a numeric scalar for dorr matrix."); | |
| 972 endif | |
| 973 | |
| 974 c = zeros (n, 1); | |
| 975 e = c; | |
| 976 d = c; | |
| 977 ## All length n for convenience. Make c, e of length n-1 later. | |
| 978 | |
| 979 h = 1/(n+1); | |
| 980 m = floor ((n+1)/2); | |
| 981 term = theta/h^2; | |
| 982 | |
| 983 i = (1:m)'; | |
| 984 c(i) = -term * ones (m, 1); | |
| 985 e(i) = c(i) - (0.5-i*h)/h; | |
| 986 d(i) = -(c(i) + e(i)); | |
| 987 | |
| 988 i = (m+1:n)'; | |
| 989 e(i) = -term * ones (n-m, 1); | |
| 990 c(i) = e(i) + (0.5-i*h)/h; | |
| 991 d(i) = -(c(i) + e(i)); | |
| 992 | |
| 993 c = c(2:n); | |
| 994 e = e(1:n-1); | |
| 995 | |
| 996 if (nargout <= 1) | |
| 997 c = tridiag (c, d, e); | |
| 998 endif | |
| 999 endfunction | |
| 1000 | |
| 1001 function A = dramadah (n, k = 1) | |
| 1002 ## DRAMADAH A (0,1) matrix whose inverse has large integer entries. | |
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1003 ## An anti-Hadamard matrix A is a matrix with elements 0 or 1 for |
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1004 ## which MU(A) := NORM(INV(A),'FRO') is maximal. |
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1005 ## A = DRAMADAH(N, K) is an N-by-N (0,1) matrix for which MU(A) is |
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1006 ## relatively large, although not necessarily maximal. |
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1007 ## Available types (the default is K = 1): |
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1008 ## K = 1: A is Toeplitz, with ABS(DET(A)) = 1, and MU(A) > c(1.75)^N, |
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1009 ## where c is a constant. |
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1010 ## K = 2: A is upper triangular and Toeplitz. |
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1011 ## The inverses of both types have integer entries. |
| 16634 | 1012 ## |
|
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1013 ## Another interesting (0,1) matrix: |
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1014 ## K = 3: A has maximal determinant among (0,1) lower Hessenberg |
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1015 ## matrices: det(A) = the n'th Fibonacci number. A is Toeplitz. |
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1016 ## The eigenvalues have an interesting distribution in the complex |
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1017 ## plane. |
| 16634 | 1018 ## |
|
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1019 ## References: |
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1020 ## R.L. Graham and N.J.A. Sloane, Anti-Hadamard matrices, |
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1021 ## Linear Algebra and Appl., 62 (1984), pp. 113-137. |
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1022 ## L. Ching, The maximum determinant of an nxn lower Hessenberg |
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|
1023 ## (0,1) matrix, Linear Algebra and Appl., 183 (1993), pp. 147-153. |
| 16634 | 1024 |
| 1025 if (nargin < 1 || nargin > 2) | |
| 1026 error ("gallery: 1 to 2 arguments are required for dramadah matrix."); | |
| 1027 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1028 error ("gallery: N must be an integer for dramadah matrix."); | |
| 1029 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 1030 error ("gallery: K must be a numeric scalar for dramadah matrix."); | |
| 1031 endif | |
| 1032 | |
| 1033 switch (k) | |
| 1034 case (1) # Toeplitz | |
| 1035 c = ones (n, 1); | |
| 1036 for i = 2:4:n | |
| 1037 m = min (1, n-i); | |
| 1038 c(i:i+m) = zeros (m+1, 1); | |
| 1039 endfor | |
| 1040 r = zeros (n, 1); | |
| 1041 r(1:4) = [1 1 0 1]; | |
| 1042 if (n < 4) | |
| 1043 r = r(1:n); | |
| 1044 endif | |
| 1045 A = toeplitz (c, r); | |
| 1046 | |
| 1047 case (2) # Upper triangular and Toeplitz | |
| 1048 c = zeros (n, 1); | |
| 1049 c(1) = 1; | |
| 1050 r = ones (n, 1); | |
| 1051 for i= 3:2:n | |
| 1052 r(i) = 0; | |
| 1053 endfor | |
| 1054 A = toeplitz (c, r); | |
| 1055 | |
| 1056 case (3) # Lower Hessenberg | |
| 1057 c = ones (n, 1); | |
| 1058 for i= 2:2:n | |
| 1059 c(i) = 0; | |
| 1060 endfor | |
| 1061 A = toeplitz (c, [1 1 zeros(1,n-2)]); | |
| 1062 | |
| 1063 otherwise | |
|
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|
1064 error ("gallery: unknown K '%d' for dramadah matrix.", k); |
| 16634 | 1065 endswitch |
| 1066 endfunction | |
| 1067 | |
| 1068 function A = fiedler (c) | |
| 1069 ## FIEDLER Fiedler matrix - symmetric. | |
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1070 ## FIEDLER(C), where C is an n-vector, is the n-by-n symmetric |
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1071 ## matrix with elements ABS(C(i)-C(j)). |
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1072 ## Special case: if C is a scalar, then A = FIEDLER(1:C) |
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1073 ## (i.e. A(i,j) = ABS(i-j)). |
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1074 ## Properties: |
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1075 ## FIEDLER(N) has a dominant positive eigenvalue and all the other |
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1076 ## eigenvalues are negative (Szego, 1936). |
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1077 ## Explicit formulas for INV(A) and DET(A) are given by Todd (1977) |
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1078 ## and attributed to Fiedler. These indicate that INV(A) is |
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1079 ## tridiagonal except for nonzero (1,n) and (n,1) elements. |
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1080 ## [I think these formulas are valid only if the elements of |
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1081 ## C are in increasing or decreasing order---NJH.] |
| 16634 | 1082 ## |
|
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1083 ## References: |
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1084 ## G. Szego, Solution to problem 3705, Amer. Math. Monthly, |
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1085 ## 43 (1936), pp. 246-259. |
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1086 ## J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, |
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1087 ## Birkhauser, Basel, and Academic Press, New York, 1977, p. 159. |
| 16634 | 1088 |
| 1089 if (nargin != 1) | |
| 1090 error ("gallery: 1 argument is required for fiedler matrix."); | |
| 1091 elseif (! isnumeric (c)) | |
| 1092 error ("gallery: C must be numeric for fiedler matrix."); | |
| 1093 endif | |
| 1094 | |
| 1095 n = numel (c); | |
| 1096 if (isscalar (c) && fix (c) == c) | |
| 1097 n = c; | |
| 1098 c = 1:n; | |
| 1099 elseif (n > 1 && isvector (c)) | |
| 1100 ## do nothing | |
| 1101 else | |
| 1102 error ("gallery: C must be an integer or a vector for fiedler matrix."); | |
| 1103 endif | |
| 1104 c = c(:).'; # Ensure c is a row vector. | |
| 1105 | |
|
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|
1106 A = abs (c - c.'); |
| 16634 | 1107 endfunction |
| 1108 | |
| 1109 function A = forsythe (n, alpha = sqrt (eps), lambda = 0) | |
| 1110 ## FORSYTHE Forsythe matrix - a perturbed Jordan block. | |
|
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1111 ## FORSYTHE(N, ALPHA, LAMBDA) is the N-by-N matrix equal to |
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1112 ## JORDBLOC(N, LAMBDA) except it has an ALPHA in the (N,1) position. |
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1113 ## It has the characteristic polynomial |
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1114 ## DET(A-t*EYE) = (LAMBDA-t)^N - (-1)^N ALPHA. |
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1115 ## ALPHA defaults to SQRT(EPS) and LAMBDA to 0. |
| 16634 | 1116 |
| 1117 if (nargin < 1 || nargin > 3) | |
| 1118 error ("gallery: 1 to 3 arguments are required for forsythe matrix."); | |
| 1119 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1120 error ("gallery: N must be an integer for forsythe matrix."); | |
| 1121 elseif (! isnumeric (alpha) || ! isscalar (alpha)) | |
| 1122 error ("gallery: ALPHA must be a numeric scalar for forsythe matrix."); | |
| 1123 elseif (! isnumeric (lambda) || ! isscalar (lambda)) | |
| 1124 error ("gallery: LAMBDA must be a numeric scalar for forsythe matrix."); | |
| 1125 endif | |
| 1126 | |
| 1127 A = jordbloc (n, lambda); | |
| 1128 A(n,1) = alpha; | |
| 1129 endfunction | |
| 1130 | |
| 1131 function F = frank (n, k = 0) | |
| 1132 ## FRANK Frank matrix---ill conditioned eigenvalues. | |
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1133 ## F = FRANK(N, K) is the Frank matrix of order N. It is upper |
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1134 ## Hessenberg with determinant 1. K = 0 is the default; if K = 1 the |
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1135 ## elements are reflected about the anti-diagonal (1,N)--(N,1). |
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1136 ## F has all positive eigenvalues and they occur in reciprocal pairs |
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1137 ## (so that 1 is an eigenvalue if N is odd). |
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1138 ## The eigenvalues of F may be obtained in terms of the zeros of the |
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1139 ## Hermite polynomials. |
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1140 ## The FLOOR(N/2) smallest eigenvalues of F are ill conditioned, |
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1141 ## the more so for bigger N. |
| 16634 | 1142 ## |
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1143 ## DET(FRANK(N)') comes out far from 1 for large N---see Frank (1958) |
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1144 ## and Wilkinson (1960) for discussions. |
| 16634 | 1145 ## |
|
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1146 ## This version incorporates improvements suggested by W. Kahan. |
| 16634 | 1147 ## |
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1148 ## References: |
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1149 ## W.L. Frank, Computing eigenvalues of complex matrices by determinant |
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1150 ## evaluation and by methods of Danilewski and Wielandt, J. Soc. |
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1151 ## Indust. Appl. Math., 6 (1958), pp. 378-392 (see pp. 385, 388). |
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1152 ## G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the |
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1153 ## computation of the Jordan canonical form, SIAM Review, 18 (1976), |
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1154 ## pp. 578-619 (Section 13). |
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1155 ## H. Rutishauser, On test matrices, Programmation en Mathematiques |
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1156 ## Numeriques, Editions Centre Nat. Recherche Sci., Paris, 165, |
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1157 ## 1966, pp. 349-365. Section 9. |
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1158 ## J.H. Wilkinson, Error analysis of floating-point computation, |
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1159 ## Numer. Math., 2 (1960), pp. 319-340 (Section 8). |
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1160 ## J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University |
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1161 ## Press, 1965 (pp. 92-93). |
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1162 ## The next two references give details of the eigensystem, as does |
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1163 ## Rutishauser (see above). |
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1164 ## P.J. Eberlein, A note on the matrices denoted by B_n, SIAM J. Appl. |
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1165 ## Math., 20 (1971), pp. 87-92. |
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1166 ## J.M. Varah, A generalization of the Frank matrix, SIAM J. Sci. Stat. |
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1167 ## Comput., 7 (1986), pp. 835-839. |
| 16634 | 1168 |
| 1169 if (nargin < 1 || nargin > 2) | |
| 1170 error ("gallery: 1 to 2 arguments are required for frank matrix."); | |
| 1171 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1172 error ("gallery: N must be an integer for frank matrix."); | |
| 1173 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 1174 error ("gallery: K must be a numeric scalar for frank matrix."); | |
| 1175 endif | |
| 1176 | |
| 1177 p = n:-1:1; | |
| 1178 F = triu (p(ones (n, 1), :) - diag (ones (n-1, 1), -1), -1); | |
| 1179 | |
| 1180 switch (k) | |
| 1181 case (0), # do nothing | |
| 1182 case (1), F = F(p,p)'; | |
| 1183 otherwise | |
| 1184 error ("gallery: K must have a value of 0 or 1 for frank matrix."); | |
| 1185 endswitch | |
| 1186 endfunction | |
| 1187 | |
| 1188 function c = gcdmat (n) | |
| 1189 if (nargin != 1) | |
| 1190 error ("gallery: 1 argument is required for gcdmat matrix."); | |
| 1191 elseif (! isscalar (n) || ! isnumeric (n) || fix (n) != n) | |
| 1192 error ("gallery: N must be an integer for gcdmat matrix."); | |
| 1193 endif | |
| 1194 c = gcd (repmat ((1:n)', [1 n]), repmat (1:n, [n 1])); | |
| 1195 endfunction | |
| 1196 | |
| 1197 function A = gearmat (n, i = n, j = -n) | |
| 1198 ## NOTE: this function was named gearm in the original Test Matrix Toolbox | |
| 1199 ## GEARMAT Gear matrix. | |
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19697
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changeset
|
1200 ## A = GEARMAT(N,I,J) is the N-by-N matrix with ones on the sub- and |
|
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diff
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|
1201 ## super-diagonals, SIGN(I) in the (1,ABS(I)) position, SIGN(J) |
|
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diff
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|
1202 ## in the (N,N+1-ABS(J)) position, and zeros everywhere else. |
|
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diff
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|
1203 ## Defaults: I = N, j = -N. |
|
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diff
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|
1204 ## All eigenvalues are of the form 2*COS(a) and the eigenvectors |
|
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diff
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|
1205 ## are of the form [SIN(w+a), SIN(w+2a), ..., SIN(w+Na)]. |
|
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diff
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|
1206 ## The values of a and w are given in the reference below. |
|
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19697
diff
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|
1207 ## A can have double and triple eigenvalues and can be defective. |
|
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diff
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|
1208 ## GEARMAT(N) is singular. |
| 16634 | 1209 ## |
|
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diff
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|
1210 ## (GEAR is a Simulink function, hence GEARMAT for Gear matrix.) |
|
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diff
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|
1211 ## Reference: |
|
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diff
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|
1212 ## C.W. Gear, A simple set of test matrices for eigenvalue programs, |
|
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Rik <rik@octave.org>
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diff
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|
1213 ## Math. Comp., 23 (1969), pp. 119-125. |
| 16634 | 1214 |
| 1215 if (nargin < 1 || nargin > 3) | |
| 1216 error ("gallery: 1 to 3 arguments are required for gearmat matrix."); | |
| 1217 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1218 error ("gallery: N must be an integer for gearmat matrix."); | |
|
20294
2102c4582e32
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Massimiliano Fasi <massimiliano.fasi@gmail.com>
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20231
diff
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|
1219 elseif (! isnumeric (i) || ! isscalar (i) || i == 0 || abs (i) > n) |
|
18812
9ac2357f19bc
doc: Replace "non-zero" with "nonzero" to match existing usage.
