Mercurial > octave
annotate scripts/special-matrix/gallery.m @ 20296:0b9d23557506
gallery: fix randsvd by adding missing dependency qmult().
* scripts/special-matrix/gallery.m (randsvd) was copied from the
Test Matrix toolbox by Nicholas J. Higham. It made use of qmult()
which was also part of that toolbox but was left behind. This qmult()
implementation is also recovered from the Test Matrix toolbox. Note
that Octave itself used to have qmult() which is part of the legacy
quaternion package (now also removed from the new quaternion package).
See cset 21904fe299c8 for when qmult() was removed from Octave.
Also fix the default value for KL and KU so that a 2 element vector
can be used as N.
| author | Carnë Draug <carandraug@octave.org> |
|---|---|
| date | Fri, 03 Jul 2015 16:18:33 +0100 |
| parents | 557979395ca9 |
| children | 26fc9bbb8762 |
| rev | line source |
|---|---|
| 16634 | 1 ## Copyright (C) 1989-1995 Nicholas .J. Higham |
|
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2 ## Copyright (C) 2013-2015 Carnë Draug |
| 16634 | 3 ## |
| 4 ## This file is part of Octave. | |
| 5 ## | |
| 6 ## Octave is free software; you can redistribute it and/or modify it | |
| 7 ## under the terms of the GNU General Public License as published by | |
| 8 ## the Free Software Foundation; either version 3 of the License, or (at | |
| 9 ## your option) any later version. | |
| 10 ## | |
| 11 ## Octave is distributed in the hope that it will be useful, but | |
| 12 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
| 13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
| 14 ## General Public License for more details. | |
| 15 ## | |
| 16 ## You should have received a copy of the GNU General Public License | |
| 17 ## along with Octave; see the file COPYING. If not, see | |
| 18 ## <http://www.gnu.org/licenses/>. | |
| 19 | |
| 20 ## -*- texinfo -*- | |
| 21 ## @deftypefn {Function File} {} gallery (@var{name}) | |
| 22 ## @deftypefnx {Function File} {} gallery (@var{name}, @var{args}) | |
| 23 ## Create interesting matrices for testing. | |
| 24 ## | |
| 25 ## @end deftypefn | |
| 26 ## | |
| 27 ## @deftypefn {Function File} {@var{c} =} gallery ("cauchy", @var{x}) | |
| 28 ## @deftypefnx {Function File} {@var{c} =} gallery ("cauchy", @var{x}, @var{y}) | |
| 29 ## Create a Cauchy matrix. | |
| 30 ## | |
| 31 ## @end deftypefn | |
| 32 ## | |
| 33 ## @deftypefn {Function File} {@var{c} =} gallery ("chebspec", @var{n}) | |
| 34 ## @deftypefnx {Function File} {@var{c} =} gallery ("chebspec", @var{n}, @var{k}) | |
| 35 ## Create a Chebyshev spectral differentiation matrix. | |
| 36 ## | |
| 37 ## @end deftypefn | |
| 38 ## | |
| 39 ## @deftypefn {Function File} {@var{c} =} gallery ("chebvand", @var{p}) | |
| 40 ## @deftypefnx {Function File} {@var{c} =} gallery ("chebvand", @var{m}, @var{p}) | |
| 41 ## Create a Vandermonde-like matrix for the Chebyshev polynomials. | |
| 42 ## | |
| 43 ## @end deftypefn | |
| 44 ## | |
| 45 ## @deftypefn {Function File} {@var{a} =} gallery ("chow", @var{n}) | |
| 46 ## @deftypefnx {Function File} {@var{a} =} gallery ("chow", @var{n}, @var{alpha}) | |
| 47 ## @deftypefnx {Function File} {@var{a} =} gallery ("chow", @var{n}, @var{alpha}, @var{delta}) | |
| 48 ## Create a Chow matrix -- a singular Toeplitz lower Hessenberg matrix. | |
| 49 ## | |
| 50 ## @end deftypefn | |
| 51 ## | |
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52 ## @deftypefn {Function File} {@var{c} =} gallery ("circul", @var{v}) |
| 16634 | 53 ## Create a circulant matrix. |
| 54 ## | |
| 55 ## @end deftypefn | |
| 56 ## | |
| 57 ## @deftypefn {Function File} {@var{a} =} gallery ("clement", @var{n}) | |
| 58 ## @deftypefnx {Function File} {@var{a} =} gallery ("clement", @var{n}, @var{k}) | |
| 59 ## Create a tridiagonal matrix with zero diagonal entries. | |
| 60 ## | |
| 61 ## @end deftypefn | |
| 62 ## | |
| 63 ## @deftypefn {Function File} {@var{c} =} gallery ("compar", @var{a}) | |
| 64 ## @deftypefnx {Function File} {@var{c} =} gallery ("compar", @var{a}, @var{k}) | |
| 65 ## Create a comparison matrix. | |
| 66 ## | |
| 67 ## @end deftypefn | |
| 68 ## | |
| 69 ## @deftypefn {Function File} {@var{a} =} gallery ("condex", @var{n}) | |
| 70 ## @deftypefnx {Function File} {@var{a} =} gallery ("condex", @var{n}, @var{k}) | |
| 71 ## @deftypefnx {Function File} {@var{a} =} gallery ("condex", @var{n}, @var{k}, @var{theta}) | |
| 72 ## Create a `counterexample' matrix to a condition estimator. | |
| 73 ## | |
| 74 ## @end deftypefn | |
| 75 ## | |
| 76 ## @deftypefn {Function File} {@var{a} =} gallery ("cycol", [@var{m} @var{n}]) | |
| 77 ## @deftypefnx {Function File} {@var{a} =} gallery ("cycol", @var{n}) | |
| 78 ## @deftypefnx {Function File} {@var{a} =} gallery (@dots{}, @var{k}) | |
| 79 ## Create a matrix whose columns repeat cyclically. | |
| 80 ## | |
| 81 ## @end deftypefn | |
| 82 ## | |
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83 ## @deftypefn {Function File} {[@var{c}, @var{d}, @var{e}] =} gallery ("dorr", @var{n}) |
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84 ## @deftypefnx {Function File} {[@var{c}, @var{d}, @var{e}] =} gallery ("dorr", @var{n}, @var{theta}) |
| 16634 | 85 ## @deftypefnx {Function File} {@var{a} =} gallery ("dorr", @dots{}) |
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86 ## Create a diagonally dominant, ill-conditioned, tridiagonal matrix. |
| 16634 | 87 ## |
| 88 ## @end deftypefn | |
| 89 ## | |
| 90 ## @deftypefn {Function File} {@var{a} =} gallery ("dramadah", @var{n}) | |
| 91 ## @deftypefnx {Function File} {@var{a} =} gallery ("dramadah", @var{n}, @var{k}) | |
| 92 ## Create a (0, 1) matrix whose inverse has large integer entries. | |
| 93 ## | |
| 94 ## @end deftypefn | |
| 95 ## | |
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96 ## @deftypefn {Function File} {@var{a} =} gallery ("fiedler", @var{c}) |
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97 ## Create a symmetric @nospell{Fiedler} matrix. |
| 16634 | 98 ## |
| 99 ## @end deftypefn | |
| 100 ## | |
| 101 ## @deftypefn {Function File} {@var{a} =} gallery ("forsythe", @var{n}) | |
| 102 ## @deftypefnx {Function File} {@var{a} =} gallery ("forsythe", @var{n}, @var{alpha}) | |
| 103 ## @deftypefnx {Function File} {@var{a} =} gallery ("forsythe", @var{n}, @var{alpha}, @var{lambda}) | |
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104 ## Create a @nospell{Forsythe} matrix (a perturbed Jordan block). |
| 16634 | 105 ## |
| 106 ## @end deftypefn | |
| 107 ## | |
| 108 ## @deftypefn {Function File} {@var{f} =} gallery ("frank", @var{n}) | |
| 109 ## @deftypefnx {Function File} {@var{f} =} gallery ("frank", @var{n}, @var{k}) | |
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110 ## Create a Frank matrix (ill-conditioned eigenvalues). |
| 16634 | 111 ## |
| 112 ## @end deftypefn | |
| 113 ## | |
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114 ## @deftypefn {Function File} {@var{c} =} gallery ("gcdmat", @var{n}) |
| 16634 | 115 ## Create a greatest common divisor matrix. |
| 116 ## | |
| 117 ## @var{c} is an @var{n}-by-@var{n} matrix whose values correspond to the | |
| 118 ## greatest common divisor of its coordinate values, i.e., @var{c}(i,j) | |
| 119 ## correspond @code{gcd (i, j)}. | |
| 120 ## @end deftypefn | |
| 121 ## | |
| 122 ## @deftypefn {Function File} {@var{a} =} gallery ("gearmat", @var{n}) | |
| 123 ## @deftypefnx {Function File} {@var{a} =} gallery ("gearmat", @var{n}, @var{i}) | |
| 124 ## @deftypefnx {Function File} {@var{a} =} gallery ("gearmat", @var{n}, @var{i}, @var{j}) | |
| 125 ## Create a Gear matrix. | |
| 126 ## | |
| 127 ## @end deftypefn | |
| 128 ## | |
| 129 ## @deftypefn {Function File} {@var{g} =} gallery ("grcar", @var{n}) | |
| 130 ## @deftypefnx {Function File} {@var{g} =} gallery ("grcar", @var{n}, @var{k}) | |
| 131 ## Create a Toeplitz matrix with sensitive eigenvalues. | |
| 132 ## | |
| 133 ## @end deftypefn | |
| 134 ## | |
| 135 ## @deftypefn {Function File} {@var{a} =} gallery ("hanowa", @var{n}) | |
| 136 ## @deftypefnx {Function File} {@var{a} =} gallery ("hanowa", @var{n}, @var{d}) | |
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137 ## Create a matrix whose eigenvalues lie on a vertical line in the complex |
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138 ## plane. |
| 16634 | 139 ## |
| 140 ## @end deftypefn | |
| 141 ## | |
| 142 ## @deftypefn {Function File} {@var{v} =} gallery ("house", @var{x}) | |
| 143 ## @deftypefnx {Function File} {[@var{v}, @var{beta}] =} gallery ("house", @var{x}) | |
| 144 ## Create a householder matrix. | |