Rik <rik@octave.org>
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|
1220 error ("gallery: I must be a nonzero scalar, and abs (I) <= N for gearmat matrix."); |
|
20294
2102c4582e32
gallery: fix chebspec, cycol, gearmat, hanowa, lauchli, and pei (bug #45466)
Massimiliano Fasi <massimiliano.fasi@gmail.com>
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20231
diff
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|
1221 elseif (! isnumeric (j) || ! isscalar (j) || i == 0 || abs (j) > n) |
|
18812
9ac2357f19bc
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|
1222 error ("gallery: J must be a nonzero scalar, and abs (J) <= N for gearmat matrix."); |
| 16634 | 1223 endif |
| 1224 | |
| 1225 A = diag (ones (n-1, 1), -1) + diag (ones (n-1, 1), 1); | |
| 1226 A(1, abs (i)) = sign (i); | |
| 1227 A(n, n+1 - abs (j)) = sign (j); | |
| 1228 endfunction | |
| 1229 | |
| 1230 function G = grcar (n, k = 3) | |
| 1231 ## GRCAR Grcar matrix - a Toeplitz matrix with sensitive eigenvalues. | |
|
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|
1232 ## GRCAR(N, K) is an N-by-N matrix with -1s on the |
|
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diff
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|
1233 ## subdiagonal, 1s on the diagonal, and K superdiagonals of 1s. |
|
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|
1234 ## The default is K = 3. The eigenvalues of this matrix form an |
|
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|
1235 ## interesting pattern in the complex plane (try PS(GRCAR(32))). |
| 16634 | 1236 ## |
|
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|
1237 ## References: |
|
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|
1238 ## J.F. Grcar, Operator coefficient methods for linear equations, |
|
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|
1239 ## Report SAND89-8691, Sandia National Laboratories, Albuquerque, |
|
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|
1240 ## New Mexico, 1989 (Appendix 2). |
|
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|
1241 ## N.M. Nachtigal, L. Reichel and L.N. Trefethen, A hybrid GMRES |
|
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|
1242 ## algorithm for nonsymmetric linear systems, SIAM J. Matrix Anal. |
|
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diff
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|
1243 ## Appl., 13 (1992), pp. 796-825. |
| 16634 | 1244 |
| 1245 if (nargin < 1 || nargin > 2) | |
| 1246 error ("gallery: 1 to 2 arguments are required for grcar matrix."); | |
| 1247 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1248 error ("gallery: N must be an integer for grcar matrix."); | |
| 1249 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 1250 error ("gallery: K must be a numeric scalar for grcar matrix."); | |
| 1251 endif | |
| 1252 | |
| 1253 G = tril (triu (ones (n)), k) - diag (ones (n-1, 1), -1); | |
| 1254 endfunction | |
| 1255 | |
| 1256 function A = hanowa (n, d = -1) | |
| 1257 ## HANOWA A matrix whose eigenvalues lie on a vertical line in the complex plane. | |
|
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|
1258 ## HANOWA(N, d) is the N-by-N block 2x2 matrix (thus N = 2M must be even) |
|
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diff
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|
1259 ## [d*EYE(M) -DIAG(1:M) |
|
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|
1260 ## DIAG(1:M) d*EYE(M)] |
|
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diff
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|
1261 ## It has complex eigenvalues lambda(k) = d +/- k*i (1 <= k <= M). |
|
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|
1262 ## Parameter d defaults to -1. |
| 16634 | 1263 ## |
|
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|
1264 ## Reference: |
|
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diff
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|
1265 ## E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary |
|
9fc020886ae9
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|
1266 ## Differential Equations I: Nonstiff Problems, Springer-Verlag, |
|
9fc020886ae9
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diff
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|
1267 ## Berlin, 1987. (pp. 86-87) |
| 16634 | 1268 |
| 1269 if (nargin < 1 || nargin > 2) | |
| 1270 error ("gallery: 1 to 2 arguments are required for hanowa matrix."); | |
| 1271 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1272 error ("gallery: N must be an integer for hanowa matrix."); | |
| 1273 elseif (rem (n, 2) != 0) | |
| 1274 error ("gallery: N must be even for hanowa matrix."); | |
|
20294
2102c4582e32
gallery: fix chebspec, cycol, gearmat, hanowa, lauchli, and pei (bug #45466)
Massimiliano Fasi <massimiliano.fasi@gmail.com>
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20231
diff
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|
1275 elseif (! isnumeric (d) || ! isscalar (d)) |
| 16634 | 1276 error ("gallery: D must be a numeric scalar for hanowa matrix."); |
| 1277 endif | |
| 1278 | |
| 1279 m = n/2; | |
| 1280 A = [ d*eye(m) -diag(1:m) | |
| 1281 diag(1:m) d*eye(m) ]; | |
| 1282 endfunction | |
| 1283 | |
| 1284 function [v, beta] = house (x) | |
| 1285 ## HOUSE Householder matrix. | |
|
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|
1286 ## If [v, beta] = HOUSE(x) then H = EYE - beta*v*v' is a Householder |
|
9fc020886ae9
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|
1287 ## matrix such that Hx = -sign(x(1))*norm(x)*e_1. |
|
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|
1288 ## NB: If x = 0 then v = 0, beta = 1 is returned. |
|
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diff
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|
1289 ## x can be real or complex. |
|
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|
1290 ## sign(x) := exp(i*arg(x)) ( = x./abs(x) when x ~= 0). |
| 16634 | 1291 ## |
|
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|
1292 ## Theory: (textbook references Golub & Van Loan 1989, 38-43; |
|
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|
1293 ## Stewart 1973, 231-234, 262; Wilkinson 1965, 48-50). |
|
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|
1294 ## Hx = y: (I - beta*v*v')x = -s*e_1. |
|
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|
1295 ## Must have |s| = norm(x), v = x+s*e_1, and |
|
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|
1296 ## x'y = x'Hx =(x'Hx)' real => arg(s) = arg(x(1)). |
|
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|
1297 ## So take s = sign(x(1))*norm(x) (which avoids cancellation). |
|
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|
1298 ## v'v = (x(1)+s)^2 + x(2)^2 + ... + x(n)^2 |
|
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|
1299 ## = 2*norm(x)*(norm(x) + |x(1)|). |
|
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|
1300 ## |
|
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diff
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|
1301 ## References: |
|
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|
1302 ## G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, |
|
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|
1303 ## Johns Hopkins University Press, Baltimore, Maryland, 1989. |
|
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|
1304 ## G.W. Stewart, Introduction to Matrix Computations, Academic Press, |
|
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|
1305 ## New York, 1973, |
|
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|
1306 ## J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University |
|
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|
1307 ## Press, 1965. |
| 16634 | 1308 |
| 1309 if (nargin != 1) | |
| 1310 error ("gallery: 1 argument is required for house matrix."); | |
|
20295
557979395ca9
gallery: allow single element vectors for 'house'.
Carnë Draug <carandraug@octave.org>
parents:
20294
diff
changeset
|
1311 elseif (! isnumeric (x) || ! isvector (x)) |
| 16634 | 1312 error ("gallery: X must be a vector for house matrix."); |
| 1313 endif | |
| 1314 | |
| 1315 ## must be a column vector | |
| 1316 x = x(:); | |
| 1317 | |
| 1318 s = norm (x) * (sign (x(1)) + (x(1) == 0)); # Modification for sign (0) == 1. | |
| 1319 v = x; | |
| 1320 if (s == 0) | |
| 1321 ## Quit if x is the zero vector. | |
| 1322 beta = 1; | |
| 1323 else | |
| 1324 v(1) = v(1) + s; | |
| 1325 beta = 1/(s'*v(1)); # NB the conjugated s. | |
| 1326 ## beta = 1/(abs (s) * (abs (s) +abs(x(1)) would guarantee beta real. | |
| 1327 ## But beta as above can be non-real (due to rounding) only when x is complex. | |
| 1328 endif | |
| 1329 endfunction | |
| 1330 | |
|
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|
1331 function A = integerdata (varargin) |
|
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|
1332 |
|
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|
1333 if (nargin < 3) |
|
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|
1334 error ("gallery: At least 3 arguments required for integerdata matrix."); |
|
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|
1335 endif |
|
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diff
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|
1336 |
|
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|
1337 if (isnumeric (varargin{end})) |
|
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|
1338 jidx = varargin{end}; |
|
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|
1339 svec = [varargin{:}]; |
|
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|
1340 varargin(end) = []; |
|
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|
1341 elseif (ischar (varargin{end})) |
|
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|
1342 if (nargin < 4) |
|
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|
1343 error (["gallery: CLASS argument requires 4 inputs " ... |
|
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|
1344 "for integerdata matrix."]); |
|
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|
1345 endif |
|
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|
1346 jidx = varargin{end-1}; |
|
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|
1347 svec = [varargin{1:end-1}]; |
|
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|
1348 varargin(end-1) = []; |
|
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|
1349 else |
|
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|
1350 error (["gallery: J must be an integer in the range [0, 2^32-1] " ... |
|
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|
1351 "for integerdata matrix"]); |
|
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|
1352 endif |
|
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16979
diff
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|
1353 |
|
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|
1354 if (! (isnumeric (jidx) && isscalar (jidx) |
|
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|
1355 && jidx == fix (jidx) |
|
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diff
changeset
|
1356 && jidx >= 0 && jidx <= 0xFFFFFFFF)) |
|
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16979
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changeset
|
1357 error (["gallery: J must be an integer in the range [0, 2^32-1] " ... |
|
1909e1ed63e6
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16979
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|
1358 "for integerdata matrix"]); |
|
1909e1ed63e6
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16979
diff
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|
1359 endif |
|
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|
1360 |
|
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Rik <rik@octave.org>
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|
1361 ## Save and restore random state. Initialization done so that reproducible |
|
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|
1362 ## data is available from gallery depending on the jidx and size vector. |
|
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|
1363 randstate = rand ("state"); |
|
16980
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|
1364 unwind_protect |
|
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|
1365 rand ("state", svec); |
|
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|
1366 A = randi (varargin{:}); |
|
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|
1367 unwind_protect_cleanup |
|
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|
1368 rand ("state", randstate); |
|
1909e1ed63e6
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|
1369 end_unwind_protect |
|
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|
1370 |
|
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|
1371 endfunction |
|
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Rik <rik@octave.org>
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|
1372 |
| 16634 | 1373 function A = invhess (x, y) |
| 1374 ## INVHESS Inverse of an upper Hessenberg matrix. | |
|
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|
1375 ## INVHESS(X, Y), where X is an N-vector and Y an N-1 vector, |
|
9fc020886ae9
maint: Clean up m-files to follow Octave coding conventions.
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|
1376 ## is the matrix whose lower triangle agrees with that of |
|
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|
1377 ## ONES(N,1)*X' and whose strict upper triangle agrees with |
|
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|
1378 ## that of [1 Y]*ONES(1,N). |
|
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|
1379 ## The matrix is nonsingular if X(1) ~= 0 and X(i+1) ~= Y(i) |
|
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Rik <rik@octave.org>
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|
1380 ## for all i, and its inverse is an upper Hessenberg matrix. |
|
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|
1381 ## If Y is omitted it defaults to -X(1:N-1). |
|
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|
1382 ## Special case: if X is a scalar INVHESS(X) is the same as |
|
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|
1383 ## INVHESS(1:X). |
| 16634 | 1384 ## |
|
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|
1385 ## References: |
|
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|
1386 ## F.N. Valvi and V.S. Geroyannis, Analytic inverses and |
|
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|
1387 ## determinants for a class of matrices, IMA Journal of Numerical |
|
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|
1388 ## Analysis, 7 (1987), pp. 123-128. |
|
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|
1389 ## W.-L. Cao and W.J. Stewart, A note on inverses of Hessenberg-like |
|
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|
1390 ## matrices, Linear Algebra and Appl., 76 (1986), pp. 233-240. |
|
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|
1391 ## Y. Ikebe, On inverses of Hessenberg matrices, Linear Algebra and |
|
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|
1392 ## Appl., 24 (1979), pp. 93-97. |
|
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|
1393 ## P. Rozsa, On the inverse of band matrices, Integral Equations and |
|
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|
1394 ## Operator Theory, 10 (1987), pp. 82-95. |
| 16634 | 1395 |
| 1396 if (nargin < 1 || nargin > 2) | |
| 1397 error ("gallery: 1 to 2 arguments are required for invhess matrix."); | |
| 1398 elseif (! isnumeric (x)) | |
| 1399 error ("gallery: X must be numeric for invhess matrix."); | |
| 1400 endif | |
| 1401 | |
| 1402 if (isscalar (x) && fix (x) == x) | |
| 1403 n = x; | |
| 1404 x = 1:n; | |
| 1405 elseif (! isscalar (x) && isvector (x)) | |
|
20297
26fc9bbb8762
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Carnë Draug <carandraug@octave.org>
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|
1406 n = numel (x); |
| 16634 | 1407 else |
| 1408 error ("gallery: X must be an integer scalar, or a vector for invhess matrix."); | |
| 1409 endif | |
| 1410 | |
| 1411 if (nargin < 2) | |
| 1412 y = -x(1:end-1); | |
| 1413 elseif (! isvector (y) || numel (y) != numel (x) -1) | |
| 1414 error ("gallery: Y must be a vector of length -1 than X for invhess matrix."); | |
| 1415 endif | |
| 1416 | |
| 1417 x = x(:); | |
| 1418 y = y(:); | |
| 1419 | |
|
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|
1420 ## FIXME: On next line, z = x'; A = z(ones(n,1),:) would be more efficient. |
| 16634 | 1421 A = ones (n, 1) * x'; |
| 1422 for j = 2:n | |
| 1423 A(1:j-1,j) = y(1:j-1); | |
| 1424 endfor | |
| 1425 endfunction | |
| 1426 | |
| 1427 function A = invol (n) | |
| 1428 ## INVOL An involutory matrix. | |
|
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|
1429 ## A = INVOL(N) is an N-by-N involutory (A*A = EYE(N)) and |
|
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|
1430 ## ill-conditioned matrix. |
|
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|
1431 ## It is a diagonally scaled version of HILB(N). |
|
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|
1432 ## NB: B = (EYE(N)-A)/2 and B = (EYE(N)+A)/2 are idempotent (B*B = B). |
| 16634 | 1433 ## |
|
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|
1434 ## Reference: |
|
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|
1435 ## A.S. Householder and J.A. Carpenter, The singular values |
|
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|
1436 ## of involutory and of idempotent matrices, Numer. Math. 5 (1963), |
|
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|
1437 ## pp. 234-237. |
| 16634 | 1438 |
|
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27931
diff
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|
1439 if (nargin < 1) |
| 16634 | 1440 error ("gallery: 1 argument is required for invol matrix."); |
| 1441 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1442 error ("gallery: N must be an integer for invol matrix."); | |
| 1443 endif | |
| 1444 | |
| 1445 A = hilb (n); | |
| 1446 | |
| 1447 d = -n; | |
| 1448 A(:, 1) = d * A(:, 1); | |
| 1449 | |
| 1450 for i = 1:n-1 | |
| 1451 d = -(n+i)*(n-i)*d/(i*i); | |
| 1452 A(i+1,:) = d * A(i+1,:); | |
| 1453 endfor | |
| 1454 endfunction | |
| 1455 | |
| 1456 function [A, detA] = ipjfact (n, k = 0) | |
| 1457 ## IPJFACT A Hankel matrix with factorial elements. | |
|
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|
1458 ## A = IPJFACT(N, K) is the matrix with |
|
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|
1459 ## A(i,j) = (i+j)! (K = 0, default) |
|
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|
1460 ## A(i,j) = 1/(i+j)! (K = 1) |
|
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|
1461 ## Both are Hankel matrices. |
|
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|
1462 ## The determinant and inverse are known explicitly. |
|
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|
1463 ## If a second output argument is present, d = DET(A) is returned: |
|