| 145 ## | |
| 146 ## @end deftypefn | |
| 147 ## | |
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148 ## @deftypefn {Function File} {@var{a} =} gallery ("integerdata", @var{imax}, [@var{M} @var{N} @dots{}], @var{j}) |
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149 ## @deftypefnx {Function File} {@var{a} =} gallery ("integerdata", @var{imax}, @var{M}, @var{N}, @dots{}, @var{j}) |
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150 ## @deftypefnx {Function File} {@var{a} =} gallery ("integerdata", [@var{imin}, @var{imax}], [@var{M} @var{N} @dots{}], @var{j}) |
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151 ## @deftypefnx {Function File} {@var{a} =} gallery ("integerdata", [@var{imin}, @var{imax}], @var{M}, @var{N}, @dots{}, @var{j}) |
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152 ## @deftypefnx {Function File} {@var{a} =} gallery ("integerdata", @dots{}, "@var{class}") |
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153 ## Create a matrix with random integers in the range [1, @var{imax}]. |
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154 ## If @var{imin} is given then the integers are in the range |
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155 ## [@var{imin}, @var{imax}]. |
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156 ## |
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157 ## The second input is a matrix of dimensions describing the size of the output. |
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158 ## The dimensions can also be input as comma-separated arguments. |
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159 ## |
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160 ## The input @var{j} is an integer index in the range [0, 2^32-1]. The values |
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161 ## of the output matrix are always exactly the same (reproducibility) for a |
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162 ## given size input and @var{j} index. |
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163 ## |
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164 ## The final optional argument determines the class of the resulting matrix. |
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165 ## Possible values for @var{class}: @qcode{"uint8"}, @qcode{"uint16"}, |
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166 ## @qcode{"uint32"}, @qcode{"int8"}, @qcode{"int16"}, int32", @qcode{"single"}, |
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167 ## @qcode{"double"}. The default is @qcode{"double"}. |
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168 ## |
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169 ## @end deftypefn |
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170 ## |
| 16634 | 171 ## @deftypefn {Function File} {@var{a} =} gallery ("invhess", @var{x}) |
| 172 ## @deftypefnx {Function File} {@var{a} =} gallery ("invhess", @var{x}, @var{y}) | |
| 173 ## Create the inverse of an upper Hessenberg matrix. | |
| 174 ## | |
| 175 ## @end deftypefn | |
| 176 ## | |
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177 ## @deftypefn {Function File} {@var{a} =} gallery ("invol", @var{n}) |
| 16634 | 178 ## Create an involutory matrix. |
| 179 ## | |
| 180 ## @end deftypefn | |
| 181 ## | |
| 182 ## @deftypefn {Function File} {@var{a} =} gallery ("ipjfact", @var{n}) | |
| 183 ## @deftypefnx {Function File} {@var{a} =} gallery ("ipjfact", @var{n}, @var{k}) | |
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184 ## Create a Hankel matrix with factorial elements. |
| 16634 | 185 ## |
| 186 ## @end deftypefn | |
| 187 ## | |
| 188 ## @deftypefn {Function File} {@var{a} =} gallery ("jordbloc", @var{n}) | |
| 189 ## @deftypefnx {Function File} {@var{a} =} gallery ("jordbloc", @var{n}, @var{lambda}) | |
| 190 ## Create a Jordan block. | |
| 191 ## | |
| 192 ## @end deftypefn | |
| 193 ## | |
| 194 ## @deftypefn {Function File} {@var{u} =} gallery ("kahan", @var{n}) | |
| 195 ## @deftypefnx {Function File} {@var{u} =} gallery ("kahan", @var{n}, @var{theta}) | |
| 196 ## @deftypefnx {Function File} {@var{u} =} gallery ("kahan", @var{n}, @var{theta}, @var{pert}) | |
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197 ## Create a @nospell{Kahan} matrix (upper trapezoidal). |
| 16634 | 198 ## |
| 199 ## @end deftypefn | |
| 200 ## | |
| 201 ## @deftypefn {Function File} {@var{a} =} gallery ("kms", @var{n}) | |
| 202 ## @deftypefnx {Function File} {@var{a} =} gallery ("kms", @var{n}, @var{rho}) | |
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203 ## Create a @nospell{Kac-Murdock-Szego} Toeplitz matrix. |
| 16634 | 204 ## |
| 205 ## @end deftypefn | |
| 206 ## | |
| 207 ## @deftypefn {Function File} {@var{b} =} gallery ("krylov", @var{a}) | |
| 208 ## @deftypefnx {Function File} {@var{b} =} gallery ("krylov", @var{a}, @var{x}) | |
| 209 ## @deftypefnx {Function File} {@var{b} =} gallery ("krylov", @var{a}, @var{x}, @var{j}) | |
| 210 ## Create a Krylov matrix. | |
| 211 ## | |
| 212 ## @end deftypefn | |
| 213 ## | |
| 214 ## @deftypefn {Function File} {@var{a} =} gallery ("lauchli", @var{n}) | |
| 215 ## @deftypefnx {Function File} {@var{a} =} gallery ("lauchli", @var{n}, @var{mu}) | |
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216 ## Create a @nospell{Lauchli} matrix (rectangular). |
| 16634 | 217 ## |
| 218 ## @end deftypefn | |
| 219 ## | |
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220 ## @deftypefn {Function File} {@var{a} =} gallery ("lehmer", @var{n}) |
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221 ## Create a @nospell{Lehmer} matrix (symmetric positive definite). |
| 16634 | 222 ## |
| 223 ## @end deftypefn | |
| 224 ## | |
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225 ## @deftypefn {Function File} {@var{t} =} gallery ("lesp", @var{n}) |
| 16634 | 226 ## Create a tridiagonal matrix with real, sensitive eigenvalues. |
| 227 ## | |
| 228 ## @end deftypefn | |
| 229 ## | |
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230 ## @deftypefn {Function File} {@var{a} =} gallery ("lotkin", @var{n}) |
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231 ## Create a @nospell{Lotkin} matrix. |
| 16634 | 232 ## |
| 233 ## @end deftypefn | |
| 234 ## | |
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235 ## @deftypefn {Function File} {@var{a} =} gallery ("minij", @var{n}) |
| 16634 | 236 ## Create a symmetric positive definite matrix MIN(i,j). |
| 237 ## | |
| 238 ## @end deftypefn | |
| 239 ## | |
| 240 ## @deftypefn {Function File} {@var{a} =} gallery ("moler", @var{n}) | |
| 241 ## @deftypefnx {Function File} {@var{a} =} gallery ("moler", @var{n}, @var{alpha}) | |
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242 ## Create a @nospell{Moler} matrix (symmetric positive definite). |
| 16634 | 243 ## |
| 244 ## @end deftypefn | |
| 245 ## | |
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246 ## @deftypefn {Function File} {[@var{a}, @var{t}] =} gallery ("neumann", @var{n}) |
| 16634 | 247 ## Create a singular matrix from the discrete Neumann problem (sparse). |
| 248 ## | |
| 249 ## @end deftypefn | |
| 250 ## | |
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251 ## @deftypefn {Function File} {@var{a} =} gallery ("normaldata", [@var{M} @var{N} @dots{}], @var{j}) |
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252 ## @deftypefnx {Function File} {@var{a} =} gallery ("normaldata", @var{M}, @var{N}, @dots{}, @var{j}) |
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253 ## @deftypefnx {Function File} {@var{a} =} gallery ("normaldata", @dots{}, "@var{class}") |
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254 ## Create a matrix with random samples from the standard normal distribution |
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255 ## (mean = 0, std = 1). |
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256 ## |
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257 ## The first input is a matrix of dimensions describing the size of the output. |
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258 ## The dimensions can also be input as comma-separated arguments. |
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259 ## |
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260 ## The input @var{j} is an integer index in the range [0, 2^32-1]. The values |
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261 ## of the output matrix are always exactly the same (reproducibility) for a |
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262 ## given size input and @var{j} index. |
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263 ## |
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264 ## The final optional argument determines the class of the resulting matrix. |
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265 ## Possible values for @var{class}: @qcode{"single"}, @qcode{"double"}. |
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266 ## The default is @qcode{"double"}. |
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267 ## |
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268 ## @end deftypefn |
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269 ## |
| 16634 | 270 ## @deftypefn {Function File} {@var{q} =} gallery ("orthog", @var{n}) |
| 271 ## @deftypefnx {Function File} {@var{q} =} gallery ("orthog", @var{n}, @var{k}) | |
| 272 ## Create orthogonal and nearly orthogonal matrices. | |
| 273 ## | |
| 274 ## @end deftypefn | |
| 275 ## | |
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276 ## @deftypefn {Function File} {@var{a} =} gallery ("parter", @var{n}) |
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277 ## Create a @nospell{Parter} matrix (a Toeplitz matrix with singular values |
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278 ## near pi). |
| 16634 | 279 ## |
| 280 ## @end deftypefn | |
| 281 ## | |
| 282 ## @deftypefn {Function File} {@var{p} =} gallery ("pei", @var{n}) | |
| 283 ## @deftypefnx {Function File} {@var{p} =} gallery ("pei", @var{n}, @var{alpha}) | |
| 284 ## Create a Pei matrix. | |
| 285 ## | |
| 286 ## @end deftypefn | |
| 287 ## | |
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288 ## @deftypefn {Function File} {@var{a} =} gallery ("Poisson", @var{n}) |
| 16634 | 289 ## Create a block tridiagonal matrix from Poisson's equation (sparse). |
| 290 ## | |
| 291 ## @end deftypefn | |
| 292 ## | |
| 293 ## @deftypefn {Function File} {@var{a} =} gallery ("prolate", @var{n}) | |
| 294 ## @deftypefnx {Function File} {@var{a} =} gallery ("prolate", @var{n}, @var{w}) | |
| 295 ## Create a prolate matrix (symmetric, ill-conditioned Toeplitz matrix). | |
| 296 ## | |
| 297 ## @end deftypefn | |
| 298 ## | |
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299 ## @deftypefn {Function File} {@var{h} =} gallery ("randhess", @var{x}) |
| 16634 | 300 ## Create a random, orthogonal upper Hessenberg matrix. |
| 301 ## | |
| 302 ## @end deftypefn | |
| 303 ## | |
| 304 ## @deftypefn {Function File} {@var{a} =} gallery ("rando", @var{n}) | |
| 305 ## @deftypefnx {Function File} {@var{a} =} gallery ("rando", @var{n}, @var{k}) | |
| 306 ## Create a random matrix with elements -1, 0 or 1. | |
| 307 ## | |
| 308 ## @end deftypefn | |
| 309 ## | |
| 310 ## @deftypefn {Function File} {@var{a} =} gallery ("randsvd", @var{n}) | |
| 311 ## @deftypefnx {Function File} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa}) | |
| 312 ## @deftypefnx {Function File} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa}, @var{mode}) | |
| 313 ## @deftypefnx {Function File} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa}, @var{mode}, @var{kl}) | |
| 314 ## @deftypefnx {Function File} {@var{a} =} gallery ("randsvd", @var{n}, @var{kappa}, @var{mode}, @var{kl}, @var{ku}) | |
| 315 ## Create a random matrix with pre-assigned singular values. | |
| 316 ## | |
| 317 ## @end deftypefn | |
| 318 ## | |
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319 ## @deftypefn {Function File} {@var{a} =} gallery ("redheff", @var{n}) |
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320 ## Create a zero and ones matrix of @nospell{Redheffer} associated with the |
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321 ## Riemann hypothesis. |
| 16634 | 322 ## |
| 323 ## @end deftypefn | |
| 324 ## | |
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325 ## @deftypefn {Function File} {@var{a} =} gallery ("riemann", @var{n}) |
| 16634 | 326 ## Create a matrix associated with the Riemann hypothesis. |
| 327 ## | |
| 328 ## @end deftypefn | |
| 329 ## | |
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330 ## @deftypefn {Function File} {@var{a} =} gallery ("ris", @var{n}) |
| 16634 | 331 ## Create a symmetric Hankel matrix. |
| 332 ## | |
| 333 ## @end deftypefn | |
| 334 ## | |
| 335 ## @deftypefn {Function File} {@var{a} =} gallery ("smoke", @var{n}) | |
| 336 ## @deftypefnx {Function File} {@var{a} =} gallery ("smoke", @var{n}, @var{k}) | |
| 337 ## Create a complex matrix, with a `smoke ring' pseudospectrum. | |
| 338 ## | |
| 339 ## @end deftypefn | |
| 340 ## | |
| 341 ## @deftypefn {Function File} {@var{t} =} gallery ("toeppd", @var{n}) | |
| 342 ## @deftypefnx {Function File} {@var{t} =} gallery ("toeppd", @var{n}, @var{m}) | |
| 343 ## @deftypefnx {Function File} {@var{t} =} gallery ("toeppd", @var{n}, @var{m}, @var{w}) | |
| 344 ## @deftypefnx {Function File} {@var{t} =} gallery ("toeppd", @var{n}, @var{m}, @var{w}, @var{theta}) | |
| 345 ## Create a symmetric positive definite Toeplitz matrix. | |
| 346 ## | |
| 347 ## @end deftypefn | |
| 348 ## | |
| 349 ## @deftypefn {Function File} {@var{p} =} gallery ("toeppen", @var{n}) | |
| 350 ## @deftypefnx {Function File} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}) | |
| 351 ## @deftypefnx {Function File} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b}) | |
| 352 ## @deftypefnx {Function File} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b}, @var{c}) | |
| 353 ## @deftypefnx {Function File} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b}, @var{c}, @var{d}) | |
| 354 ## @deftypefnx {Function File} {@var{p} =} gallery ("toeppen", @var{n}, @var{a}, @var{b}, @var{c}, @var{d}, @var{e}) | |
| 355 ## Create a pentadiagonal Toeplitz matrix (sparse). | |
| 356 ## | |
| 357 ## @end deftypefn | |
| 358 ## | |
| 359 ## @deftypefn {Function File} {@var{a} =} gallery ("tridiag", @var{x}, @var{y}, @var{z}) | |
| 360 ## @deftypefnx {Function File} {@var{a} =} gallery ("tridiag", @var{n}) | |
| 361 ## @deftypefnx {Function File} {@var{a} =} gallery ("tridiag", @var{n}, @var{c}, @var{d}, @var{e}) | |
| 362 ## Create a tridiagonal matrix (sparse). | |
| 363 ## | |
| 364 ## @end deftypefn | |
| 365 ## | |
| 366 ## @deftypefn {Function File} {@var{t} =} gallery ("triw", @var{n}) | |
| 367 ## @deftypefnx {Function File} {@var{t} =} gallery ("triw", @var{n}, @var{alpha}) | |
| 368 ## @deftypefnx {Function File} {@var{t} =} gallery ("triw", @var{n}, @var{alpha}, @var{k}) | |
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369 ## Create an upper triangular matrix discussed by |
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370 ## @nospell{Kahan, Golub, and Wilkinson}. |
| 16634 | 371 ## |
| 372 ## @end deftypefn | |
| 373 ## | |
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374 ## @deftypefn {Function File} {@var{a} =} gallery ("uniformdata", [@var{M} @var{N} @dots{}], @var{j}) |
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375 ## @deftypefnx {Function File} {@var{a} =} gallery ("uniformdata", @var{M}, @var{N}, @dots{}, @var{j}) |
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376 ## @deftypefnx {Function File} {@var{a} =} gallery ("uniformdata", @dots{}, "@var{class}") |
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377 ## Create a matrix with random samples from the standard uniform distribution |
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378 ## (range [0,1]). |
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379 ## |
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380 ## The first input is a matrix of dimensions describing the size of the output. |
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381 ## The dimensions can also be input as comma-separated arguments. |
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382 ## |
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383 ## The input @var{j} is an integer index in the range [0, 2^32-1]. The values |
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384 ## of the output matrix are always exactly the same (reproducibility) for a |
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385 ## given size input and @var{j} index. |
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386 ## |
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387 ## The final optional argument determines the class of the resulting matrix. |
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388 ## Possible values for @var{class}: @qcode{"single"}, @qcode{"double"}. |
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389 ## The default is @qcode{"double"}. |
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390 ## |