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|
1464 ## [A, d] = IPJFACT(N, K); |
| 16634 | 1465 ## |
|
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|
1466 ## Suggested by P. R. Graves-Morris. |
|
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|
1467 ## |
|
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|
1468 ## Reference: |
|
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|
1469 ## M.J.C. Gover, The explicit inverse of factorial Hankel matrices, |
|
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|
1470 ## Dept. of Mathematics, University of Bradford, 1993. |
| 16634 | 1471 |
| 1472 if (nargin < 1 || nargin > 2) | |
| 1473 error ("gallery: 1 to 2 arguments are required for ipjfact matrix."); | |
| 1474 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1475 error ("gallery: N must be an integer for ipjfact matrix."); | |
| 1476 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 1477 error ("gallery: K must be a numeric scalar for ipjfact matrix."); | |
| 1478 endif | |
| 1479 | |
| 1480 c = cumprod (2:n+1); | |
| 1481 d = cumprod (n+1:2*n) * c(n-1); | |
| 1482 | |
| 1483 A = hankel (c, d); | |
| 1484 | |
| 1485 switch (k) | |
| 1486 case (0), # do nothing | |
| 1487 case (1), A = ones (n) ./ A; | |
| 1488 otherwise | |
| 1489 error ("gallery: K must have a value of 0 or 1 for ipjfact matrix."); | |
| 1490 endswitch | |
| 1491 | |
| 1492 if (nargout == 2) | |
| 1493 d = 1; | |
| 1494 | |
| 1495 if (k == 0) | |
| 1496 for i = 1:n-1 | |
|
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|
1497 d *= prod (1:i+1) * prod (1:n-i); |
| 16634 | 1498 endfor |
|
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|
1499 d *= prod (1:n+1); |
| 16634 | 1500 |
| 1501 elseif (k == 1) | |
| 1502 for i = 0:n-1 | |
|
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|
1503 d *= prod (1:i) / prod (1:n+1+i); |
| 16634 | 1504 endfor |
| 1505 if (rem (n*(n-1)/2, 2)) | |
| 1506 d = -d; | |
| 1507 endif | |
| 1508 | |
| 1509 else | |
| 1510 error ("gallery: K must have a value of 0 or 1 for ipjfact matrix."); | |
| 1511 endif | |
| 1512 | |
| 1513 detA = d; | |
| 1514 endif | |
| 1515 endfunction | |
| 1516 | |
| 1517 function J = jordbloc (n, lambda = 1) | |
| 1518 ## JORDBLOC Jordan block. | |
|
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|
1519 ## JORDBLOC(N, LAMBDA) is the N-by-N Jordan block with eigenvalue |
|
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|
1520 ## LAMBDA. LAMBDA = 1 is the default. |
| 16634 | 1521 |
| 1522 if (nargin < 1 || nargin > 2) | |
| 1523 error ("gallery: 1 to 2 arguments are required for jordbloc matrix."); | |
| 1524 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1525 error ("gallery: N must be an integer for jordbloc matrix."); | |
| 1526 elseif (! isnumeric (lambda) || ! isscalar (lambda)) | |
| 1527 error ("gallery: LAMBDA must be a numeric scalar for jordbloc matrix."); | |
| 1528 endif | |
| 1529 | |
| 1530 J = lambda * eye (n) + diag (ones (n-1, 1), 1); | |
| 1531 endfunction | |
| 1532 | |
| 1533 function U = kahan (n, theta = 1.2, pert = 25) | |
| 1534 ## KAHAN Kahan matrix - upper trapezoidal. | |
|
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|
1535 ## KAHAN(N, THETA) is an upper trapezoidal matrix |
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|
1536 ## that has some interesting properties regarding estimation of |
|
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|
1537 ## condition and rank. |
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|
1538 ## The matrix is N-by-N unless N is a 2-vector, in which case it |
|
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|
1539 ## is N(1)-by-N(2). |
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|
1540 ## The parameter THETA defaults to 1.2. |
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|
1541 ## The useful range of THETA is 0 < THETA < PI. |
| 16634 | 1542 ## |
|
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|
1543 ## To ensure that the QR factorization with column pivoting does not |
|
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|
1544 ## interchange columns in the presence of rounding errors, the diagonal |
|
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|
1545 ## is perturbed by PERT*EPS*diag( [N:-1:1] ). |
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|
1546 ## The default is PERT = 25, which ensures no interchanges for KAHAN(N) |
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|
1547 ## up to at least N = 90 in IEEE arithmetic. |
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|
1548 ## KAHAN(N, THETA, PERT) uses the given value of PERT. |
| 16634 | 1549 ## |
|
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|
1550 ## The inverse of KAHAN(N, THETA) is known explicitly: see |
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|
1551 ## Higham (1987, p. 588), for example. |
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|
1552 ## The diagonal perturbation was suggested by Christian Bischof. |
| 16634 | 1553 ## |
|
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|
1554 ## References: |
|
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|
1555 ## W. Kahan, Numerical linear algebra, Canadian Math. Bulletin, |
|
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|
1556 ## 9 (1966), pp. 757-801. |
|
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|
1557 ## N.J. Higham, A survey of condition number estimation for |
|
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|
1558 ## triangular matrices, SIAM Review, 29 (1987), pp. 575-596. |
| 16634 | 1559 |
| 1560 if (nargin < 1 || nargin > 3) | |
| 1561 error ("gallery: 1 to 3 arguments are required for kahan matrix."); | |
| 1562 elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n) | |
| 1563 error ("gallery: N must be a 1 or 2 element integer for kahan matrix."); | |
| 1564 elseif (! isnumeric (theta) || ! isscalar (theta)) | |
| 1565 error ("gallery: THETA must be a numeric scalar for kahan matrix."); | |
| 1566 elseif (! isnumeric (pert) || ! isscalar (pert)) | |
| 1567 error ("gallery: PERT must be a numeric scalar for kahan matrix."); | |
| 1568 endif | |
| 1569 | |
| 1570 ## Parameter n specifies dimension: r-by-n | |
| 1571 r = n(1); | |
| 1572 n = n(end); | |
| 1573 | |
| 1574 s = sin (theta); | |
| 1575 c = cos (theta); | |
| 1576 | |
| 1577 U = eye (n) - c * triu (ones (n), 1); | |
| 1578 U = diag (s.^[0:n-1]) * U + pert*eps* diag ([n:-1:1]); | |
| 1579 if (r > n) | |
| 1580 U(r,n) = 0; # Extend to an r-by-n matrix | |
| 1581 else | |
| 1582 U = U(1:r,:); # Reduce to an r-by-n matrix | |
| 1583 endif | |
| 1584 endfunction | |
| 1585 | |
| 1586 function A = kms (n, rho = 0.5) | |
| 1587 ## KMS Kac-Murdock-Szego Toeplitz matrix. | |
|
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|
1588 ## A = KMS(N, RHO) is the N-by-N Kac-Murdock-Szego Toeplitz matrix with |
|
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|
1589 ## A(i,j) = RHO^(ABS((i-j))) (for real RHO). |
|
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|
1590 ## If RHO is complex, then the same formula holds except that elements |
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|
1591 ## below the diagonal are conjugated. |
|
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|
1592 ## RHO defaults to 0.5. |
|
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|
1593 ## Properties: |
|
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|
1594 ## A has an LDL' factorization with |
|
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|
1595 ## L = INV(TRIW(N,-RHO,1)'), |
|
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|
1596 ## D(i,i) = (1-ABS(RHO)^2)*EYE(N) except D(1,1) = 1. |
|
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|
1597 ## A is positive definite if and only if 0 < ABS(RHO) < 1. |
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|
1598 ## INV(A) is tridiagonal. |
| 16634 | 1599 ## |
|
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|
1600 ## Reference: |
|
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|
1601 ## W.F. Trench, Numerical solution of the eigenvalue problem |
|
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|
1602 ## for Hermitian Toeplitz matrices, SIAM J. Matrix Analysis and Appl., |
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|
1603 ## 10 (1989), pp. 135-146 (and see the references therein). |
| 16634 | 1604 |
| 1605 if (nargin < 1 || nargin > 2) | |
| 1606 error ("gallery: 1 to 2 arguments are required for lauchli matrix."); | |
| 1607 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
|
21385
89fa0694aa2e
Fix check-missing-semicolon QA target (bug #47277).
Rik <rik@octave.org>
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diff
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|
1608 error ("gallery: N must be an integer for lauchli matrix."); |
|
20294
2102c4582e32
gallery: fix chebspec, cycol, gearmat, hanowa, lauchli, and pei (bug #45466)
Massimiliano Fasi <massimiliano.fasi@gmail.com>
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20231
diff
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|
1609 elseif (! isscalar (rho)) |
|
21385
89fa0694aa2e
Fix check-missing-semicolon QA target (bug #47277).
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diff
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|
1610 error ("gallery: RHO must be a scalar for lauchli matrix."); |
| 16634 | 1611 endif |
| 1612 | |
|
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|
1613 A = (1:n)'*ones (1,n); |
|
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|
1614 A = abs (A - A'); |
| 16634 | 1615 A = rho .^ A; |
|
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|
1616 if (imag (rho)) |
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|
1617 A = conj (tril (A,-1)) + triu (A); |
| 16634 | 1618 endif |
| 1619 endfunction | |
| 1620 | |
| 1621 function B = krylov (A, x, j) | |
| 1622 ## KRYLOV Krylov matrix. | |
|
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|
1623 ## KRYLOV(A, x, j) is the Krylov matrix |
|
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|
1624 ## [x, Ax, A^2x, ..., A^(j-1)x], |
|
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|
1625 ## where A is an n-by-n matrix and x is an n-vector. |
|
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|
1626 ## Defaults: x = ONES(n,1), j = n. |
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|
1627 ## KRYLOV(n) is the same as KRYLOV(RANDN(n)). |
| 16634 | 1628 ## |
|
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|
1629 ## Reference: |
|
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|
1630 ## G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, |
|
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|
1631 ## Johns Hopkins University Press, Baltimore, Maryland, 1989, p. 369. |
| 16634 | 1632 |
| 1633 if (nargin < 1 || nargin > 3) | |
| 1634 error ("gallery: 1 to 3 arguments are required for krylov matrix."); | |
| 1635 elseif (! isnumeric (A) || ! issquare (A) || ndims (A) != 2) | |
| 17199 | 1636 error ("gallery: A must be a square 2-D matrix for krylov matrix."); |
| 16634 | 1637 endif |
| 1638 | |
| 1639 n = length (A); | |
| 1640 if (isscalar (A)) | |
| 1641 n = A; | |
| 1642 A = randn (n); | |
| 1643 endif | |
| 1644 | |
| 1645 if (nargin < 2) | |
| 1646 x = ones (n, 1); | |
| 1647 elseif (! isvector (x) || numel (x) != n) | |
| 1648 error ("gallery: X must be a vector of length equal to A for krylov matrix."); | |
| 1649 endif | |
| 1650 | |
| 1651 if (nargin < 3) | |
| 1652 j = n; | |
| 1653 elseif (! isnumeric (j) || ! isscalar (j) || fix (j) != j) | |
| 1654 error ("gallery: J must be an integer for krylov matrix."); | |
| 1655 endif | |
| 1656 | |
| 1657 B = ones (n, j); | |
| 1658 B(:,1) = x(:); | |
| 1659 for i = 2:j | |
| 1660 B(:,i) = A*B(:,i-1); | |
| 1661 endfor | |
| 1662 endfunction | |
| 1663 | |
| 1664 function A = lauchli (n, mu = sqrt (eps)) | |
| 1665 ## LAUCHLI Lauchli matrix - rectangular. | |
|
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|
1666 ## LAUCHLI(N, MU) is the (N+1)-by-N matrix [ONES(1,N); MU*EYE(N))]. |
|
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|
1667 ## It is a well-known example in least squares and other problems |
|
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|
1668 ## that indicates the dangers of forming A'*A. |
|
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|
1669 ## MU defaults to SQRT(EPS). |
| 16634 | 1670 ## |
|
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|
1671 ## Reference: |
|
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|
1672 ## P. Lauchli, Jordan-Elimination und Ausgleichung nach |
|
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diff
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|
1673 ## kleinsten Quadraten, Numer. Math, 3 (1961), pp. 226-240. |
| 16634 | 1674 |
| 1675 if (nargin < 1 || nargin > 2) | |
| 1676 error ("gallery: 1 to 2 arguments are required for lauchli matrix."); | |
| 1677 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1678 error ("gallery: N must be an integer for lauchli matrix."); | |
| 1679 elseif (! isscalar (mu)) | |
| 1680 error ("gallery: MU must be a scalar for lauchli matrix."); | |
| 1681 endif | |
| 1682 | |
| 1683 A = [ones(1, n) | |
| 1684 mu*eye(n) ]; | |
| 1685 endfunction | |
| 1686 | |
| 1687 function A = lehmer (n) | |
| 1688 ## LEHMER Lehmer matrix - symmetric positive definite. | |
|
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|
1689 ## A = LEHMER(N) is the symmetric positive definite N-by-N matrix with |
|
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|
1690 ## A(i,j) = i/j for j >= i. |
|
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|
1691 ## A is totally nonnegative. INV(A) is tridiagonal, and explicit |
|
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|
1692 ## formulas are known for its entries. |
|
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|
1693 ## N <= COND(A) <= 4*N*N. |
| 16634 | 1694 ## |
|
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|
1695 ## References: |
|
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|
1696 ## M. Newman and J. Todd, The evaluation of matrix inversion |
|
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|
1697 ## programs, J. Soc. Indust. Appl. Math., 6 (1958), pp. 466-476. |
|
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|
1698 ## Solutions to problem E710 (proposed by D.H. Lehmer): The inverse |
|
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|
1699 ## of a matrix, Amer. Math. Monthly, 53 (1946), pp. 534-535. |
|
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|
1700 ## J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, |
|
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|
1701 ## Birkhauser, Basel, and Academic Press, New York, 1977, p. 154. |
| 16634 | 1702 |
|
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|
1703 if (nargin < 1) |
| 16634 | 1704 error ("gallery: 1 argument is required for lehmer matrix."); |
| 1705 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1706 error ("gallery: N must be an integer for lehmer matrix."); | |
| 1707 endif | |
| 1708 | |
| 1709 A = ones (n, 1) * (1:n); | |
| 1710 A = A./A'; | |
| 1711 A = tril (A) + tril (A, -1)'; | |
| 1712 endfunction | |
| 1713 | |
| 1714 function T = lesp (n) | |
| 1715 ## LESP A tridiagonal matrix with real, sensitive eigenvalues. | |
|
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|
1716 ## LESP(N) is an N-by-N matrix whose eigenvalues are real and smoothly |
|
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|
1717 ## distributed in the interval approximately [-2*N-3.5, -4.5]. |
|
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|
1718 ## The sensitivities of the eigenvalues increase exponentially as |
|
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|
1719 ## the eigenvalues grow more negative. |
|
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|
1720 ## The matrix is similar to the symmetric tridiagonal matrix with |
|
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diff
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|
1721 ## the same diagonal entries and with off-diagonal entries 1, |
|
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diff
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|
1722 ## via a similarity transformation with D = diag(1!,2!,...,N!). |
| 16634 | 1723 ## |
|
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|
1724 ## References: |
|
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|
1725 ## H.W.J. Lenferink and M.N. Spijker, On the use of stability regions in |
|
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|
1726 ## the numerical analysis of initial value problems, |
|
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|
1727 ## Math. Comp., 57 (1991), pp. 221-237. |
|
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|
1728 ## L.N. Trefethen, Pseudospectra of matrices, in Numerical Analysis 1991, |
|
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|
1729 ## Proceedings of the 14th Dundee Conference, |
|
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|
1730 ## D.F. Griffiths and G.A. Watson, eds, Pitman Research Notes in |
|
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|
1731 ## Mathematics, volume 260, Longman Scientific and Technical, Essex, |
|
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|
1732 ## UK, 1992, pp. 234-266. |
| 16634 | 1733 |
|
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|
1734 if (nargin < 1) |
| 16634 | 1735 error ("gallery: 1 argument is required for lesp matrix."); |
| 1736 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1737 error ("gallery: N must be an integer for lesp matrix."); | |
| 1738 endif | |
| 1739 | |
| 1740 x = 2:n; | |
| 1741 T = full (tridiag (ones (size (x)) ./x, -(2*[x n+1]+1), x)); | |
| 1742 endfunction | |
| 1743 | |
| 1744 function A = lotkin (n) | |
| 1745 ## LOTKIN Lotkin matrix. | |
|
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|
1746 ## A = LOTKIN(N) is the Hilbert matrix with its first row altered to |
|
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|
1747 ## all ones. A is unsymmetric, ill-conditioned, and has many negative |
|
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|
1748 ## eigenvalues of small magnitude. |
|
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|
1749 ## The inverse has integer entries and is known explicitly. |
| 16634 | 1750 ## |
|
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|
1751 ## Reference: |
|
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|
1752 ## M. Lotkin, A set of test matrices, MTAC, 9 (1955), pp. 153-161. |
| 16634 | 1753 |
|
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|
1754 if (nargin < 1) |
| 16634 | 1755 error ("gallery: 1 argument is required for lotkin matrix."); |
| 1756 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1757 error ("gallery: N must be an integer for lotkin matrix."); | |
| 1758 endif | |
| 1759 | |
| 1760 A = hilb (n); | |
| 1761 A(1,:) = ones (1, n); | |
| 1762 endfunction | |
| 1763 | |
| 1764 function A = minij (n) | |
| 1765 ## MINIJ Symmetric positive definite matrix MIN(i,j). | |
|
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|
1766 ## A = MINIJ(N) is the N-by-N symmetric positive definite matrix with |
|
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|
1767 ## A(i,j) = MIN(i,j). |
|
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|
1768 ## Properties, variations: |
|
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|
1769 ## INV(A) is tridiagonal: it is minus the second difference matrix |
|
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|
1770 ## except its (N,N) element is 1. |
|
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|
1771 ## 2*A-ONES(N) (Givens' matrix) has tridiagonal inverse and |
|
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|
1772 ## eigenvalues .5*sec^2([2r-1)PI/4N], r=1:N. |
|
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|
1773 ## (N+1)*ONES(N)-A also has a tridiagonal inverse. |
| 16634 | 1774 ## |
|
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|
1775 ## References: |
|
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|
1776 ## J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, |
|
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|
1777 ## Birkhauser, Basel, and Academic Press, New York, 1977, p. 158. |
|
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|
1778 ## D.E. Rutherford, Some continuant determinants arising in physics and |
|
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|
1779 ## chemistry---II, Proc. Royal Soc. Edin., 63, A (1952), pp. 232-241. |
|
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|
1780 ## (For the eigenvalues of Givens' matrix.) |
| 16634 | 1781 |
|
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|
1782 if (nargin < 1) |
| 16634 | 1783 error ("gallery: 1 argument is required for minij matrix."); |
| 1784 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1785 error ("gallery: N must be an integer for minij matrix."); | |
| 1786 endif | |
| 1787 | |
|
19944
941e782d0429
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diff
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|
1788 A = bsxfun (@min, 1:n, (1:n)'); |
| 16634 | 1789 endfunction |
| 1790 | |
| 1791 function A = moler (n, alpha = -1) | |
| 1792 ## MOLER Moler matrix - symmetric positive definite. | |
|
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|
1793 ## A = MOLER(N, ALPHA) is the symmetric positive definite N-by-N matrix |
|
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|
1794 ## U'*U where U = TRIW(N, ALPHA). |
|
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|
1795 ## For ALPHA = -1 (the default) A(i,j) = MIN(i,j)-2, A(i,i) = i. |
|
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|
1796 ## A has one small eigenvalue. |
| 16634 | 1797 ## |
|
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|
1798 ## Nash (1990) attributes the ALPHA = -1 matrix to Moler. |
| 16634 | 1799 ## |
|
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|
1800 ## Reference: |
|
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|
1801 ## J.C. Nash, Compact Numerical Methods for Computers: Linear |
|
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|
1802 ## Algebra and Function Minimisation, second edition, Adam Hilger, |
|
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|
1803 ## Bristol, 1990 (Appendix 1). |
| 16634 | 1804 |
| 1805 if (nargin < 1 || nargin > 2) | |
| 1806 error ("gallery: 1 to 2 arguments are required for moler matrix."); | |
| 1807 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1808 error ("gallery: N must be an integer for moler matrix."); | |
| 1809 elseif (! isscalar (alpha)) | |
| 1810 error ("gallery: ALPHA must be a scalar for moler matrix."); | |
| 1811 endif | |
| 1812 | |
| 1813 A = triw (n, alpha)' * triw (n, alpha); | |
| 1814 endfunction | |
| 1815 | |
| 1816 function [A, T] = neumann (n) | |
| 1817 ## NEUMANN Singular matrix from the discrete Neumann problem (sparse). | |
|
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|
1818 ## NEUMANN(N) is the singular, row diagonally dominant matrix resulting |
|
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|
1819 ## from discretizing the Neumann problem with the usual five point |
|
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|
1820 ## operator on a regular mesh. |
|
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|
1821 ## It has a one-dimensional null space with null vector ONES(N,1). |
|
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|
1822 ## The dimension N should be a perfect square, or else a 2-vector, |
|
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|
1823 ## in which case the dimension of the matrix is N(1)*N(2). |
| 16634 | 1824 ## |
|
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|
1825 ## Reference: |
|
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|
1826 ## R.J. Plemmons, Regular splittings and the discrete Neumann |
|
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|
1827 ## problem, Numer. Math., 25 (1976), pp. 153-161. |
| 16634 | 1828 |
|
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|
1829 if (nargin < 1) |
| 16634 | 1830 error ("gallery: 1 argument is required for neumann matrix."); |
| 1831 elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n) | |
| 1832 error ("gallery: N must be a 1 or 2 element integer for neumann matrix."); | |
| 1833 endif | |
| 1834 | |
| 1835 if (isscalar (n)) | |
| 1836 m = sqrt (n); | |
| 1837 if (m^2 != n) | |
| 1838 error ("gallery: N must be a perfect square for neumann matrix."); | |
| 1839 endif | |
| 1840 n(1) = m; | |
| 1841 n(2) = m; | |
| 1842 endif | |
| 1843 | |
| 1844 T = tridiag (n(1), -1, 2, -1); | |
| 1845 T(1,2) = -2; | |
| 1846 T(n(1),n(1)-1) = -2; | |
| 1847 | |
| 1848 A = kron (T, eye (n(2))) + kron (eye (n(2)), T); | |
| 1849 endfunction | |
| 1850 | |
|
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|
1851 function A = normaldata (varargin) |
|
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|
1852 |
|
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|
1853 if (nargin < 2) |
|
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|
1854 error ("gallery: At least 2 arguments required for normaldata matrix."); |
|
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|
1855 endif |
|
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|
1856 if (isnumeric (varargin{end})) |
|
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|
1857 jidx = varargin{end}; |
|
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|
1858 svec = [varargin{:}]; |
|
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|
1859 varargin(end) = []; |
|
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|
1860 elseif (ischar (varargin{end})) |
|
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|
1861 if (nargin < 3) |
|
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|
1862 error (["gallery: CLASS argument requires 3 inputs " ... |
|
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|
1863 "for normaldata matrix."]); |
|
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|
1864 endif |
|
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|
1865 jidx = varargin{end-1}; |
|
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|
1866 svec = [varargin{1:end-1}]; |
|
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|
1867 varargin(end-1) = []; |
|
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|
1868 else |
|
16979
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|
1869 error (["gallery: J must be an integer in the range [0, 2^32-1] " ... |
|
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|
1870 "for normaldata matrix"]); |
|
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diff
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|
1871 endif |
|
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diff
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|
1872 |
|
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|
1873 if (! (isnumeric (jidx) && isscalar (jidx) |
|
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|
1874 && jidx == fix (jidx) |
|
9aa293e00475
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|
1875 && jidx >= 0 && jidx <= 0xFFFFFFFF)) |
|
9aa293e00475
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diff
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|
1876 error (["gallery: J must be an integer in the range [0, 2^32-1] " ... |
|
9aa293e00475
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|
1877 "for normaldata matrix"]); |
|
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diff
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|
1878 endif |
|
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diff
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|
1879 |
|
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|
1880 ## Save and restore random state. Initialization done so that reproducible |
|
9aa293e00475
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diff
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|
1881 ## data is available from gallery depending on the jidx and size vector. |
|
19593
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17386
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|
1882 randstate = randn ("state"); |
|
16979
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diff
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|
1883 unwind_protect |
|
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|
1884 randn ("state", svec); |
|
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|
1885 A = randn (varargin{:}); |
|
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diff
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|
1886 unwind_protect_cleanup |
|
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diff
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|
1887 randn ("state", randstate); |
|
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|
1888 end_unwind_protect |
|
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diff
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|
1889 |
|
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diff
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|
1890 endfunction |
|
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|
1891 |
| 16634 | 1892 function Q = orthog (n, k = 1) |
| 1893 ## ORTHOG Orthogonal and nearly orthogonal matrices. | |
|
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|
1894 ## Q = ORTHOG(N, K) selects the K'th type of matrix of order N. |
|
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|
1895 ## K > 0 for exactly orthogonal matrices, K < 0 for diagonal scalings of |
|
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|
1896 ## orthogonal matrices. |
|
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|
1897 ## Available types: (K = 1 is the default) |
|
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|
1898 ## K = 1: Q(i,j) = SQRT(2/(n+1)) * SIN( i*j*PI/(n+1) ) |
|
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|
1899 ## Symmetric eigenvector matrix for second difference matrix. |
|
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|
1900 ## K = 2: Q(i,j) = 2/SQRT(2*n+1)) * SIN( 2*i*j*PI/(2*n+1) ) |
|
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1901 ## Symmetric. |
|
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|
1902 ## K = 3: Q(r,s) = EXP(2*PI*i*(r-1)*(s-1)/n) / SQRT(n) (i=SQRT(-1)) |
|
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|
1903 ## Unitary, the Fourier matrix. Q^4 is the identity. |
|
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|
1904 ## This is essentially the same matrix as FFT(EYE(N))/SQRT(N)! |
|
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|
1905 ## K = 4: Helmert matrix: a permutation of a lower Hessenberg matrix, |
|
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|
1906 ## whose first row is ONES(1:N)/SQRT(N). |
|
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|
1907 ## K = 5: Q(i,j) = SIN( 2*PI*(i-1)*(j-1)/n ) + COS( 2*PI*(i-1)*(j-1)/n ). |
|
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|
1908 ## Symmetric matrix arising in the Hartley transform. |
|
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|
1909 ## K = -1: Q(i,j) = COS( (i-1)*(j-1)*PI/(n-1) ) |
|
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|
1910 ## Chebyshev Vandermonde-like matrix, based on extrema of T(n-1). |
|
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|
1911 ## K = -2: Q(i,j) = COS( (i-1)*(j-1/2)*PI/n) ) |
|
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|
1912 ## Chebyshev Vandermonde-like matrix, based on zeros of T(n). |
| 16634 | 1913 ## |
|
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|
1914 ## References: |
|
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|
1915 ## N.J. Higham and D.J. Higham, Large growth factors in Gaussian |
|
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|
1916 ## elimination with pivoting, SIAM J. Matrix Analysis and Appl., |
|
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|
1917 ## 10 (1989), pp. 155-164. |
|
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|
1918 ## P. Morton, On the eigenvectors of Schur's matrix, J. Number Theory, |
|
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|
1919 ## 12 (1980), pp. 122-127. (Re. ORTHOG(N, 3)) |
|
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|
1920 ## H.O. Lancaster, The Helmert Matrices, Amer. Math. Monthly, 72 (1965), |
|
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1921 ## pp. 4-12. |
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1922 ## D. Bini and P. Favati, On a matrix algebra related to the discrete |
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1923 ## Hartley transform, SIAM J. Matrix Anal. Appl., 14 (1993), |
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1924 ## pp. 500-507. |
| 16634 | 1925 |
| 1926 if (nargin < 1 || nargin > 2) | |
| 1927 error ("gallery: 1 to 2 arguments are required for orthog matrix."); | |
| 1928 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1929 error ("gallery: N must be an integer for orthog matrix."); | |
| 1930 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 1931 error ("gallery: K must be a numeric scalar for orthog matrix."); | |
| 1932 endif | |
| 1933 | |
| 1934 switch (k) | |
| 1935 case (1) | |
| 1936 ## E'vectors second difference matrix | |
| 1937 m = (1:n)'*(1:n) * (pi/(n+1)); | |
| 1938 Q = sin (m) * sqrt (2/(n+1)); | |
| 1939 | |
| 1940 case (2) | |
| 1941 m = (1:n)'*(1:n) * (2*pi/(2*n+1)); | |
| 1942 Q = sin (m) * (2/ sqrt (2*n+1)); | |
| 1943 | |
| 1944 case (3) | |
| 1945 ## Vandermonde based on roots of unity | |
| 1946 m = 0:n-1; | |
| 1947 Q = exp (m'*m*2*pi* sqrt (-1) / n) / sqrt (n); | |
| 1948 | |
| 1949 case (4) | |
| 1950 ## Helmert matrix | |
| 1951 Q = tril (ones (n)); | |
| 1952 Q(1,2:n) = ones (1, n-1); | |
| 1953 for i = 2:n | |
| 1954 Q(i,i) = -(i-1); | |
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1955 endfor |
| 16634 | 1956 Q = diag (sqrt ([n 1:n-1] .* [1:n])) \ Q; |
| 1957 | |
| 1958 case (5) | |
| 1959 ## Hartley matrix | |
| 1960 m = (0:n-1)'*(0:n-1) * (2*pi/n); | |
| 1961 Q = (cos (m) + sin (m)) / sqrt (n); | |
| 1962 | |
| 1963 case (-1) | |
| 1964 ## extrema of T(n-1) | |
| 1965 m = (0:n-1)'*(0:n-1) * (pi/(n-1)); | |
| 1966 Q = cos (m); | |
| 1967 | |
| 1968 case (-2) | |
| 1969 ## zeros of T(n) | |
| 1970 m = (0:n-1)'*(.5:n-.5) * (pi/n); | |
| 1971 Q = cos (m); | |
| 1972 | |
| 1973 otherwise | |
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1974 error ("gallery: unknown K '%d' for orthog matrix.", k); |
| 16634 | 1975 endswitch |
| 1976 endfunction | |
| 1977 | |
| 1978 function A = parter (n) | |
| 1979 ## PARTER Parter matrix - a Toeplitz matrix with singular values near PI. | |
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1980 ## PARTER(N) is the matrix with (i,j) element 1/(i-j+0.5). |
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1981 ## It is a Cauchy matrix and a Toeplitz matrix. |
| 16634 | 1982 ## |
|
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1983 ## At the Second SIAM Conference on Linear Algebra, Raleigh, N.C., |
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1984 ## 1985, Cleve Moler noted that most of the singular values of |
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1985 ## PARTER(N) are very close to PI. An explanation of the phenomenon |
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1986 ## was given by Parter; see also the paper by Tyrtyshnikov. |
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1987 ## |
|
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1988 ## References: |
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1989 ## The MathWorks Newsletter, Volume 1, Issue 1, March 1986, page 2. |
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1990 ## S.V. Parter, On the distribution of the singular values of Toeplitz |
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1991 ## matrices, Linear Algebra and Appl., 80 (1986), pp. 115-130. |
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1992 ## E.E. Tyrtyshnikov, Cauchy-Toeplitz matrices and some applications, |
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1993 ## Linear Algebra and Appl., 149 (1991), pp. 1-18. |
| 16634 | 1994 |
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1995 if (nargin < 1) |
| 16634 | 1996 error ("gallery: 1 argument is required for parter matrix."); |
| 1997 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1998 error ("gallery: N must be an integer for parter matrix."); | |
| 1999 endif | |
| 2000 | |
| 2001 A = cauchy ((1:n) + 0.5, -(1:n)); | |
| 2002 endfunction | |
| 2003 | |
| 2004 function P = pei (n, alpha = 1) | |
| 2005 ## PEI Pei matrix. | |
|
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2006 ## PEI(N, ALPHA), where ALPHA is a scalar, is the symmetric matrix |
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2007 ## ALPHA*EYE(N) + ONES(N). |
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2008 ## If ALPHA is omitted then ALPHA = 1 is used. |
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2009 ## The matrix is singular for ALPHA = 0, -N. |
| 16634 | 2010 ## |
|
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2011 ## Reference: |
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2012 ## M.L. Pei, A test matrix for inversion procedures, |
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2013 ## Comm. ACM, 5 (1962), p. 508. |
| 16634 | 2014 |
| 2015 if (nargin < 1 || nargin > 2) | |
| 2016 error ("gallery: 1 to 2 arguments are required for pei matrix."); | |
| 2017 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2018 error ("gallery: N must be an integer for pei matrix."); | |
|
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|
2019 elseif (! isnumeric (alpha) || ! isscalar (alpha)) |
| 16634 | 2020 error ("gallery: ALPHA must be a scalar for pei matrix."); |
| 2021 endif | |
| 2022 | |
| 2023 P = alpha * eye (n) + ones (n); | |
| 2024 endfunction | |
| 2025 | |
| 2026 function A = poisson (n) | |