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391 ## @end deftypefn |
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392 ## |
| 16634 | 393 ## @deftypefn {Function File} {@var{a} =} gallery ("wathen", @var{nx}, @var{ny}) |
| 394 ## @deftypefnx {Function File} {@var{a} =} gallery ("wathen", @var{nx}, @var{ny}, @var{k}) | |
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395 ## Create the @nospell{Wathen} matrix. |
| 16634 | 396 ## |
| 397 ## @end deftypefn | |
| 398 ## | |
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399 ## @deftypefn {Function File} {[@var{a}, @var{b}] =} gallery ("wilk", @var{n}) |
| 16634 | 400 ## Create various specific matrices devised/discussed by Wilkinson. |
| 401 ## | |
| 402 ## @end deftypefn | |
| 403 | |
| 404 ## Code for most of the individual matrices (except binomial, gcdmat, | |
| 405 ## integerdata, leslie, normaldata, randcolu, randcorr, randjorth, sampling, | |
| 406 ## uniformdata) by Nicholas .J. Higham <Nicholas.J.Higham@manchester.ac.uk> | |
| 407 ## Adapted for Octave and into single gallery function by Carnë Draug | |
| 408 | |
| 409 function [varargout] = gallery (name, varargin) | |
| 410 | |
| 411 if (nargin < 1) | |
| 412 print_usage (); | |
| 413 elseif (! ischar (name)) | |
| 414 error ("gallery: NAME must be a string."); | |
| 415 endif | |
| 416 | |
| 417 ## NOTE: there isn't a lot of input check in the individual functions | |
| 418 ## that actually build the functions. This is by design. The original | |
| 419 ## code by Higham did not perform it and was propagated to Matlab, so | |
| 420 ## for compatibility, we also don't make it. For example, arguments | |
| 421 ## that behave as switches, and in theory accepting a value of 0 or 1, | |
| 422 ## will use a value of 0, for any value other than 1 (only check made | |
| 423 ## is if the value is equal to 1). It will often also accept string | |
| 424 ## values instead of numeric. Only input check added was where it | |
| 425 ## would be causing an error anyway. | |
| 426 | |
| 427 ## we will always want to return at least 1 output | |
| 428 n_out = nargout; | |
| 429 if (n_out == 0) | |
| 430 n_out = 1; | |
| 431 endif | |
| 432 | |
| 433 switch (tolower (name)) | |
| 434 case "binomial" | |
| 435 error ("gallery: matrix %s not implemented.", name); | |
| 436 case "cauchy" , [varargout{1:n_out}] = cauchy (varargin{:}); | |
| 437 case "chebspec" , [varargout{1:n_out}] = chebspec (varargin{:}); | |
| 438 case "chebvand" , [varargout{1:n_out}] = chebvand (varargin{:}); | |
| 439 case "chow" , [varargout{1:n_out}] = chow (varargin{:}); | |
| 440 case "circul" , [varargout{1:n_out}] = circul (varargin{:}); | |
| 441 case "clement" , [varargout{1:n_out}] = clement (varargin{:}); | |
| 442 case "compar" , [varargout{1:n_out}] = compar (varargin{:}); | |
| 443 case "condex" , [varargout{1:n_out}] = condex (varargin{:}); | |
| 444 case "cycol" , [varargout{1:n_out}] = cycol (varargin{:}); | |
| 445 case "dorr" , [varargout{1:n_out}] = dorr (varargin{:}); | |
| 446 case "dramadah" , [varargout{1:n_out}] = dramadah (varargin{:}); | |
| 447 case "fiedler" , [varargout{1:n_out}] = fiedler (varargin{:}); | |
| 448 case "forsythe" , [varargout{1:n_out}] = forsythe (varargin{:}); | |
| 449 case "frank" , [varargout{1:n_out}] = frank (varargin{:}); | |
| 450 case "gearmat" , [varargout{1:n_out}] = gearmat (varargin{:}); | |
| 451 case "gcdmat" , [varargout{1:n_out}] = gcdmat (varargin{:}); | |
| 452 case "grcar" , [varargout{1:n_out}] = grcar (varargin{:}); | |
| 453 case "hanowa" , [varargout{1:n_out}] = hanowa (varargin{:}); | |
| 454 case "house" , [varargout{1:n_out}] = house (varargin{:}); | |
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455 case "integerdata", [varargout{1:n_out}] = integerdata (varargin{:}); |
| 16634 | 456 case "invhess" , [varargout{1:n_out}] = invhess (varargin{:}); |
| 457 case "invol" , [varargout{1:n_out}] = invol (varargin{:}); | |
| 458 case "ipjfact" , [varargout{1:n_out}] = ipjfact (varargin{:}); | |
| 459 case "jordbloc" , [varargout{1:n_out}] = jordbloc (varargin{:}); | |
| 460 case "kahan" , [varargout{1:n_out}] = kahan (varargin{:}); | |
| 461 case "kms" , [varargout{1:n_out}] = kms (varargin{:}); | |
| 462 case "krylov" , [varargout{1:n_out}] = krylov (varargin{:}); | |
| 463 case "lauchli" , [varargout{1:n_out}] = lauchli (varargin{:}); | |
| 464 case "lehmer" , [varargout{1:n_out}] = lehmer (varargin{:}); | |
| 465 case "leslie" | |
| 466 error ("gallery: matrix %s not implemented.", name); | |
| 467 case "lesp" , [varargout{1:n_out}] = lesp (varargin{:}); | |
| 468 case "lotkin" , [varargout{1:n_out}] = lotkin (varargin{:}); | |
| 469 case "minij" , [varargout{1:n_out}] = minij (varargin{:}); | |
| 470 case "moler" , [varargout{1:n_out}] = moler (varargin{:}); | |
| 471 case "neumann" , [varargout{1:n_out}] = neumann (varargin{:}); | |
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472 case "normaldata" , [varargout{1:n_out}] = normaldata (varargin{:}); |
| 16634 | 473 case "orthog" , [varargout{1:n_out}] = orthog (varargin{:}); |
| 474 case "parter" , [varargout{1:n_out}] = parter (varargin{:}); | |
| 475 case "pei" , [varargout{1:n_out}] = pei (varargin{:}); | |
| 476 case "poisson" , [varargout{1:n_out}] = poisson (varargin{:}); | |
| 477 case "prolate" , [varargout{1:n_out}] = prolate (varargin{:}); | |
| 478 case "randcolu" | |
| 479 error ("gallery: matrix %s not implemented.", name); | |
| 480 case "randcorr" | |
| 481 error ("gallery: matrix %s not implemented.", name); | |
| 482 case "randhess" , [varargout{1:n_out}] = randhess (varargin{:}); | |
| 483 case "randjorth" | |
| 484 error ("gallery: matrix %s not implemented.", name); | |
| 485 case "rando" , [varargout{1:n_out}] = rando (varargin{:}); | |
| 486 case "randsvd" , [varargout{1:n_out}] = randsvd (varargin{:}); | |
| 487 case "redheff" , [varargout{1:n_out}] = redheff (varargin{:}); | |
| 488 case "riemann" , [varargout{1:n_out}] = riemann (varargin{:}); | |
| 489 case "ris" , [varargout{1:n_out}] = ris (varargin{:}); | |
| 490 case "sampling" | |
| 491 error ("gallery: matrix %s not implemented.", name); | |
| 492 case "smoke" , [varargout{1:n_out}] = smoke (varargin{:}); | |
| 493 case "toeppd" , [varargout{1:n_out}] = toeppd (varargin{:}); | |
| 494 case "toeppen" , [varargout{1:n_out}] = toeppen (varargin{:}); | |
| 495 case "tridiag" , [varargout{1:n_out}] = tridiag (varargin{:}); | |
| 496 case "triw" , [varargout{1:n_out}] = triw (varargin{:}); | |
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497 case "uniformdata" , [varargout{1:n_out}] = uniformdata (varargin{:}); |
| 16634 | 498 case "wathen" , [varargout{1:n_out}] = wathen (varargin{:}); |
| 499 case "wilk" , [varargout{1:n_out}] = wilk (varargin{:}); | |
| 500 otherwise | |
| 501 error ("gallery: unknown matrix with NAME %s", name); | |
| 502 endswitch | |
| 503 | |
| 504 endfunction | |
| 505 | |
| 506 function C = cauchy (x, y) | |
| 507 ##CAUCHY Cauchy matrix. | |
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508 ## C = CAUCHY(X, Y), where X, Y are N-vectors, is the N-by-N matrix |
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509 ## with C(i,j) = 1/(X(i)+Y(j)). By default, Y = X. |
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510 ## Special case: if X is a scalar CAUCHY(X) is the same as CAUCHY(1:X). |
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511 ## Explicit formulas are known for DET(C) (which is nonzero if X and Y |
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512 ## both have distinct elements) and the elements of INV(C). |
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513 ## C is totally positive if 0 < X(1) < ... < X(N) and |
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514 ## 0 < Y(1) < ... < Y(N). |
| 16634 | 515 ## |
|
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516 ## References: |
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517 ## N.J. Higham, Accuracy and Stability of Numerical Algorithms, |
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518 ## Society for Industrial and Applied Mathematics, Philadelphia, PA, |
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519 ## USA, 1996; sec. 26.1. |
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520 ## D.E. Knuth, The Art of Computer Programming, Volume 1, |
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521 ## Fundamental Algorithms, second edition, Addison-Wesley, Reading, |
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522 ## Massachusetts, 1973, p. 36. |