| 2027 ## POISSON Block tridiagonal matrix from Poisson's equation (sparse). | |
|
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2028 ## POISSON(N) is the block tridiagonal matrix of order N^2 |
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2029 ## resulting from discretizing Poisson's equation with the |
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2030 ## 5-point operator on an N-by-N mesh. |
| 16634 | 2031 ## |
|
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2032 ## Reference: |
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2033 ## G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, |
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2034 ## Johns Hopkins University Press, Baltimore, Maryland, 1989 |
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2035 ## (Section 4.5.4). |
| 16634 | 2036 |
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|
2037 if (nargin < 1) |
| 16634 | 2038 error ("gallery: 1 argument is required for poisson matrix."); |
| 2039 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2040 error ("gallery: N must be an integer for poisson matrix."); | |
| 2041 endif | |
| 2042 | |
| 2043 S = tridiag (n, -1, 2, -1); | |
| 2044 I = speye (n); | |
| 2045 A = kron (I, S) + kron (S, I); | |
| 2046 endfunction | |
| 2047 | |
| 2048 function A = prolate (n, w = 0.25) | |
| 2049 ## PROLATE Prolate matrix - symmetric, ill-conditioned Toeplitz matrix. | |
|
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2050 ## A = PROLATE(N, W) is the N-by-N prolate matrix with parameter W. |
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2051 ## It is a symmetric Toeplitz matrix. |
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2052 ## If 0 < W < 0.5 then |
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2053 ## - A is positive definite |
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2054 ## - the eigenvalues of A are distinct, lie in (0, 1), and |
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2055 ## tend to cluster around 0 and 1. |
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2056 ## W defaults to 0.25. |
| 16634 | 2057 ## |
|
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2058 ## Reference: |
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2059 ## J.M. Varah. The Prolate matrix. Linear Algebra and Appl., |
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2060 ## 187:269--278, 1993. |
| 16634 | 2061 |
| 2062 if (nargin < 1 || nargin > 2) | |
| 2063 error ("gallery: 1 to 2 arguments are required for prolate matrix."); | |
| 2064 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2065 error ("gallery: N must be an integer for prolate matrix."); | |
| 2066 elseif (! isnumeric (w) || ! isscalar (w)) | |
| 2067 error ("gallery: W must be a scalar for prolate matrix."); | |
| 2068 endif | |
| 2069 | |
| 2070 a = zeros (n, 1); | |
| 2071 a(1) = 2*w; | |
| 2072 a(2:n) = sin (2*pi*w*(1:n-1)) ./ (pi*(1:n-1)); | |
| 2073 | |
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|
2074 A = toeplitz (a); |
| 16634 | 2075 endfunction |
| 2076 | |
| 2077 function H = randhess (x) | |
| 2078 ## NOTE: this function was named ohess in the original Test Matrix Toolbox | |
| 2079 ## RANDHESS Random, orthogonal upper Hessenberg matrix. | |
|
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2080 ## H = RANDHESS(N) is an N-by-N real, random, orthogonal |
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|
2081 ## upper Hessenberg matrix. |
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|
2082 ## Alternatively, H = RANDHESS(X), where X is an arbitrary real |
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2083 ## N-vector (N > 1) constructs H non-randomly using the elements |
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2084 ## of X as parameters. |
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2085 ## In both cases H is constructed via a product of N-1 Givens rotations. |
| 16634 | 2086 ## |
|
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2087 ## Note: See Gragg (1986) for how to represent an N-by-N (complex) |
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2088 ## unitary Hessenberg matrix with positive subdiagonal elements in terms |
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2089 ## of 2N-1 real parameters (the Schur parametrization). |
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2090 ## This M-file handles the real case only and is intended simply as a |
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2091 ## convenient way to generate random or non-random orthogonal Hessenberg |
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2092 ## matrices. |
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2093 ## |
|
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2094 ## Reference: |
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2095 ## W.B. Gragg, The QR algorithm for unitary Hessenberg matrices, |
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2096 ## J. Comp. Appl. Math., 16 (1986), pp. 1-8. |
| 16634 | 2097 |
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2098 if (nargin < 1) |
| 16634 | 2099 error ("gallery: 1 argument is required for randhess matrix."); |
| 2100 elseif (! isnumeric (x) || ! isreal (x)) | |
| 2101 error ("gallery: N or X must be numeric real values for randhess matrix."); | |
| 2102 endif | |
| 2103 | |
| 2104 if (isscalar (x)) | |
| 2105 n = x; | |
| 2106 x = rand (n-1, 1) * 2*pi; | |
| 2107 H = eye (n); | |
| 2108 H(n,n) = sign (randn); | |
| 2109 elseif (isvector (x)) | |
| 2110 n = numel (x); | |
| 2111 H = eye (n); | |
| 2112 H(n,n) = sign (x(n)) + (x(n) == 0); # Second term ensures H(n,n) nonzero. | |
| 2113 else | |
| 2114 error ("gallery: N or X must be a scalar or a vector for randhess matrix."); | |
| 2115 endif | |
| 2116 | |
| 2117 for i = n:-1:2 | |
| 2118 ## Apply Givens rotation through angle x(i-1). | |
| 2119 theta = x(i-1); | |
| 2120 c = cos (theta); | |
| 2121 s = sin (theta); | |
| 2122 H([i-1 i], :) = [ c*H(i-1,:)+s*H(i,:) | |
| 2123 -s*H(i-1,:)+c*H(i,:) ]; | |
| 2124 endfor | |
| 2125 endfunction | |
| 2126 | |
| 2127 function A = rando (n, k = 1) | |
| 2128 ## RANDO Random matrix with elements -1, 0 or 1. | |
|
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2129 ## A = RANDO(N, K) is a random N-by-N matrix with elements from |
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2130 ## one of the following discrete distributions (default K = 1): |
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2131 ## K = 1: A(i,j) = 0 or 1 with equal probability, |
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2132 ## K = 2: A(i,j) = -1 or 1 with equal probability, |
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2133 ## K = 3: A(i,j) = -1, 0 or 1 with equal probability. |
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2134 ## N may be a 2-vector, in which case the matrix is N(1)-by-N(2). |
| 16634 | 2135 |
| 2136 if (nargin < 1 || nargin > 2) | |
| 2137 error ("gallery: 1 to 2 arguments are required for rando matrix."); | |
| 2138 elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n) | |
| 2139 error ("gallery: N must be an integer for rando matrix."); | |
| 2140 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 2141 error ("gallery: K must be a numeric scalar for smoke matrix."); | |
| 2142 endif | |
| 2143 | |
| 2144 ## Parameter n specifies dimension: m-by-n. | |
| 2145 m = n(1); | |
| 2146 n = n(end); | |
| 2147 | |
| 2148 switch (k) | |
| 2149 case (1), A = floor ( rand(m, n) + 0.5); # {0, 1} | |
| 2150 case (2), A = 2*floor ( rand(m, n) + 0.5) -1; # {-1, 1} | |
| 2151 case (3), A = round (3*rand(m, n) - 1.5); # {-1, 0, 1} | |
| 2152 otherwise | |
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2153 error ("gallery: unknown K '%d' for smoke matrix.", k); |
| 16634 | 2154 endswitch |
| 2155 | |
| 2156 endfunction | |
| 2157 | |
|
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|
2158 function A = randsvd (n, kappa = sqrt (1/eps), mode = 3, kl = max (n) -1, |
|
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|
2159 ku = kl) |
| 16634 | 2160 ## RANDSVD Random matrix with pre-assigned singular values. |
|
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2161 ## RANDSVD(N, KAPPA, MODE, KL, KU) is a (banded) random matrix of order N |
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2162 ## with COND(A) = KAPPA and singular values from the distribution MODE. |
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2163 ## N may be a 2-vector, in which case the matrix is N(1)-by-N(2). |
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2164 ## Available types: |
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2165 ## MODE = 1: one large singular value, |
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2166 ## MODE = 2: one small singular value, |
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2167 ## MODE = 3: geometrically distributed singular values, |
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|
2168 ## MODE = 4: arithmetically distributed singular values, |
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2169 ## MODE = 5: random singular values with unif. dist. logarithm. |
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2170 ## If omitted, MODE defaults to 3, and KAPPA defaults to SQRT(1/EPS). |
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2171 ## If MODE < 0 then the effect is as for ABS(MODE) except that in the |
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2172 ## original matrix of singular values the order of the diagonal entries |
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2173 ## is reversed: small to large instead of large to small. |
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2174 ## KL and KU are the lower and upper bandwidths respectively; if they |
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2175 ## are omitted a full matrix is produced. |
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|
2176 ## If only KL is present, KU defaults to KL. |
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|
2177 ## Special case: if KAPPA < 0 then a random full symmetric positive |
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|
2178 ## definite matrix is produced with COND(A) = -KAPPA and |
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|
2179 ## eigenvalues distributed according to MODE. |
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2180 ## KL and KU, if present, are ignored. |
| 16634 | 2181 ## |
|
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|
2182 ## Reference: |
|
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|
2183 ## N.J. Higham, Accuracy and Stability of Numerical Algorithms, |
|
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|
2184 ## Society for Industrial and Applied Mathematics, Philadelphia, PA, |
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|
2185 ## USA, 1996; sec. 26.3. |
| 16634 | 2186 ## |
|
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|
2187 ## This routine is similar to the more comprehensive Fortran routine xLATMS |
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|
2188 ## in the following reference: |
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|
2189 ## J.W. Demmel and A. McKenney, A test matrix generation suite, |
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|
2190 ## LAPACK Working Note #9, Courant Institute of Mathematical Sciences, |
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|
2191 ## New York, 1989. |
| 16634 | 2192 |
| 2193 if (nargin < 1 || nargin > 5) | |
| 2194 error ("gallery: 1 to 5 arguments are required for randsvd matrix."); | |
| 2195 elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n) | |
| 2196 error ("gallery: N must be a 1 or 2 element integer vector for randsvd matrix."); | |
| 2197 elseif (! isnumeric (kappa) || ! isscalar (kappa)) | |
| 2198 error ("gallery: KAPPA must be a numeric scalar for randsvd matrix."); | |
| 2199 elseif (abs (kappa) < 1) | |
| 2200 error ("gallery: KAPPA must larger than or equal to 1 for randsvd matrix."); | |
| 2201 elseif (! isnumeric (mode) || ! isscalar (mode)) | |
| 2202 error ("gallery: MODE must be a numeric scalar for randsvd matrix."); | |
| 2203 elseif (! isnumeric (kl) || ! isscalar (kl)) | |
| 2204 error ("gallery: KL must be a numeric scalar for randsvd matrix."); | |
| 2205 elseif (! isnumeric (ku) || ! isscalar (ku)) | |
| 2206 error ("gallery: KU must be a numeric scalar for randsvd matrix."); | |
| 2207 endif | |
| 2208 | |
| 2209 posdef = 0; | |
| 2210 if (kappa < 0) | |
| 2211 posdef = 1; | |
| 2212 kappa = -kappa; | |
| 2213 endif | |
| 2214 | |
| 2215 ## Parameter n specifies dimension: m-by-n. | |
| 2216 m = n(1); | |
| 2217 n = n(end); | |
| 2218 p = min ([m n]); | |
| 2219 | |
| 2220 ## If A will be a vector | |
| 2221 if (p == 1) | |
| 2222 A = randn (m, n); | |
|
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|
2223 A /= norm (A); |
|
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|
2224 return; |
|
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|
2225 endif |
| 16634 | 2226 |
| 2227 ## Set up vector sigma of singular values. | |
| 2228 switch (abs (mode)) | |
| 2229 case (1) | |
| 2230 sigma = ones (p, 1) ./ kappa; | |
| 2231 sigma(1) = 1; | |
| 2232 case (2) | |
| 2233 sigma = ones (p, 1); | |
| 2234 sigma(p) = 1 / kappa; | |
| 2235 case (3) | |
| 2236 factor = kappa^(-1/(p-1)); | |
| 2237 sigma = factor.^[0:p-1]; | |
| 2238 case (4) | |
| 2239 sigma = ones (p, 1) - (0:p-1)'/(p-1)*(1-1/kappa); | |
| 2240 case (5) | |
| 2241 ## In this case cond (A) <= kappa. | |
| 2242 rand ("uniform"); | |
| 2243 sigma = exp (-rand (p, 1) * log (kappa)); | |
| 2244 otherwise | |
|
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|
2245 error ("gallery: unknown MODE '%d' for randsvd matrix.", mode); |
| 16634 | 2246 endswitch |
| 2247 | |
| 2248 ## Convert to diagonal matrix of singular values. | |
| 2249 if (mode < 0) | |
|
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|
2250 sigma = sigma(p:-1:1); |
|
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|
2251 endif |
| 16634 | 2252 sigma = diag (sigma); |
| 2253 | |
| 2254 if (posdef) | |
| 2255 ## handle case where KAPPA was negative | |
| 2256 Q = qmult (p); | |
| 2257 A = Q' * sigma * Q; | |
| 2258 A = (A + A') / 2; # Ensure matrix is symmetric. | |
|
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|
2259 return; |
| 16634 | 2260 endif |
| 2261 | |
| 2262 if (m != n) | |
| 2263 ## Expand to m-by-n diagonal matrix | |
| 2264 sigma(m, n) = 0; | |
|
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|
2265 endif |
| 16634 | 2266 |
| 2267 if (kl == 0 && ku == 0) | |
| 2268 ## Diagonal matrix requested - nothing more to do. | |
| 2269 A = sigma; | |
| 2270 else | |
| 2271 ## A = U*sigma*V, where U, V are random orthogonal matrices from the | |
| 2272 ## Haar distribution. | |
| 2273 A = qmult (sigma'); | |
| 2274 A = qmult (A'); | |
| 2275 | |
| 2276 if (kl < n-1 || ku < n-1) | |
| 2277 ## Bandwidth reduction | |
| 2278 A = bandred (A, kl, ku); | |
| 2279 endif | |
| 2280 endif | |
| 2281 endfunction | |
| 2282 | |
| 2283 function A = redheff (n) | |
| 2284 ## REDHEFF A (0,1) matrix of Redheffer associated with the Riemann hypothesis. | |
|
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|
2285 ## A = REDHEFF(N) is an N-by-N matrix of 0s and 1s defined by |
|
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|
2286 ## A(i,j) = 1 if j = 1 or if i divides j, |
|
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|
2287 ## A(i,j) = 0 otherwise. |
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|
2288 ## It has N - FLOOR(LOG2(N)) - 1 eigenvalues equal to 1, |
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|
2289 ## a real eigenvalue (the spectral radius) approximately SQRT(N), |
|
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|
2290 ## a negative eigenvalue approximately -SQRT(N), |
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|
2291 ## and the remaining eigenvalues are provably ``small''. |
|
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|
2292 ## Barrett and Jarvis (1992) conjecture that |
|
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|
2293 ## ``the small eigenvalues all lie inside the unit circle |
|
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|
2294 ## ABS(Z) = 1'', |
|
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|
2295 ## and a proof of this conjecture, together with a proof that some |
|
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|
2296 ## eigenvalue tends to zero as N tends to infinity, would yield |
|
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|
2297 ## a new proof of the prime number theorem. |
|
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|
2298 ## The Riemann hypothesis is true if and only if |
|
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|
2299 ## DET(A) = O( N^(1/2+epsilon) ) for every epsilon > 0 |
|
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|
2300 ## ('!' denotes factorial). |
|
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|
2301 ## See also RIEMANN. |
| 16634 | 2302 ## |
|
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|
2303 ## Reference: |
|
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|
2304 ## W.W. Barrett and T.J. Jarvis, |
|
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|
2305 ## Spectral Properties of a Matrix of Redheffer, |
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|
2306 ## Linear Algebra and Appl., 162 (1992), pp. 673-683. |
| 16634 | 2307 |
|
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|
2308 if (nargin < 1) |
| 16634 | 2309 error ("gallery: 1 argument is required for redheff matrix."); |
| 2310 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2311 error ("gallery: N must be an integer for redheff matrix."); | |
| 2312 endif | |
| 2313 | |
| 2314 i = (1:n)' * ones (1, n); | |
| 2315 A = ! rem (i', i); | |
| 2316 A(:,1) = ones (n, 1); | |
| 2317 endfunction | |
| 2318 | |
| 2319 function A = riemann (n) | |
| 2320 ## RIEMANN A matrix associated with the Riemann hypothesis. | |
|
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|
2321 ## A = RIEMANN(N) is an N-by-N matrix for which the |
|
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2322 ## Riemann hypothesis is true if and only if |
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|
2323 ## DET(A) = O( N! N^(-1/2+epsilon) ) for every epsilon > 0 |
|
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2324 ## ('!' denotes factorial). |
|
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|
2325 ## A = B(2:N+1, 2:N+1), where |
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|
2326 ## B(i,j) = i-1 if i divides j and -1 otherwise. |
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2327 ## Properties include, with M = N+1: |
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|
2328 ## Each eigenvalue E(i) satisfies ABS(E(i)) <= M - 1/M. |
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|
2329 ## i <= E(i) <= i+1 with at most M-SQRT(M) exceptions. |
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|