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523 ## E.E. Tyrtyshnikov, Cauchy-Toeplitz matrices and some applications, |
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524 ## Linear Algebra and Appl., 149 (1991), pp. 1-18. |
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525 ## O. Taussky and M. Marcus, Eigenvalues of finite matrices, in |
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526 ## Survey of Numerical Analysis, J. Todd, ed., McGraw-Hill, New York, |
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527 ## pp. 279-313, 1962. (States the totally positive property on p. 295.) |
| 16634 | 528 |
| 529 if (nargin < 1 || nargin > 2) | |
| 530 error ("gallery: 1 or 2 arguments are required for cauchy matrix."); | |
| 531 elseif (! isnumeric (x)) | |
| 532 error ("gallery: X must be numeric for cauchy matrix."); | |
| 533 elseif (nargin == 2 && ! isnumeric (y)) | |
| 534 error ("gallery: Y must be numeric for cauchy matrix."); | |
| 535 endif | |
| 536 | |
| 537 n = numel (x); | |
| 538 if (isscalar (x) && fix (x) == x) | |
| 539 n = x; | |
| 540 x = 1:n; | |
| 541 elseif (n > 1 && isvector (x)) | |
| 542 ## do nothing | |
| 543 else | |
| 544 error ("gallery: X be an integer or a vector for cauchy matrix."); | |
| 545 endif | |
| 546 | |
| 547 if (nargin == 1) | |
| 548 y = x; | |
| 549 endif | |
| 550 | |
| 551 ## Ensure x and y are column vectors | |
| 552 x = x(:); | |
| 553 y = y(:); | |
| 554 if (numel (x) != numel (y)) | |
| 555 error ("gallery: X and Y must be vectors of same length for cauchy matrix."); | |
| 556 endif | |
| 557 | |
| 558 C = x * ones (1, n) + ones (n, 1) * y.'; | |
| 559 C = ones (n) ./ C; | |
| 560 endfunction | |
| 561 | |
| 562 function C = chebspec (n, k = 0) | |
| 563 ## CHEBSPEC Chebyshev spectral differentiation matrix. | |
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564 ## C = CHEBSPEC(N, K) is a Chebyshev spectral differentiation |
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565 ## matrix of order N. K = 0 (the default) or 1. |
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566 ## For K = 0 (`no boundary conditions'), C is nilpotent, with |
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567 ## C^N = 0 and it has the null vector ONES(N,1). |
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568 ## C is similar to a Jordan block of size N with eigenvalue zero. |
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569 ## For K = 1, C is nonsingular and well-conditioned, and its eigenvalues |
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570 ## have negative real parts. |
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571 ## For both K, the computed eigenvector matrix X from EIG is |
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572 ## ill-conditioned (MESH(REAL(X)) is interesting). |
| 16634 | 573 ## |
|
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574 ## References: |
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575 ## C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral |
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576 ## Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988; p. 69. |
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577 ## L.N. Trefethen and M.R. Trummer, An instability phenomenon in |
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578 ## spectral methods, SIAM J. Numer. Anal., 24 (1987), pp. 1008-1023. |
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579 ## D. Funaro, Computing the inverse of the Chebyshev collocation |
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580 ## derivative, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 1050-1057. |
| 16634 | 581 |
| 582 if (nargin < 1 || nargin > 2) | |
| 583 error ("gallery: 1 to 2 arguments are required for chebspec matrix."); | |
| 584 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 585 error ("gallery: N must be an integer for chebspec matrix."); | |
| 586 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 587 error ("gallery: K must be a scalar for chebspec matrix."); | |
| 588 endif | |
| 589 | |
| 590 ## k = 1 case obtained from k = 0 case with one bigger n. | |
| 591 switch (k) | |
| 592 case (0), # do nothing | |
| 593 case (1), n = n + 1; | |
| 594 otherwise | |
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595 error ("gallery: K should be either 0 or 1 for chebspec matrix."); |
| 16634 | 596 endswitch |
| 597 | |
|
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598 n -= 1; |
| 16634 | 599 C = zeros (n+1); |
| 600 | |
| 601 one = ones (n+1, 1); | |
| 602 x = cos ((0:n)' * (pi/n)); | |
| 603 d = ones (n+1, 1); | |
| 604 d(1) = 2; | |
| 605 d(n+1) = 2; | |
| 606 | |
| 607 ## eye(size(C)) on next line avoids div by zero. | |
| 608 C = (d * (one./d)') ./ (x*one'-one*x' + eye (size (C))); | |
| 609 | |
| 610 ## Now fix diagonal and signs. | |
| 611 C(1,1) = (2*n^2+1)/6; | |
| 612 for i = 2:n+1 | |
| 613 if (rem (i, 2) == 0) | |
| 614 C(:,i) = -C(:,i); | |
| 615 C(i,:) = -C(i,:); | |
| 616 endif | |
| 617 if (i < n+1) | |
| 618 C(i,i) = -x(i)/(2*(1-x(i)^2)); | |
| 619 else | |
| 620 C(n+1,n+1) = -C(1,1); | |
| 621 endif | |
| 622 endfor | |
| 623 | |
| 624 if (k == 1) | |
| 625 C = C(2:n+1,2:n+1); | |
| 626 endif | |
| 627 endfunction | |
| 628 | |
| 629 function C = chebvand (m, p) | |
| 630 ## CHEBVAND Vandermonde-like matrix for the Chebyshev polynomials. | |
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631 ## C = CHEBVAND(P), where P is a vector, produces the (primal) |
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632 ## Chebyshev Vandermonde matrix based on the points P, |
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633 ## i.e., C(i,j) = T_{i-1}(P(j)), where T_{i-1} is the Chebyshev |
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634 ## polynomial of degree i-1. |
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635 ## CHEBVAND(M,P) is a rectangular version of CHEBVAND(P) with M rows. |
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636 ## Special case: If P is a scalar then P equally spaced points on |
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637 ## [0,1] are used. |
| 16634 | 638 ## |
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639 ## Reference: |
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640 ## N.J. Higham, Stability analysis of algorithms for solving confluent |
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641 ## Vandermonde-like systems, SIAM J. Matrix Anal. Appl., 11 (1990), |
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642 ## pp. 23-41. |
| 16634 | 643 |
| 644 if (nargin < 1 || nargin > 2) | |
| 645 error ("gallery: 1 or 2 arguments are required for chebvand matrix."); | |
| 646 endif | |
| 647 | |
| 648 ## because the order of the arguments changes if nargin is 1 or 2 ... | |
| 649 | |
| 650 if (nargin == 1) | |
| 651 p = m; | |
| 652 endif | |
| 653 | |
| 654 n = numel (p); | |
| 655 if (! isnumeric (p)) | |
| 656 error ("gallery: P must be numeric for chebvand matrix."); | |
| 657 elseif (isscalar (p) && fix (p) == p) | |
| 658 n = p; | |
| 659 p = linspace (0, 1, n); | |
| 660 elseif (n > 1 && isvector (p)) | |
| 661 ## do nothing | |
| 662 endif | |
| 663 p = p(:).'; # Ensure p is a row vector. | |
| 664 | |
| 665 if (nargin == 1) | |
| 666 m = n; | |
| 667 elseif (! isnumeric (m) || ! isscalar (m)) | |
| 668 error ("gallery: M must be a scalar for chebvand matrix."); | |
| 669 endif | |
| 670 | |
| 671 C = ones (m, n); | |
| 672 if (m != 1) | |
| 673 C(2,:) = p; | |
| 674 ## Use Chebyshev polynomial recurrence. | |
| 675 for i = 3:m | |
| 676 C(i,:) = 2.*p.*C(i-1,:) - C(i-2,:); | |
| 677 endfor | |
| 678 endif | |
| 679 endfunction | |
| 680 | |
| 681 function A = chow (n, alpha = 1, delta = 0) | |
| 682 ## CHOW Chow matrix - a singular Toeplitz lower Hessenberg matrix. | |
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683 ## A = CHOW(N, ALPHA, DELTA) is a Toeplitz lower Hessenberg matrix |
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684 ## A = H(ALPHA) + DELTA*EYE, where H(i,j) = ALPHA^(i-j+1). |
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685 ## H(ALPHA) has p = FLOOR(N/2) zero eigenvalues, the rest being |
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686 ## 4*ALPHA*COS( k*PI/(N+2) )^2, k=1:N-p. |
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687 ## Defaults: ALPHA = 1, DELTA = 0. |
| 16634 | 688 ## |
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689 ## References: |
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690 ## T.S. Chow, A class of Hessenberg matrices with known |