2330 ## All integers in the interval (M/3, M/2] are eigenvalues. |
| 16634 | 2331 ## |
|
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2332 ## See also REDHEFF. |
| 16634 | 2333 ## |
|
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2334 ## Reference: |
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|
2335 ## F. Roesler, Riemann's hypothesis as an eigenvalue problem, |
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|
2336 ## Linear Algebra and Appl., 81 (1986), pp. 153-198. |
| 16634 | 2337 |
|
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|
2338 if (nargin < 1) |
| 16634 | 2339 error ("gallery: 1 argument is required for riemann matrix."); |
| 2340 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2341 error ("gallery: N must be an integer for riemann matrix."); | |
| 2342 endif | |
| 2343 | |
|
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|
2344 n += 1; |
| 16634 | 2345 i = (2:n)' * ones (1, n-1); |
| 2346 j = i'; | |
| 2347 A = i .* (! rem (j, i)) - ones (n-1); | |
| 2348 endfunction | |
| 2349 | |
| 2350 function A = ris (n) | |
| 2351 ## NOTE: this function was named dingdong in the original Test Matrix Toolbox | |
| 2352 ## RIS Dingdong matrix - a symmetric Hankel matrix. | |
|
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|
2353 ## A = RIS(N) is the symmetric N-by-N Hankel matrix with |
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|
2354 ## A(i,j) = 0.5/(N-i-j+1.5). |
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|
2355 ## The eigenvalues of A cluster around PI/2 and -PI/2. |
| 16634 | 2356 ## |
|
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|
2357 ## Invented by F.N. Ris. |
| 16634 | 2358 ## |
|
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|
2359 ## Reference: |
|
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|
2360 ## J.C. Nash, Compact Numerical Methods for Computers: Linear |
|
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|
2361 ## Algebra and Function Minimisation, second edition, Adam Hilger, |
|
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|
2362 ## Bristol, 1990 (Appendix 1). |
| 16634 | 2363 |
|
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|
2364 if (nargin < 1) |
| 16634 | 2365 error ("gallery: 1 argument is required for ris matrix."); |
| 2366 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2367 error ("gallery: N must be an integer for ris matrix."); | |
| 2368 endif | |
| 2369 | |
| 2370 p = -2*(1:n) + (n+1.5); | |
| 2371 A = cauchy (p); | |
| 2372 endfunction | |
| 2373 | |
| 2374 function A = smoke (n, k = 0) | |
|
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|
2375 ## SMOKE Smoke matrix - complex, with a 'smoke ring' pseudospectrum. |
|
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|
2376 ## SMOKE(N) is an N-by-N matrix with 1s on the |
|
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|
2377 ## superdiagonal, 1 in the (N,1) position, and powers of |
|
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|
2378 ## roots of unity along the diagonal. |
|
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|
2379 ## SMOKE(N, 1) is the same except for a zero (N,1) element. |
|
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|
2380 ## The eigenvalues of SMOKE(N, 1) are the N'th roots of unity; |
|
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|
2381 ## those of SMOKE(N) are the N'th roots of unity times 2^(1/N). |
| 16634 | 2382 ## |
|
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|
2383 ## Try PS(SMOKE(32)). For SMOKE(N, 1) the pseudospectrum looks |
|
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|
2384 ## like a sausage folded back on itself. |
|
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|
2385 ## GERSH(SMOKE(N, 1)) is interesting. |
| 16634 | 2386 ## |
|
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|
2387 ## Reference: |
|
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|
2388 ## L. Reichel and L.N. Trefethen, Eigenvalues and pseudo-eigenvalues of |
|
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|
2389 ## Toeplitz matrices, Linear Algebra and Appl., 162-164:153-185, 1992. |
| 16634 | 2390 |
| 2391 if (nargin < 1 || nargin > 2) | |
| 2392 error ("gallery: 1 to 2 arguments are required for smoke matrix."); | |
| 2393 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2394 error ("gallery: N must be an integer for smoke matrix."); | |
| 2395 elseif (! isnumeric (n) || ! isscalar (n)) | |
| 2396 error ("gallery: K must be a numeric scalar for smoke matrix."); | |
| 2397 endif | |
| 2398 | |
|
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|
2399 w = exp (2*pi*i/n); |
|
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|
2400 A = diag ( [w.^(1:n-1) 1] ) + diag (ones (n-1,1), 1); |
| 16634 | 2401 |
| 2402 switch (k) | |
| 2403 case (0), A(n,1) = 1; | |
| 2404 case (1), # do nothing | |
| 2405 otherwise, | |
| 2406 error ("gallery: K must have a value of 0 or 1 for smoke matrix."); | |
| 2407 endswitch | |
| 2408 endfunction | |
| 2409 | |
| 2410 function T = toeppd (n, m = n, w = rand (m,1), theta = rand (m,1)) | |
| 2411 ## NOTE: this function was named pdtoep in the original Test Matrix Toolbox | |
| 2412 ## TOEPPD Symmetric positive definite Toeplitz matrix. | |
|
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|
2413 ## TOEPPD(N, M, W, THETA) is an N-by-N symmetric positive (semi-) |
|
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|
2414 ## definite (SPD) Toeplitz matrix, comprised of the sum of M rank 2 |
|
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|
2415 ## (or, for certain THETA, rank 1) SPD Toeplitz matrices. |
|
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|
2416 ## Specifically, |
|
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|
2417 ## T = W(1)*T(THETA(1)) + ... + W(M)*T(THETA(M)), |
|
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|
2418 ## where T(THETA(k)) has (i,j) element COS(2*PI*THETA(k)*(i-j)). |
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|
2419 ## Defaults: M = N, W = RAND(M,1), THETA = RAND(M,1). |
| 16634 | 2420 ## |
|
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|
2421 ## Reference: |
|
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|
2422 ## G. Cybenko and C.F. Van Loan, Computing the minimum eigenvalue of |
|
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|
2423 ## a symmetric positive definite Toeplitz matrix, SIAM J. Sci. Stat. |
|
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|
2424 ## Comput., 7 (1986), pp. 123-131. |
| 16634 | 2425 |
| 2426 if (nargin < 1 || nargin > 4) | |
| 2427 error ("gallery: 1 to 4 arguments are required for toeppd matrix."); | |
| 2428 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2429 error ("gallery: N must be a numeric integer for toeppd matrix."); | |
| 2430 elseif (! isnumeric (m) || ! isscalar (m) || fix (m) != m) | |
| 2431 error ("gallery: M must be a numeric integer for toeppd matrix."); | |
| 2432 elseif (numel (w) != m || numel (theta) != m) | |
| 2433 error ("gallery: W and THETA must be vectors of length M for toeppd matrix."); | |
| 2434 endif | |
| 2435 | |
| 2436 T = zeros (n); | |
| 2437 E = 2*pi * ((1:n)' * ones (1, n) - ones (n, 1) * (1:n)); | |
| 2438 | |
| 2439 for i = 1:m | |
|
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|
2440 T += w(i) * cos (theta(i)*E); |
| 16634 | 2441 endfor |
| 2442 endfunction | |
| 2443 | |
| 2444 function P = toeppen (n, a = 1, b = -10, c = 0, d = 10, e = 1) | |
| 2445 ## NOTE: this function was named pentoep in the original Test Matrix Toolbox | |
| 2446 ## TOEPPEN Pentadiagonal Toeplitz matrix (sparse). | |
|
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|
2447 ## P = TOEPPEN(N, A, B, C, D, E) is the N-by-N pentadiagonal |
|
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|
2448 ## Toeplitz matrix with diagonals composed of the numbers |
|
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|
2449 ## A =: P(3,1), B =: P(2,1), C =: P(1,1), D =: P(1,2), E =: P(1,3). |
|
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|
2450 ## Default: (A,B,C,D,E) = (1,-10,0,10,1) (a matrix of Rutishauser). |
|
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|
2451 ## This matrix has eigenvalues lying approximately on |
|
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|
2452 ## the line segment 2*cos(2*t) + 20*i*sin(t). |
| 16634 | 2453 ## |
|
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|
2454 ## Interesting plots are |
|
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John W. Eaton <jwe@octave.org>
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27085
diff
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|
2455 ## PS(FULL(TOEPPEN(32,0,1,0,0,1/4))) - 'triangle' |
|
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|
2456 ## PS(FULL(TOEPPEN(32,0,1/2,0,0,1))) - 'propeller' |
|
823b4bcf79fc
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John W. Eaton <jwe@octave.org>
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diff
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|
2457 ## PS(FULL(TOEPPEN(32,0,1/2,1,1,1))) - 'fish' |
| 16634 | 2458 ## |
|
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|
2459 ## References: |
|
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|
2460 ## R.M. Beam and R.F. Warming, The asymptotic spectra of |
|
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|
2461 ## banded Toeplitz and quasi-Toeplitz matrices, SIAM J. Sci. |
|
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|
2462 ## Comput. 14 (4), 1993, pp. 971-1006. |
|
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|
2463 ## H. Rutishauser, On test matrices, Programmation en Mathematiques |
|
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|
2464 ## Numeriques, Editions Centre Nat. Recherche Sci., Paris, 165, |
|
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|
2465 ## 1966, pp. 349-365. |
| 16634 | 2466 |
| 2467 if (nargin < 1 || nargin > 6) | |
| 2468 error ("gallery: 1 to 6 arguments are required for toeppen matrix."); | |
| 2469 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2470 error ("gallery: N must be a numeric integer for toeppen matrix."); | |
|
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Replace cellfun() occurrences with faster code where possible.
Rik <rik@octave.org>
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17338
diff
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|
2471 elseif (any (! cellfun ("isnumeric", {a b c d e})) || any (cellfun ("numel", {a b c d e}) != 1)) |
| 16634 | 2472 error ("gallery: A, B, C, D and E must be numeric scalars for toeppen matrix."); |
| 2473 endif | |
| 2474 | |
| 2475 P = spdiags ([a*ones(n,1) b*ones(n,1) c*ones(n,1) d*ones(n,1) e*ones(n,1)], | |
| 2476 -2:2, n, n); | |
| 2477 endfunction | |
| 2478 | |
| 2479 function T = tridiag (n, x = -1, y = 2, z = -1) | |
| 2480 ## TRIDIAG Tridiagonal matrix (sparse). | |
|
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|
2481 ## TRIDIAG(X, Y, Z) is the tridiagonal matrix with subdiagonal X, |
|
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|
2482 ## diagonal Y, and superdiagonal Z. |
|
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|
2483 ## X and Z must be vectors of dimension one less than Y. |
|
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|
2484 ## Alternatively TRIDIAG(N, C, D, E), where C, D, and E are all |
|
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|
2485 ## scalars, yields the Toeplitz tridiagonal matrix of order N |
|
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|
2486 ## with subdiagonal elements C, diagonal elements D, and superdiagonal |
|
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|
2487 ## elements E. This matrix has eigenvalues (Todd 1977) |
|
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diff
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|
2488 ## D + 2*SQRT(C*E)*COS(k*PI/(N+1)), k=1:N. |
|
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diff
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|
2489 ## TRIDIAG(N) is the same as TRIDIAG(N,-1,2,-1), which is |
|
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|
2490 ## a symmetric positive definite M-matrix (the negative of the |
|
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diff
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|
2491 ## second difference matrix). |
| 16634 | 2492 ## |
|
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|
2493 ## References: |
|
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|
2494 ## J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, |
|
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|
2495 ## Birkhauser, Basel, and Academic Press, New York, 1977, p. 155. |
|
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|
2496 ## D.E. Rutherford, Some continuant determinants arising in physics and |
|
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diff
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|
2497 ## chemistry---II, Proc. Royal Soc. Edin., 63, A (1952), pp. 232-241. |
| 16634 | 2498 |
| 2499 if (nargin != 1 && nargin != 3 && nargin != 4) | |
| 2500 error ("gallery: 1, 3, or 4 arguments are required for tridiag matrix."); | |
| 2501 elseif (nargin == 3) | |
| 2502 z = y; | |
| 2503 y = x; | |
| 2504 x = n; | |
| 2505 endif | |
| 2506 | |
| 2507 ## Force column vectors | |
| 2508 x = x(:); | |
| 2509 y = y(:); | |
| 2510 z = z(:); | |
| 2511 | |
| 2512 if (isscalar (x) && isscalar (y) && isscalar (z)) | |
| 2513 x *= ones (n-1, 1); | |
| 2514 z *= ones (n-1, 1); | |
| 2515 y *= ones (n, 1); | |
| 2516 elseif (numel (y) != numel (x) + 1) | |
| 2517 error ("gallery: X must have one element less than Y for tridiag matrix."); | |
| 2518 elseif (numel (y) != numel (z) + 1) | |
| 2519 error ("gallery: Z must have one element less than Y for tridiag matrix."); | |
| 2520 endif | |
| 2521 | |
| 2522 ## T = diag (x, -1) + diag (y) + diag (z, 1); # For non-sparse matrix. | |
| 2523 n = numel (y); | |
| 2524 T = spdiags ([[x;0] y [0;z]], -1:1, n, n); | |
| 2525 endfunction | |
| 2526 | |
|
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|
2527 function t = triw (n, alpha = -1, k = n(end) - 1) |
| 16634 | 2528 ## TRIW Upper triangular matrix discussed by Wilkinson and others. |
|
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|
2529 ## TRIW(N, ALPHA, K) is the upper triangular matrix with ones on |
|
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|
2530 ## the diagonal and ALPHAs on the first K >= 0 superdiagonals. |
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|
2531 ## N may be a 2-vector, in which case the matrix is N(1)-by-N(2) and |
|
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|
2532 ## upper trapezoidal. |
|
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|
2533 ## Defaults: ALPHA = -1, |
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|
2534 ## K = N - 1 (full upper triangle). |
|
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|
2535 ## TRIW(N) is a matrix discussed by Kahan, Golub and Wilkinson. |
| 16634 | 2536 ## |
|
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|
2537 ## Ostrowski (1954) shows that |
|
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|
2538 ## COND(TRIW(N,2)) = COT(PI/(4*N))^2, |
|
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|
2539 ## and for large ABS(ALPHA), |
|
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|
2540 ## COND(TRIW(N,ALPHA)) is approximately ABS(ALPHA)^N*SIN(PI/(4*N-2)). |
| 16634 | 2541 ## |
|
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|
2542 ## Adding -2^(2-N) to the (N,1) element makes TRIW(N) singular, |
|
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|
2543 ## as does adding -2^(1-N) to all elements in the first column. |
| 16634 | 2544 ## |
|
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Rik <rik@octave.org>
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diff
changeset
|
2545 ## References: |
|
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maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
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diff
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|
2546 ## G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the |
|
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Rik <rik@octave.org>
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diff
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|
2547 ## computation of the Jordan canonical form, SIAM Review, |
|
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maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
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diff
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|
2548 ## 18(4), 1976, pp. 578-619. |
|
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Rik <rik@octave.org>
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19697
diff
changeset
|
2549 ## W. Kahan, Numerical linear algebra, Canadian Math. Bulletin, |
|
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Rik <rik@octave.org>
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diff
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|
2550 ## 9 (1966), pp. 757-801. |
|
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Rik <rik@octave.org>
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diff
changeset
|
2551 ## A.M. Ostrowski, On the spectrum of a one-parametric family of |
|
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maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
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diff
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|
2552 ## matrices, J. Reine Angew. Math., 193 (3/4), 1954, pp. 143-160. |
|
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diff
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|
2553 ## J.H. Wilkinson, Singular-value decomposition---basic aspects, |
|
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Rik <rik@octave.org>
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diff
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|
2554 ## in D.A.H. Jacobs, ed., Numerical Software---Needs and Availability, |
|
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Rik <rik@octave.org>
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diff
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|
2555 ## Academic Press, London, 1978, pp. 109-135. |
| 16634 | 2556 |
| 2557 if (nargin < 1 || nargin > 3) | |
| 2558 error ("gallery: 1 to 3 arguments are required for triw matrix."); | |
| 2559 elseif (! isnumeric (n) || all (numel (n) != [1 2])) | |
| 2560 error ("gallery: N must be a 1 or 2 elements vector for triw matrix."); | |
| 2561 elseif (! isscalar (alpha)) | |
| 2562 error ("gallery: ALPHA must be a scalar for triw matrix."); | |
|
18846
3d33fe79816c
gallery.m: Return correct matrix if 3rd argument is not specified.
Rik <rik@octave.org>
parents:
18812
diff
changeset
|
2563 elseif (! isscalar (k) || ! isnumeric (k) || fix (k) != k || k < 0) |
|
3d33fe79816c
gallery.m: Return correct matrix if 3rd argument is not specified.
Rik <rik@octave.org>
parents:
18812
diff
changeset
|
2564 error ("gallery: K must be a numeric integer >= 0 for triw matrix."); |
| 16634 | 2565 endif |
| 2566 | |
| 2567 m = n(1); # Parameter n specifies dimension: m-by-n. | |
| 2568 n = n(end); | |
| 2569 | |
| 2570 t = tril (eye (m, n) + alpha * triu (ones (m, n), 1), k); | |
| 2571 endfunction | |
| 2572 | |
|
16978
00379f9f8773
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Rik <rik@octave.org>
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diff
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|
2573 function A = uniformdata (varargin) |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
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16933
diff
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|
2574 |
|
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gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
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16933
diff
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|
2575 if (nargin < 2) |
|
16979
9aa293e00475
gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents:
16978
diff
changeset
|
2576 error ("gallery: At least 2 arguments required for uniformdata matrix."); |
|
16978
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
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|
2577 endif |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2578 if (isnumeric (varargin{end})) |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
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16933
diff
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|
2579 jidx = varargin{end}; |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2580 svec = [varargin{:}]; |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
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16933
diff
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|
2581 varargin(end) = []; |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
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|
2582 elseif (ischar (varargin{end})) |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
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16933
diff
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|
2583 if (nargin < 3) |
|
16979
9aa293e00475
gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents:
16978
diff
changeset
|
2584 error (["gallery: CLASS argument requires 3 inputs " ... |
|
9aa293e00475
gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents:
16978
diff
changeset
|
2585 "for uniformdata matrix."]); |
|
16978
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
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16933
diff
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|
2586 endif |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2587 jidx = varargin{end-1}; |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2588 svec = [varargin{1:end-1}]; |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
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diff
changeset
|
2589 varargin(end-1) = []; |
|
19593
446c46af4b42
strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
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17386
diff
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|
2590 else |
|
16979
9aa293e00475
gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents:
16978
diff
changeset
|
2591 error (["gallery: J must be an integer in the range [0, 2^32-1] " ... |
|
9aa293e00475
gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents:
16978
diff
changeset
|
2592 "for uniformdata matrix"]); |
|
16978
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2593 endif |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2594 |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
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diff
changeset
|
2595 if (! (isnumeric (jidx) && isscalar (jidx) |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
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|
2596 && jidx == fix (jidx) |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2597 && jidx >= 0 && jidx <= 0xFFFFFFFF)) |
|
16979
9aa293e00475
gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents:
16978
diff
changeset
|
2598 error (["gallery: J must be an integer in the range [0, 2^32-1] " ... |
|
9aa293e00475
gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents:
16978
diff
changeset
|
2599 "for uniformdata matrix"]); |
|
16978
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2600 endif |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2601 |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2602 ## Save and restore random state. Initialization done so that reproducible |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2603 ## data is available from gallery depending on the jidx and size vector. |
|
19593
446c46af4b42
strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents:
17386
diff
changeset
|
2604 randstate = rand ("state"); |
|
16978
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2605 unwind_protect |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2606 rand ("state", svec); |
|
00379f9f8773
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Rik <rik@octave.org>
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16933
diff
changeset
|
2607 A = rand (varargin{:}); |
|
00379f9f8773
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Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2608 unwind_protect_cleanup |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
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|
2609 rand ("state", randstate); |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2610 end_unwind_protect |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2611 |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2612 endfunction |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2613 |
| 16634 | 2614 function A = wathen (nx, ny, k = 0) |
|
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Rik <rik@octave.org>
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diff
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|
2615 ## WATHEN returns the Wathen matrix. |
| 16634 | 2616 ## |
|
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Rik <rik@octave.org>
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diff
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|
2617 ## Discussion: |
| 16634 | 2618 ## |
|
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Rik <rik@octave.org>
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19697
diff
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|
2619 ## The Wathen matrix is a finite element matrix which is sparse. |
| 16634 | 2620 ## |
|
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Rik <rik@octave.org>
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19697
diff
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|
2621 ## The entries of the matrix depend in part on a physical quantity |
|
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Rik <rik@octave.org>
parents:
19697
diff
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|
2622 ## related to density. That density is here assigned random values between |
|
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Rik <rik@octave.org>
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19697
diff
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|
2623 ## 0 and 100. |
| 16634 | 2624 ## |
|
19833
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Rik <rik@octave.org>
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19697
diff
changeset
|
2625 ## A = WATHEN ( NX, NY ) is a sparse random N-by-N finite element matrix |
|
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Rik <rik@octave.org>
parents:
19697
diff
changeset
|
2626 ## where N = 3*NX*NY + 2*NX + 2*NY + 1. |
| 16634 | 2627 ## |
|
19833
9fc020886ae9
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Rik <rik@octave.org>
parents:
19697
diff
changeset
|
2628 ## A is the consistent mass matrix for a regular NX-by-NY |
|
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Rik <rik@octave.org>
parents:
19697
diff
changeset
|
2629 ## grid of 8-node (serendipity) elements in 2 space dimensions. |
| 16634 | 2630 ## |
|
19833
9fc020886ae9
maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents:
19697
diff
changeset
|
2631 ## Here is an illustration for NX = 3, NX = 2: |
| 16634 | 2632 ## |
|
19833
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Rik <rik@octave.org>
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19697
diff
changeset
|
2633 ## 23-24-25-26-27-28-29 |
|
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maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents:
19697
diff
changeset
|
2634 ## | | | | |
|
9fc020886ae9
maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents:
19697
diff
changeset
|
2635 ## 19 20 21 22 |
|
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maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents:
19697
diff
changeset
|
2636 ## | | | | |
|
9fc020886ae9
maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents:
19697
diff
changeset
|
2637 ## 12-13-14-15-16-17-18 |
|
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Rik <rik@octave.org>
parents:
19697
diff
changeset
|
2638 ## | | | | |
|
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maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents:
19697
diff
changeset
|
2639 ## 8 9 10 11 |
|
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Rik <rik@octave.org>
parents:
19697
diff
changeset
|
2640 ## | | | | |
|
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maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents:
19697
diff
changeset
|
2641 ## 1--2--3--4--5--6--7 |
| 16634 | 2642 ## |
|
19833
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maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
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19697
diff
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|
2643 ## For this example, the total number of nodes is, as expected, |
| 16634 | 2644 ## |
|
19833
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Rik <rik@octave.org>
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19697
diff
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|
2645 ## N = 3 * 3 * 2 + 2 * 2 + 2 * 3 + 1 = 29. |
| 16634 | 2646 ## |
|
19833
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Rik <rik@octave.org>
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19697
diff
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|
2647 ## A is symmetric positive definite for any (positive) values of |
|
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Rik <rik@octave.org>
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19697
diff
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|
2648 ## the density, RHO(NX,NY), which is chosen randomly in this routine. |
| 16634 | 2649 ## |
|
19833
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diff
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|
2650 ## In particular, if D = DIAG(DIAG(A)), then |
|
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diff
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|
2651 ## 0.25 <= EIG(INV(D)*A) <= 4.5 |
|
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19697
diff
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|
2652 ## for any positive integers NX and NY and any densities RHO(NX,NY). |
| 16634 | 2653 ## |
|
19833
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Rik <rik@octave.org>
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diff
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|
2654 ## A = WATHEN ( NX, NY, 1 ) returns the diagonally scaled matrix. |
| 16634 | 2655 ## |
|
19833
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diff
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|
2656 ## Modified: |
| 16634 | 2657 ## |
|
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Rik <rik@octave.org>
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diff
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|
2658 ## 17 September 2007 |
| 16634 | 2659 ## |
|
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diff
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|
2660 ## Author: |
| 16634 | 2661 ## |
|
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diff
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|
2662 ## Nicholas Higham |
| 16634 | 2663 ## |
|
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diff
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|
2664 ## Reference: |
| 16634 | 2665 ## |
|
19833
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diff
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|
2666 ## Nicholas Higham, |
|
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diff
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|
2667 ## Algorithm 694: A Collection of Test Matrices in MATLAB, |
|
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19697
diff
changeset
|
2668 ## ACM Transactions on Mathematical Software, |
|
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19697
diff
changeset
|
2669 ## Volume 17, Number 3, September 1991, pages 289-305. |
| 16634 | 2670 ## |
|
19833
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diff
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|
2671 ## Andrew Wathen, |
|
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19697
diff
changeset
|
2672 ## Realistic eigenvalue bounds for the Galerkin mass matrix, |
|
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diff
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|
2673 ## IMA Journal of Numerical Analysis, |
|
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diff
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|
2674 ## Volume 7, 1987, pages 449-457. |
| 16634 | 2675 ## |
|
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|
2676 ## Parameters: |
| 16634 | 2677 ## |
|
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|
2678 ## Input, integer NX, NY, the number of elements in the X and Y directions |
|
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|
2679 ## of the finite element grid. NX and NY must each be at least 1. |
| 16634 | 2680 ## |
|
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|
2681 ## Optional input, integer K, is used to request that the diagonally scaled |
|
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|
2682 ## version of the matrix be returned. This happens if K is specified with |
|
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|
2683 ## the value 1. |
| 16634 | 2684 ## |
|
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|
2685 ## Output, sparse real A(N,N), the matrix. The dimension N is determined by |
|
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|
2686 ## NX and NY, as described above. A is stored in the MATLAB sparse matrix |
|
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|
2687 ## format. |
| 16634 | 2688 |
| 2689 if (nargin < 2 || nargin > 3) | |
| 2690 error ("gallery: 2 or 3 arguments are required for wathen matrix."); | |
| 2691 elseif (! isnumeric (nx) || ! isscalar (nx) || nx < 1) | |
| 2692 error ("gallery: NX must be a positive scalar for wathen matrix."); | |
| 2693 elseif (! isnumeric (ny) || ! isscalar (ny) || ny < 1) | |
| 2694 error ("gallery: NY must be a positive scalar for wathen matrix."); | |
| 2695 elseif (! isscalar (k)) | |
| 2696 error ("gallery: K must be a scalar for wathen matrix."); | |
| 2697 endif | |
| 2698 | |
| 2699 e1 = [ 6 -6 2 -8 | |
| 2700 -6 32 -6 20 | |
| 2701 2 -6 6 -6 | |
| 2702 -8 20 -6 32 ]; | |
| 2703 | |
| 2704 e2 = [ 3 -8 2 -6 | |
| 2705 -8 16 -8 20 | |
| 2706 2 -8 3 -8 | |
| 2707 -6 20 -8 16 ]; | |
| 2708 | |
| 2709 e = [ e1 e2 | |
| 2710 e2' e1] / 45; | |
| 2711 | |
| 2712 n = 3*nx*ny + 2*nx + 2*ny + 1; | |
| 2713 | |
| 2714 A = sparse (n, n); | |
| 2715 | |
| 2716 rho = 100 * rand (nx, ny); | |
| 2717 | |
| 2718 for j = 1:ny | |
| 2719 for i = 1:nx | |
|
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2720 ## |
| 16634 | 2721 ## For the element (I,J), determine the indices of the 8 nodes. |
|
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2722 ## |
| 16634 | 2723 nn(1) = 3*j*nx + 2*i + 2*j + 1; |
| 2724 nn(2) = nn(1) - 1; | |
| 2725 nn(3) = nn(2) - 1; | |
| 2726 nn(4) = (3*j - 1) * nx + 2*j + i - 1; | |
| 2727 nn(5) = 3 * (j-1) * nx + 2*i + 2*j - 3; | |
| 2728 nn(6) = nn(5) + 1; | |
| 2729 nn(7) = nn(6) + 1; | |
| 2730 nn(8) = nn(4) + 1; | |
| 2731 | |
| 2732 em = e * rho(i,j); | |
| 2733 | |
| 2734 for krow = 1:8 | |
| 2735 for kcol = 1:8 | |
| 2736 A(nn(krow),nn(kcol)) = A(nn(krow),nn(kcol)) + em(krow,kcol); | |
| 2737 endfor | |
| 2738 endfor | |
| 2739 | |
| 2740 endfor | |
| 2741 endfor | |
| 2742 | |
| 2743 ## If requested, return A with diagonal scaling. | |
| 2744 if (k) | |
| 2745 A = diag (diag (A)) \ A; | |
| 2746 endif | |
| 2747 endfunction | |
| 2748 | |
| 2749 function [A, b] = wilk (n) | |
| 2750 ## WILK Various specific matrices devised/discussed by Wilkinson. | |
|
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2751 ## [A, b] = WILK(N) is the matrix or system of order N. |
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2752 ## N = 3: upper triangular system Ux=b illustrating inaccurate solution. |
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2753 ## N = 4: lower triangular system Lx=b, ill-conditioned. |
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2754 ## N = 5: HILB(6)(1:5,2:6)*1.8144. Symmetric positive definite. |
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2755 ## N = 21: W21+, tridiagonal. Eigenvalue problem. |
| 16634 | 2756 ## |
|
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2757 ## References: |
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2758 ## J.H. Wilkinson, Error analysis of direct methods of matrix inversion, |
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2759 ## J. Assoc. Comput. Mach., 8 (1961), pp. 281-330. |
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2760 ## J.H. Wilkinson, Rounding Errors in Algebraic Processes, Notes on Applied |
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2761 ## Science No. 32, Her Majesty's Stationery Office, London, 1963. |
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2762 ## J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University |
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2763 ## Press, 1965. |
| 16634 | 2764 |
|
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2765 if (nargin < 1) |
| 16634 | 2766 error ("gallery: 1 argument is required for wilk matrix."); |
| 2767 elseif (! isnumeric (n) || ! isscalar (n)) | |
| 2768 error ("gallery: N must be a numeric scalar for wilk matrix."); | |
| 2769 endif | |
| 2770 | |
| 2771 if (n == 3) | |
| 2772 ## Wilkinson (1961) p.323. | |
| 2773 A = [ 1e-10 0.9 -0.4 | |
| 2774 0 0.9 -0.4 | |
| 2775 0 0 1e-10 ]; | |
| 2776 | |
| 2777 b = [ 0 | |
| 2778 0 | |
| 2779 1]; | |
| 2780 | |
| 2781 elseif (n == 4) | |
| 2782 ## Wilkinson (1963) p.105. | |
| 2783 A = [0.9143e-4 0 0 0 | |
| 2784 0.8762 0.7156e-4 0 0 | |
| 2785 0.7943 0.8143 0.9504e-4 0 | |
| 2786 0.8017 0.6123 0.7165 0.7123e-4]; | |
| 2787 | |
| 2788 b = [0.6524 | |
| 2789 0.3127 | |
| 2790 0.4186 | |
| 2791 0.7853]; | |
| 2792 | |
| 2793 elseif (n == 5) | |
| 2794 ## Wilkinson (1965), p.234. | |
| 2795 A = hilb (6); | |
| 2796 A = A(1:5, 2:6) * 1.8144; | |
| 2797 | |
| 2798 elseif (n == 21) | |
| 2799 ## Wilkinson (1965), p.308. | |
| 2800 E = diag (ones (n-1, 1), 1); | |
| 2801 m = (n-1)/2; | |
| 2802 A = diag (abs (-m:m)) + E + E'; | |
| 2803 | |
| 2804 else | |
|
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2805 error ("gallery: unknown N '%d' for wilk matrix.", n); |
| 16634 | 2806 endif |
| 2807 endfunction | |
| 2808 | |
| 2809 ## NOTE: bandred is part of the Test Matrix Toolbox and is used by randsvd() | |
| 2810 function A = bandred (A, kl, ku) | |
| 2811 ## BANDRED Band reduction by two-sided unitary transformations. | |
|
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2812 ## B = BANDRED(A, KL, KU) is a matrix unitarily equivalent to A |
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2813 ## with lower bandwidth KL and upper bandwidth KU |
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2814 ## (i.e. B(i,j) = 0 if i > j+KL or j > i+KU). |
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2815 ## The reduction is performed using Householder transformations. |
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2816 ## If KU is omitted it defaults to KL. |
| 16634 | 2817 ## |
|
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2818 ## Called by RANDSVD. |
|
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2819 ## This is a 'standard' reduction. Cf. reduction to bidiagonal form |
|
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2820 ## prior to computing the SVD. This code is a little wasteful in that |
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2821 ## it computes certain elements which are immediately set to zero! |
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2822 ## |
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2823 ## Reference: |
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2824 ## G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, |
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2825 ## Johns Hopkins University Press, Baltimore, Maryland, 1989. |
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2826 ## Section 5.4.3. |
| 16634 | 2827 |
| 2828 ## Check for special case where order of left/right transformations matters. | |
| 2829 ## Easiest approach is to work on the transpose, flipping back at the end. | |
| 2830 flip = false; | |
| 2831 if (ku == 0) | |
| 2832 flip = true; | |
| 2833 A = A'; | |
| 2834 [ku, kl] = deal (kl, ku); | |
| 2835 endif | |
| 2836 | |
| 2837 [m, n] = size (A); | |
| 2838 | |
| 2839 for j = 1:min (min (m, n), max (m-kl-1, n-ku-1)) | |
| 2840 if (j+kl+1 <= m) | |
| 2841 [v, beta] = house (A(j+kl:m,j)); | |
| 2842 temp = A(j+kl:m,j:n); | |
| 2843 A(j+kl:m,j:n) = temp - beta*v*(v'*temp); | |
| 2844 A(j+kl+1:m,j) = zeros (m-j-kl, 1); | |
| 2845 endif | |
| 2846 | |
| 2847 if (j+ku+1 <= n) | |
| 2848 [v, beta] = house (A(j,j+ku:n)'); | |
| 2849 temp = A(j:m,j+ku:n); | |
| 2850 A(j:m,j+ku:n) = temp - beta*(temp*v)*v'; | |
| 2851 A(j,j+ku+1:n) = zeros (1, n-j-ku); | |
| 2852 endif | |
| 2853 endfor | |
| 2854 | |
| 2855 if (flip) | |
| 2856 A = A'; | |
| 2857 endif | |
| 2858 endfunction | |
|
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2859 |
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2860 ## NOTE: qmult is part of the Test Matrix Toolbox and is used by randsvd() |
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2861 function B = qmult (A) |
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2862 ## QMULT Pre-multiply by random orthogonal matrix. |
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2863 ## QMULT(A) is Q*A where Q is a random real orthogonal matrix from |
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2864 ## the Haar distribution, of dimension the number of rows in A. |
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2865 ## Special case: if A is a scalar then QMULT(A) is the same as |
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2866 ## QMULT(EYE(A)). |
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2867 ## |
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2868 ## Called by RANDSVD. |
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2869 ## |
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2870 ## Reference: |
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2871 ## G.W. Stewart, The efficient generation of random |
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2872 ## orthogonal matrices with an application to condition estimators, |
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2873 ## SIAM J. Numer. Anal., 17 (1980), 403-409. |
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2874 |
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2875 [n, m] = size (A); |
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2876 |
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2877 ## Handle scalar A |
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2878 if (isscalar (A)) |
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2879 n = A; |
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2880 A = eye (n); |
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2881 endif |
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2882 |
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2883 d = zeros (n); |
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2884 |
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2885 for k = n-1:-1:1 |
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2886 ## Generate random Householder transformation. |
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2887 x = randn (n-k+1, 1); |
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2888 s = norm (x); |
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2889 sgn = sign (x(1)) + (x(1) == 0); # Modification for sign(1)=1. |
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2890 s = sgn*s; |
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2891 d(k) = -sgn; |
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2892 x(1) = x(1) + s; |
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2893 beta = s*x(1); |
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2894 |
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2895 ## Apply the transformation to A. |
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2896 y = x'*A(k:n,:); |
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2897 A(k:n,:) = A(k:n,:) - x*(y/beta); |
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2898 endfor |
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2899 |
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2900 ## Tidy up signs |
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2901 for i = 1:n-1 |
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2902 A(i,:) = d(i)*A(i,:); |
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2903 endfor |
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2904 A(n,:) = A(n,:) * sign (randn); |
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2905 B = A; |
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20295
diff
changeset
|
2906 endfunction |
|
19226
cdfc8bc9ab62
gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents:
19040
diff
changeset
|
2907 |
|
21759
b002b4331a12
maint: Use two newlines after endfunction and start of BIST tests.