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691 ## eigenvalues and inverses, SIAM Review, 11 (1969), pp. 391-395. |
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692 ## G. Fairweather, On the eigenvalues and eigenvectors of a class of |
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693 ## Hessenberg matrices, SIAM Review, 13 (1971), pp. 220-221. |
| 16634 | 694 |
| 695 if (nargin < 1 || nargin > 3) | |
| 696 error ("gallery: 1 to 3 arguments are required for chow matrix."); | |
| 697 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 698 error ("gallery: N must be an integer for chow matrix."); | |
| 699 elseif (! isnumeric (alpha) || ! isscalar (alpha)) | |
| 700 error ("gallery: ALPHA must be a scalar for chow matrix."); | |
| 701 elseif (! isnumeric (delta) || ! isscalar (delta)) | |
| 702 error ("gallery: DELTA must be a scalar for chow matrix."); | |
| 703 endif | |
| 704 | |
| 705 A = toeplitz (alpha.^(1:n), [alpha 1 zeros(1, n-2)]) + delta * eye (n); | |
| 706 endfunction | |
| 707 | |
| 708 function C = circul (v) | |
| 709 ## CIRCUL Circulant matrix. | |
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710 ## C = CIRCUL(V) is the circulant matrix whose first row is V. |
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711 ## (A circulant matrix has the property that each row is obtained |
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712 ## from the previous one by cyclically permuting the entries one step |
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713 ## forward; it is a special Toeplitz matrix in which the diagonals |
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714 ## `wrap round'.) |
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715 ## Special case: if V is a scalar then C = CIRCUL(1:V). |
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716 ## The eigensystem of C (N-by-N) is known explicitly. If t is an Nth |
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717 ## root of unity, then the inner product of V with W = [1 t t^2 ... t^N] |
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|
718 ## is an eigenvalue of C, and W(N:-1:1) is an eigenvector of C. |
| 16634 | 719 ## |
|
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720 ## Reference: |
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|
721 ## P.J. Davis, Circulant Matrices, John Wiley, 1977. |
| 16634 | 722 |
| 723 if (nargin != 1) | |
| 724 error ("gallery: 1 argument is required for circul matrix."); | |
| 725 elseif (! isnumeric (v)) | |
| 726 error ("gallery: V must be numeric for circul matrix."); | |
| 727 endif | |
| 728 | |
|
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729 n = numel (v); |
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730 if (isscalar (v) && fix (v) == v) |
| 16634 | 731 n = v; |
| 732 v = 1:n; | |
|
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|
733 elseif (n > 1 && isvector (v)) |
| 16634 | 734 ## do nothing |
| 735 else | |
| 736 error ("gallery: X must be a scalar or a vector for circul matrix."); | |
| 737 endif | |
| 738 | |
| 739 v = v(:).'; # Make sure v is a row vector | |
| 740 C = toeplitz ([v(1) v(n:-1:2)], v); | |
| 741 endfunction | |
| 742 | |
| 743 function A = clement (n, k = 0) | |
| 744 ## CLEMENT Clement matrix - tridiagonal with zero diagonal entries. | |
|
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745 ## CLEMENT(N, K) is a tridiagonal matrix with zero diagonal entries |
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746 ## and known eigenvalues. It is singular if N is odd. About 64 |
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747 ## percent of the entries of the inverse are zero. The eigenvalues |
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748 ## are plus and minus the numbers N-1, N-3, N-5, ..., (1 or 0). |
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749 ## For K = 0 (the default) the matrix is unsymmetric, while for |
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750 ## K = 1 it is symmetric. |
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751 ## CLEMENT(N, 1) is diagonally similar to CLEMENT(N). |
| 16634 | 752 ## |
|
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753 ## Similar properties hold for TRIDIAG(X,Y,Z) where Y = ZEROS(N,1). |
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754 ## The eigenvalues still come in plus/minus pairs but they are not |
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755 ## known explicitly. |
| 16634 | 756 ## |
|
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757 ## References: |
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758 ## P.A. Clement, A class of triple-diagonal matrices for test |
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759 ## purposes, SIAM Review, 1 (1959), pp. 50-52. |
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760 ## A. Edelman and E. Kostlan, The road from Kac's matrix to Kac's |
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761 ## random polynomials. In John~G. Lewis, editor, Proceedings of |
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762 ## the Fifth SIAM Conference on Applied Linear Algebra Society |
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763 ## for Industrial and Applied Mathematics, Philadelphia, 1994, |
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764 ## pp. 503-507. |
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765 ## O. Taussky and J. Todd, Another look at a matrix of Mark Kac, |
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766 ## Linear Algebra and Appl., 150 (1991), pp. 341-360. |
| 16634 | 767 |
| 768 if (nargin < 1 || nargin > 2) | |
| 769 error ("gallery: 1 or 2 arguments are required for clement matrix."); | |
| 770 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 771 error ("gallery: N must be an integer for clement matrix."); | |
| 772 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 773 error ("gallery: K must be a numeric scalar for clement matrix."); | |
| 774 endif | |
| 775 | |
|
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|
776 n -= 1; |
| 16634 | 777 x = n:-1:1; |
| 778 z = 1:n; | |
| 779 | |
| 780 if (k == 0) | |
| 781 A = diag (x, -1) + diag (z, 1); | |
| 782 elseif (k == 1) | |
| 783 y = sqrt (x.*z); | |
| 784 A = diag (y, -1) + diag (y, 1); | |
| 785 else | |
| 786 error ("gallery: K must have a value of 0 or 1 for clement matrix."); | |
| 787 endif | |
| 788 endfunction | |
| 789 | |
| 790 function C = compar (A, k = 0) | |
| 791 ## COMP Comparison matrices. | |
|
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792 ## COMP(A) is DIAG(B) - TRIL(B,-1) - TRIU(B,1), where B = ABS(A). |
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793 ## COMP(A, 1) is A with each diagonal element replaced by its |
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794 ## absolute value, and each off-diagonal element replaced by minus |
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795 ## the absolute value of the largest element in absolute value in |
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796 ## its row. However, if A is triangular COMP(A, 1) is too. |
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797 ## COMP(A, 0) is the same as COMP(A). |
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798 ## COMP(A) is often denoted by M(A) in the literature. |
| 16634 | 799 ## |
|
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800 ## Reference (e.g.): |
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801 ## N.J. Higham, A survey of condition number estimation for |
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802 ## triangular matrices, SIAM Review, 29 (1987), pp. 575-596. |
| 16634 | 803 |
| 804 if (nargin < 1 || nargin > 2) | |
| 805 error ("gallery: 1 or 2 arguments are required for compar matrix."); | |
| 806 elseif (! isnumeric (A) || ndims (A) != 2) | |
| 17199 | 807 error ("gallery: A must be a 2-D matrix for compar matrix."); |
| 16634 | 808 elseif (! isnumeric (k) || ! isscalar (k)) |
| 809 error ("gallery: K must be a numeric scalar for compar matrix."); | |
| 810 endif | |
| 811 | |
| 812 [m, n] = size (A); | |
| 813 p = min (m, n); | |
| 814 | |
| 815 if (k == 0) | |
| 816 ## This code uses less temporary storage than | |
| 817 ## the `high level' definition above. | |
| 818 C = -abs (A); | |
| 819 for j = 1:p | |
| 820 C(j,j) = abs (A(j,j)); | |
| 821 endfor | |
| 822 | |
| 823 elseif (k == 1) | |
| 824 C = A'; | |
| 825 for j = 1:p | |
| 826 C(k,k) = 0; | |
| 827 endfor | |
| 828 mx = max (abs (C)); | |