Rik <rik@octave.org>
parents:
21751
diff
changeset
|
2908 |
|
19226
cdfc8bc9ab62
gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents:
19040
diff
changeset
|
2909 ## BIST testing for just a few functions to verify that the main gallery |
|
cdfc8bc9ab62
gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents:
19040
diff
changeset
|
2910 ## dispatch function works. |
|
20929
45a64a6c7273
fix comment character style in most .m files to be consistent
John W. Eaton <jwe@octave.org>
parents:
20852
diff
changeset
|
2911 %!assert (gallery ("clement", 3), [0 1 0; 2 0 2; 0 1 0]) |
|
45a64a6c7273
fix comment character style in most .m files to be consistent
John W. Eaton <jwe@octave.org>
parents:
20852
diff
changeset
|
2912 %!assert (gallery ("invhess", 2), [1 -1; 1 2]) |
|
19226
cdfc8bc9ab62
gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents:
19040
diff
changeset
|
2913 |
|
cdfc8bc9ab62
gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents:
19040
diff
changeset
|
2914 ## Test input validation of main dispatch function only |
|
28896
90fea9cc9caa
test: Add expected error message <Invalid call> to BIST tests for nargin.
Rik <rik@octave.org>
parents:
28891
diff
changeset
|
2915 %!error <Invalid call> gallery () |
|
19226
cdfc8bc9ab62
gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents:
19040
diff
changeset
|
2916 %!error <NAME must be a string> gallery (123) |
|
cdfc8bc9ab62
gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents:
19040
diff
changeset
|
2917 %!error <matrix binomial not implemented> gallery ("binomial") |
|
cdfc8bc9ab62
gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents:
19040
diff
changeset
|
2918 %!error <unknown matrix with NAME foobar> gallery ("foobar") |
|
cdfc8bc9ab62
gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents:
19040
diff
changeset
|
2919 |
|
21759
b002b4331a12
maint: Use two newlines after endfunction and start of BIST tests.
Rik <rik@octave.org>
parents:
21751
diff
changeset
|
2920 ## BIST testing for individual gallery functions |
|
19944
941e782d0429
gallery: performance increase for minij matrix.
Carnë Draug <carandraug@octave.org>
parents:
19833
diff
changeset
|
2921 %!assert (gallery ("minij", 4), [1 1 1 1; 1 2 2 2; 1 2 3 3; 1 2 3 4]) |
|
941e782d0429
gallery: performance increase for minij matrix.
Carnë Draug <carandraug@octave.org>
parents:
19833
diff
changeset
|
2922 %!assert (gallery ("minij", 1), 1) |
|
941e782d0429
gallery: performance increase for minij matrix.
Carnë Draug <carandraug@octave.org>
parents:
19833
diff
changeset
|
2923 %!assert (gallery ("minij", 0), []) |
|
941e782d0429
gallery: performance increase for minij matrix.
Carnë Draug <carandraug@octave.org>
parents:
19833
diff
changeset
|
2924 %!assert (gallery ("minij", -1), []) |
|
20299
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2925 |
|
23381
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2926 %!test |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2927 %! exp = 1 ./ [ |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2928 %! 2 3 4 5 6 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2929 %! 3 4 5 6 7 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2930 %! 4 5 6 7 8 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2931 %! 5 6 7 8 9 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2932 %! 6 7 8 9 10]; |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2933 %! assert (gallery ("cauchy", 5), exp) |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2934 %! assert (gallery ("cauchy", 1:5), exp) |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2935 %! assert (gallery ("cauchy", 1:5, 1:5), exp) |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2936 %! |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2937 %! exp = 1 ./ [ |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2938 %! 1 2 3 4 5 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2939 %! 2 3 4 5 6 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2940 %! 3 4 5 6 7 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2941 %! 4 5 6 7 8 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2942 %! 5 6 7 8 9]; |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2943 %! assert (gallery ("cauchy", 0:4, 1:5), exp) |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2944 %! assert (gallery ("cauchy", 1:5, 0:4), exp) |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2945 %! assert (gallery ("cauchy", 1:5, 4:-1:0), fliplr (exp)) |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2946 %! |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2947 %! exp = 1 ./ [ |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2948 %! -1 0 1 2 3 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2949 %! 0 1 2 3 4 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2950 %! 1 2 3 4 5 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2951 %! 2 3 4 5 6 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2952 %! 3 4 5 6 7]; |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2953 %! assert (gallery ("cauchy", 1:5, -2:2), exp) |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2954 %! |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2955 %! exp = 1 ./ [ |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2956 %! 8 18 -4 2 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2957 %! 13 23 1 7 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2958 %! 9 19 -3 3 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2959 %! 15 25 3 9]; |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2960 %! assert (gallery ("cauchy", [-2 3 -1 5], [10 20 -2 4]), exp) |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2961 %! assert (gallery ("cauchy", [-2 3 -1 5], [10 20 -2 4]'), exp) |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
2962 %! assert (gallery ("cauchy", [-2 3 -1 5]', [10 20 -2 4]), exp) |
|
20299
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2963 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2964 %!assert (size (gallery ("chebspec", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2965 %!assert (size (gallery ("chebspec", 5, 1)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2966 %!assert (size (gallery ("chebspec", 5, 0)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2967 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2968 %!assert (size (gallery ("chebvand", 7)), [7 7]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2969 %!assert (size (gallery ("chebvand", 1:7)), [7 7]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2970 %!assert (size (gallery ("chebvand", 5, 7)), [5 7]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2971 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2972 %!assert (size (gallery ("chow", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2973 %!assert (size (gallery ("chow", 5, 6)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2974 %!assert (size (gallery ("chow", 5, 6, 7)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2975 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2976 %!assert (gallery ("circul", 3), [1 2 3; 3 1 2; 2 3 1]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2977 %!assert (gallery ("circul", [1 3 6]), [1 3 6; 6 1 3; 3 6 1]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2978 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2979 %!assert (size (gallery ("clement", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2980 %!assert (size (gallery ("clement", 5, 1)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2981 %!assert (size (gallery ("clement", 5, 0)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2982 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2983 %!assert (size (gallery ("compar", ones (5))), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2984 %!assert (size (gallery ("compar", ones (5), 0)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2985 %!assert (size (gallery ("compar", ones (5), 1)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2986 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2987 %!assert (size (gallery ("condex", 4)), [4 4]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2988 %!assert (size (gallery ("condex", 4, 1)), [4 4]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2989 %!assert (size (gallery ("condex", 4, 1, 50)), [4 4]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2990 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2991 %!assert (size (gallery ("cycol", [4 5])), [4 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2992 %!assert (size (gallery ("cycol", [4 5], 1)), [4 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2993 %!assert (size (gallery ("cycol", 4)), [4 4]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2994 %!assert (size (gallery ("cycol", 4, 1)), [4 4]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2995 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2996 %!assert (size (gallery ("dorr", 4)), [4 4]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2997 %!assert (cellfun (@rows, nthargout (1:3, @gallery, "dorr", 4)), [3 4 3]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2998 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
2999 %!assert (size (gallery ("dramadah", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3000 %!assert (size (gallery ("dramadah", 5, 2)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3001 |
|
23381
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
3002 %!test |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
3003 %! exp = [ |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
3004 %! 0 1 2 3 4 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
3005 %! 1 0 1 2 3 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
3006 %! 2 1 0 1 2 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
3007 %! 3 2 1 0 1 |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
3008 %! 4 3 2 1 0]; |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
3009 %! assert (gallery ("fiedler", 5), exp) |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
3010 %! assert (gallery ("fiedler", 1:5), exp) |
|
e15ffa2b2262
gallery: use broadcasting to generate cauchy and fiedler matrices
Carnë Draug <carandraug@octave.org>
parents:
23084
diff
changeset
|
3011 %! assert (gallery ("fiedler", -2:2), exp) |
|
23384
d20b385d8c99
gallery.m: fix failing test due to typo on test unit.
Carnë Draug <carandraug@octave.org>
parents:
23381
diff
changeset
|
3012 %! assert (gallery ("fiedler", 2:5), exp(1:4,1:4)) |
|
20299
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3013 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3014 %!assert (size (gallery ("forsythe", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3015 %!assert (size (gallery ("forsythe", 5, 1, 0.5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3016 %!assert (size (gallery ("forsythe", 5, 4, 7)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3017 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3018 %!assert (size (gallery ("frank", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3019 %!assert (size (gallery ("frank", 5, 1)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3020 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3021 %!assert (size (gallery ("gcdmat", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3022 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3023 %!assert (size (gallery ("gearmat", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3024 %!assert (size (gallery ("gearmat", 5, 4)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3025 %!assert (size (gallery ("gearmat", 5, 4, 3)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3026 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3027 %!assert (size (gallery ("grcar", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3028 %!assert (size (gallery ("grcar", 5, 2)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3029 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3030 %!error <N must be even> gallery ("hanowa", 5) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3031 %!assert (size (gallery ("hanowa", 6, 5)), [6 6]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3032 %!assert (size (gallery ("hanowa", 6, 5)), [6 6]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3033 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3034 %!assert (size (gallery ("house", [1:5]')), [5 1]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3035 %!assert (cellfun (@rows, nthargout (1:2, @gallery, "house", [1:5]')), [5 1]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3036 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3037 %!assert (size (gallery ("integerdata", 5, [3 2], 5)), [3 2]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3038 %!assert (size (gallery ("integerdata", 5, [3 2 6], 5)), [3 2 6]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3039 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3040 %!assert (size (gallery ("invhess", 1:4, 1:3)), [4 4]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3041 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3042 %!assert (size (gallery ("invol", 4)), [4 4]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3043 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3044 %!assert (size (gallery ("ipjfact", 4)), [4 4]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3045 %!assert (size (gallery ("ipjfact", 4, 0)), [4 4]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3046 %!assert (size (gallery ("ipjfact", 4, 1)), [4 4]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3047 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3048 %!assert (size (gallery ("jordbloc", 4)), [4 4]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3049 %!assert (size (gallery ("jordbloc", 4, 1)), [4 4]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3050 %!assert (size (gallery ("jordbloc", 4, 3)), [4 4]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3051 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3052 %!assert (size (gallery ("kahan", 4)), [4 4]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3053 %!assert (size (gallery ("kahan", [4 5])), [4 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3054 %!assert (size (gallery ("kahan", [4 5], 1)), [4 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3055 %!assert (size (gallery ("kahan", [4 5], 1, 30)), [4 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3056 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3057 %!assert (size (gallery ("kms", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3058 %!assert (size (gallery ("kms", 5, 0.2)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3059 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3060 %!assert (size (gallery ("krylov", 4)), [4 4]) |
|
22021
e67ff2b11147
Check that second argument to size () is a scalar (bug #48368).
Lachlan Andrew <lachlanbis@gmail.com>
parents:
21759
diff
changeset
|
3061 %!assert (size (gallery ("krylov", ones (4))), [4 4]) |
|
e67ff2b11147
Check that second argument to size () is a scalar (bug #48368).
Lachlan Andrew <lachlanbis@gmail.com>
parents:
21759
diff
changeset
|
3062 %!assert (size (gallery ("krylov", ones (4), [.2 .3 .4 .5], 3)), [4 3]) |
|
20299
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3063 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3064 %!assert (size (gallery ("lauchli", 5)), [6 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3065 %!assert (size (gallery ("lauchli", 5, 3)), [6 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3066 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3067 %!assert (size (gallery ("lehmer", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3068 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3069 %!assert (size (gallery ("lesp", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3070 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3071 %!assert (size (gallery ("lotkin", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3072 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3073 %!assert (size (gallery ("minij", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3074 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3075 %!assert (size (gallery ("moler", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3076 %!assert (size (gallery ("moler", 5, 0.2)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3077 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3078 %!assert (size (gallery ("neumann", 4)), [4 4]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3079 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3080 %!assert (size (gallery ("normaldata", [5 4 6], 3)), [5 4 6]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3081 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3082 %!assert (size (gallery ("orthog", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3083 %!assert (size (gallery ("orthog", 5, 2)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3084 %!assert (size (gallery ("orthog", 5, -2)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3085 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3086 %!assert (size (gallery ("parter", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3087 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3088 %!assert (size (gallery ("pei", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3089 %!assert (size (gallery ("pei", 5, 4)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3090 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3091 %!assert (size (gallery ("poisson", 1)), [1 1]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3092 %!assert (size (gallery ("poisson", 4)), [16 16]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3093 %!assert (size (gallery ("poisson", 5)), [25 25]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3094 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3095 %!assert (size (gallery ("prolate", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3096 %!assert (size (gallery ("prolate", 5, 0.5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3097 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3098 %!assert (size (gallery ("randhess", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3099 %!assert (size (gallery ("randhess", 2:5)), [4 4]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3100 |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3101 %!assert (size (gallery ("rando", 5)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
Carnë Draug <carandraug@octave.org>
parents:
20298
diff
changeset
|
3102 %!assert (size (gallery ("rando", 5, 2)), [5 5]) |
|
c5a8eff5a05d
gallery: add very basic tests (check output size) for all matrix types.
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3103 |
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3104 %!assert (size (gallery ("randsvd", 5)), [5 5]) |
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3105 %!assert (size (gallery ("randsvd", [5 3])), [5 3]) |
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3106 |
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3107 %!assert (size (gallery ("redheff", 5)), [5 5]) |
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3108 %!assert (size (gallery ("riemann", 5)), [5 5]) |
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3109 %!assert (size (gallery ("ris", 5)), [5 5]) |
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3110 |
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3111 %!assert (size (gallery ("smoke", 5)), [5 5]) |
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3112 %!assert (size (gallery ("smoke", 5, 1)), [5 5]) |
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3113 %!assert (gallery ("smoke", 5, 1)(5, 1), 0) |
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|
3114 |
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3115 %!assert (size (gallery ("toeppd", 5)), [5 5]) |
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3116 |
|
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gallery: add very basic tests (check output size) for all matrix types.
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3117 %!assert (size (gallery ("toeppen", 5)), [5 5]) |
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3118 |
|
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gallery: add very basic tests (check output size) for all matrix types.
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3119 %!assert (size (gallery ("tridiag", 5)), [5 5]) |
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3120 %!assert (size (gallery ("tridiag", 1:4, 1:5, 1:4)), [5 5]) |
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3121 %!assert (gallery ("tridiag", 5), gallery ("tridiag", 5, -1, 2, -1)) |
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|
3122 |
|
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gallery: add very basic tests (check output size) for all matrix types.
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3123 %!assert (size (gallery ("triw", 5)), [5 5]) |
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3124 |
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gallery: add very basic tests (check output size) for all matrix types.
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3125 %!assert (size (gallery ("uniformdata", [5 3 4], 3)), [5 3 4]) |
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|
3126 |
|
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gallery: add very basic tests (check output size) for all matrix types.
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3127 %!assert (size (gallery ("wathen", 2, 3)), [29 29]) |
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|
3128 |
|
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gallery: add very basic tests (check output size) for all matrix types.
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3129 %!assert (cellfun (@rows, nthargout (1:2, @gallery, "wilk", 3)), [3 3]) |
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3130 %!assert (cellfun (@rows, nthargout (1:2, @gallery, "wilk", 4)), [4 4]) |
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3131 %!assert (size (gallery ("wilk", 5)), [5 5]) |
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gallery: add very basic tests (check output size) for all matrix types.
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3132 %!assert (size (gallery ("wilk", 21)), [21 21]) |