| 829 C = -mx'*ones (1, n); | |
| 830 for j = 1:p | |
| 831 C(j,j) = abs (A(j,j)); | |
| 832 endfor | |
| 833 if (all (A == tril (A))), C = tril (C); endif | |
| 834 if (all (A == triu (A))), C = triu (C); endif | |
| 835 | |
| 836 else | |
| 837 error ("gallery: K must have a value of 0 or 1 for compar matrix."); | |
| 838 endif | |
| 839 | |
| 840 endfunction | |
| 841 | |
| 842 function A = condex (n, k = 4, theta = 100) | |
| 843 ## CONDEX `Counterexamples' to matrix condition number estimators. | |
|
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844 ## CONDEX(N, K, THETA) is a `counterexample' matrix to a condition |
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845 ## estimator. It has order N and scalar parameter THETA (default 100). |
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846 ## If N is not equal to the `natural' size of the matrix then |
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|
847 ## the matrix is padded out with an identity matrix to order N. |
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848 ## The matrix, its natural size, and the estimator to which it applies |
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849 ## are specified by K (default K = 4) as follows: |
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850 ## K = 1: 4-by-4, LINPACK (RCOND) |
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851 ## K = 2: 3-by-3, LINPACK (RCOND) |
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852 ## K = 3: arbitrary, LINPACK (RCOND) (independent of THETA) |
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853 ## K = 4: N >= 4, SONEST (Higham 1988) |
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854 ## (Note that in practice the K = 4 matrix is not usually a |
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855 ## counterexample because of the rounding errors in forming it.) |
| 16634 | 856 ## |
|
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857 ## References: |
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858 ## A.K. Cline and R.K. Rew, A set of counter-examples to three |
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859 ## condition number estimators, SIAM J. Sci. Stat. Comput., |
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860 ## 4 (1983), pp. 602-611. |
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861 ## N.J. Higham, FORTRAN codes for estimating the one-norm of a real or |
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862 ## complex matrix, with applications to condition estimation |
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863 ## (Algorithm 674), ACM Trans. Math. Soft., 14 (1988), pp. 381-396. |
| 16634 | 864 |
| 865 if (nargin < 1 || nargin > 3) | |
| 866 error ("gallery: 1 to 3 arguments are required for condex matrix."); | |
| 867 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 868 error ("gallery: N must be an integer for condex matrix."); | |
| 869 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 870 error ("gallery: K must be a numeric scalar for condex matrix."); | |
| 871 elseif (! isnumeric (theta) || ! isscalar (theta)) | |
| 872 error ("gallery: THETA must be a numeric scalar for condex matrix."); | |
| 873 endif | |
| 874 | |
| 875 if (k == 1) # Cline and Rew (1983), Example B. | |
| 876 A = [1 -1 -2*theta 0 | |
| 877 0 1 theta -theta | |
| 878 0 1 1+theta -(theta+1) | |
| 879 0 0 0 theta]; | |
| 880 | |
| 881 elseif (k == 2) # Cline and Rew (1983), Example C. | |
| 882 A = [1 1-2/theta^2 -2 | |
| 883 0 1/theta -1/theta | |
| 884 0 0 1]; | |
| 885 | |
| 886 elseif (k == 3) # Cline and Rew (1983), Example D. | |
| 887 A = gallery ("triw", n, -1)'; | |
| 888 A(n,n) = -1; | |
| 889 | |
| 890 elseif (k == 4) # Higham (1988), p. 390. | |
| 891 x = ones (n, 3); # First col is e | |
| 892 x(2:n,2) = zeros (n-1, 1); # Second col is e(1) | |
| 893 | |
| 894 ## Third col is special vector b in SONEST | |
| 895 x(:, 3) = (-1).^[0:n-1]' .* ( 1 + [0:n-1]'/(n-1) ); | |
| 896 | |
| 897 Q = orth (x); # Q*Q' is now the orthogonal projector onto span(e(1),e,b)). | |
| 898 P = eye (n) - Q*Q'; | |
| 899 A = eye (n) + theta*P; | |
| 900 | |
| 901 else | |
|
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902 error ("gallery: unknown estimator K '%d' for condex matrix.", k); |
| 16634 | 903 endif |
| 904 | |
| 905 ## Pad out with identity as necessary. | |
| 906 m = columns (A); | |
| 907 if (m < n) | |
| 908 for i = n:-1:m+1 | |
| 909 A(i,i) = 1; | |
| 910 endfor | |
| 911 endif | |
| 912 endfunction | |
| 913 | |
|
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914 function A = cycol (n, k = max (round (n/4), 1)) |
| 16634 | 915 ## CYCOL Matrix whose columns repeat cyclically. |
|
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916 ## A = CYCOL([M N], K) is an M-by-N matrix of the form A = B(1:M,1:N) |
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917 ## where B = [C C C...] and C = RANDN(M, K). Thus A's columns repeat |
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918 ## cyclically, and A has rank at most K. K need not divide N. |
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919 ## K defaults to ROUND(N/4). |
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920 ## CYCOL(N, K), where N is a scalar, is the same as CYCOL([N N], K). |
| 16634 | 921 ## |
|
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922 ## This type of matrix can lead to underflow problems for Gaussian |
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923 ## elimination: see NA Digest Volume 89, Issue 3 (January 22, 1989). |
| 16634 | 924 |
| 925 if (nargin < 1 || nargin > 2) | |
| 926 error ("gallery: 1 or 2 arguments are required for cycol matrix."); | |
| 927 elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n) | |
| 928 error ("gallery: N must be a 1 or 2 element integer for cycol matrix."); | |
| 929 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 930 error ("gallery: K must be a scalar for cycol matrix."); | |
| 931 endif | |
| 932 | |
| 933 ## Parameter n specifies dimension: m-by-n | |
| 934 m = n(1); | |
| 935 n = n(end); | |
| 936 | |
| 937 A = randn (m, k); | |
| 938 for i = 2:ceil (n/k) | |
| 939 A = [A A(:,1:k)]; | |
| 940 endfor | |
| 941 A = A(:,1:n); | |
| 942 endfunction | |
| 943 | |
| 944 function [c, d, e] = dorr (n, theta = 0.01) | |
| 945 ## DORR Dorr matrix - diagonally dominant, ill conditioned, tridiagonal. | |
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946 ## [C, D, E] = DORR(N, THETA) returns the vectors defining a row diagonally |
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947 ## dominant, tridiagonal M-matrix that is ill conditioned for small |
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948 ## values of the parameter THETA >= 0. |
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949 ## If only one output parameter is supplied then |
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950 ## C = FULL(TRIDIAG(C,D,E)), i.e., the matrix iself is returned. |
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951 ## The columns of INV(C) vary greatly in norm. THETA defaults to 0.01. |
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952 ## The amount of diagonal dominance is given by (ignoring rounding errors): |
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953 ## COMP(C)*ONES(N,1) = THETA*(N+1)^2 * [1 0 0 ... 0 1]'. |
| 16634 | 954 ## |
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955 ## Reference: |
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956 ## F.W. Dorr, An example of ill-conditioning in the numerical |
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957 ## solution of singular perturbation problems, Math. Comp., 25 (1971), |
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958 ## pp. 271-283. |
| 16634 | 959 |
| 960 if (nargin < 1 || nargin > 2) | |
| 961 error ("gallery: 1 or 2 arguments are required for dorr matrix."); | |
| 962 elseif (! isscalar (n) || ! isnumeric (n) || fix (n) != n) | |
| 963 error ("gallery: N must be an integer for dorr matrix."); | |
| 964 elseif (! isscalar (theta) || ! isnumeric (theta)) | |
| 965 error ("gallery: THETA must be a numeric scalar for dorr matrix."); | |
| 966 endif | |
| 967 | |
| 968 c = zeros (n, 1); | |
| 969 e = c; | |
| 970 d = c; | |
| 971 ## All length n for convenience. Make c, e of length n-1 later. | |
| 972 | |
| 973 h = 1/(n+1); | |
| 974 m = floor ((n+1)/2); | |
| 975 term = theta/h^2; | |
| 976 | |
| 977 i = (1:m)'; | |
| 978 c(i) = -term * ones (m, 1); | |
| 979 e(i) = c(i) - (0.5-i*h)/h; | |
| 980 d(i) = -(c(i) + e(i)); | |
| 981 | |
| 982 i = (m+1:n)'; | |
| 983 e(i) = -term * ones (n-m, 1); | |
| 984 c(i) = e(i) + (0.5-i*h)/h; | |
| 985 d(i) = -(c(i) + e(i)); | |
| 986 | |
| 987 c = c(2:n); | |
| 988 e = e(1:n-1); | |
| 989 | |
| 990 if (nargout <= 1) | |
| 991 c = tridiag (c, d, e); | |
| 992 endif | |
| 993 endfunction | |
| 994 | |
| 995 function A = dramadah (n, k = 1) | |
| 996 ## DRAMADAH A (0,1) matrix whose inverse has large integer entries. | |
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997 ## An anti-Hadamard matrix A is a matrix with elements 0 or 1 for |
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998 ## which MU(A) := NORM(INV(A),'FRO') is maximal. |
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999 ## A = DRAMADAH(N, K) is an N-by-N (0,1) matrix for which MU(A) is |
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1000 ## relatively large, although not necessarily maximal. |
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1001 ## Available types (the default is K = 1): |
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1002 ## K = 1: A is Toeplitz, with ABS(DET(A)) = 1, and MU(A) > c(1.75)^N, |
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1003 ## where c is a constant. |
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1004 ## K = 2: A is upper triangular and Toeplitz. |
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1005 ## The inverses of both types have integer entries. |
| 16634 | 1006 ## |
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1007 ## Another interesting (0,1) matrix: |
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1008 ## K = 3: A has maximal determinant among (0,1) lower Hessenberg |
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1009 ## matrices: det(A) = the n'th Fibonacci number. A is Toeplitz. |
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1010 ## The eigenvalues have an interesting distribution in the complex |
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1011 ## plane. |
| 16634 | 1012 ## |
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1013 ## References: |
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1014 ## R.L. Graham and N.J.A. Sloane, Anti-Hadamard matrices, |
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1015 ## Linear Algebra and Appl., 62 (1984), pp. 113-137. |
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1016 ## L. Ching, The maximum determinant of an nxn lower Hessenberg |
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|
1017 ## (0,1) matrix, Linear Algebra and Appl., 183 (1993), pp. 147-153. |
| 16634 | 1018 |
| 1019 if (nargin < 1 || nargin > 2) | |
| 1020 error ("gallery: 1 to 2 arguments are required for dramadah matrix."); | |
| 1021 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1022 error ("gallery: N must be an integer for dramadah matrix."); | |
| 1023 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 1024 error ("gallery: K must be a numeric scalar for dramadah matrix."); | |
| 1025 endif | |
| 1026 | |
| 1027 switch (k) | |
| 1028 case (1) # Toeplitz | |
| 1029 c = ones (n, 1); | |
| 1030 for i = 2:4:n | |
| 1031 m = min (1, n-i); | |
| 1032 c(i:i+m) = zeros (m+1, 1); | |
| 1033 endfor | |
| 1034 r = zeros (n, 1); | |
| 1035 r(1:4) = [1 1 0 1]; | |
| 1036 if (n < 4) | |
| 1037 r = r(1:n); | |
| 1038 endif | |
| 1039 A = toeplitz (c, r); | |
| 1040 | |
| 1041 case (2) # Upper triangular and Toeplitz | |
| 1042 c = zeros (n, 1); | |
| 1043 c(1) = 1; | |
| 1044 r = ones (n, 1); | |
| 1045 for i= 3:2:n | |
| 1046 r(i) = 0; | |
| 1047 endfor | |
| 1048 A = toeplitz (c, r); | |
| 1049 | |
| 1050 case (3) # Lower Hessenberg | |
| 1051 c = ones (n, 1); | |
| 1052 for i= 2:2:n | |
| 1053 c(i) = 0; | |
| 1054 endfor | |
| 1055 A = toeplitz (c, [1 1 zeros(1,n-2)]); | |
| 1056 | |
| 1057 otherwise | |
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1058 error ("gallery: unknown K '%d' for dramadah matrix.", k); |
| 16634 | 1059 endswitch |
| 1060 endfunction | |
| 1061 | |
| 1062 function A = fiedler (c) | |
| 1063 ## FIEDLER Fiedler matrix - symmetric. | |
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1064 ## FIEDLER(C), where C is an n-vector, is the n-by-n symmetric |
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1065 ## matrix with elements ABS(C(i)-C(j)). |
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1066 ## Special case: if C is a scalar, then A = FIEDLER(1:C) |
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1067 ## (i.e. A(i,j) = ABS(i-j)). |
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1068 ## Properties: |
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1069 ## FIEDLER(N) has a dominant positive eigenvalue and all the other |
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1070 ## eigenvalues are negative (Szego, 1936). |
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1071 ## Explicit formulas for INV(A) and DET(A) are given by Todd (1977) |
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1072 ## and attributed to Fiedler. These indicate that INV(A) is |
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1073 ## tridiagonal except for nonzero (1,n) and (n,1) elements. |
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1074 ## [I think these formulas are valid only if the elements of |
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1075 ## C are in increasing or decreasing order---NJH.] |
| 16634 | 1076 ## |
|
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1077 ## References: |
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1078 ## G. Szego, Solution to problem 3705, Amer. Math. Monthly, |
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1079 ## 43 (1936), pp. 246-259. |
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1080 ## J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, |
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1081 ## Birkhauser, Basel, and Academic Press, New York, 1977, p. 159. |
| 16634 | 1082 |
| 1083 if (nargin != 1) | |
| 1084 error ("gallery: 1 argument is required for fiedler matrix."); | |
| 1085 elseif (! isnumeric (c)) | |
| 1086 error ("gallery: C must be numeric for fiedler matrix."); | |
| 1087 endif | |
| 1088 | |
| 1089 n = numel (c); | |
| 1090 if (isscalar (c) && fix (c) == c) | |
| 1091 n = c; | |
| 1092 c = 1:n; | |
| 1093 elseif (n > 1 && isvector (c)) | |
| 1094 ## do nothing | |
| 1095 else | |
| 1096 error ("gallery: C must be an integer or a vector for fiedler matrix."); | |
| 1097 endif | |
| 1098 c = c(:).'; # Ensure c is a row vector. | |
| 1099 | |
| 1100 A = ones (n, 1) * c; | |
| 1101 A = abs (A - A.'); # NB. array transpose. | |
| 1102 endfunction | |
| 1103 | |
| 1104 function A = forsythe (n, alpha = sqrt (eps), lambda = 0) | |
| 1105 ## FORSYTHE Forsythe matrix - a perturbed Jordan block. | |
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1106 ## FORSYTHE(N, ALPHA, LAMBDA) is the N-by-N matrix equal to |
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1107 ## JORDBLOC(N, LAMBDA) except it has an ALPHA in the (N,1) position. |
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1108 ## It has the characteristic polynomial |
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1109 ## DET(A-t*EYE) = (LAMBDA-t)^N - (-1)^N ALPHA. |
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1110 ## ALPHA defaults to SQRT(EPS) and LAMBDA to 0. |
| 16634 | 1111 |
| 1112 if (nargin < 1 || nargin > 3) | |
| 1113 error ("gallery: 1 to 3 arguments are required for forsythe matrix."); | |
| 1114 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1115 error ("gallery: N must be an integer for forsythe matrix."); | |
| 1116 elseif (! isnumeric (alpha) || ! isscalar (alpha)) | |
| 1117 error ("gallery: ALPHA must be a numeric scalar for forsythe matrix."); | |
| 1118 elseif (! isnumeric (lambda) || ! isscalar (lambda)) | |
| 1119 error ("gallery: LAMBDA must be a numeric scalar for forsythe matrix."); | |
| 1120 endif | |
| 1121 | |
| 1122 A = jordbloc (n, lambda); | |
| 1123 A(n,1) = alpha; | |
| 1124 endfunction | |
| 1125 | |
| 1126 function F = frank (n, k = 0) | |
| 1127 ## FRANK Frank matrix---ill conditioned eigenvalues. | |
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1128 ## F = FRANK(N, K) is the Frank matrix of order N. It is upper |
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1129 ## Hessenberg with determinant 1. K = 0 is the default; if K = 1 the |
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1130 ## elements are reflected about the anti-diagonal (1,N)--(N,1). |
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1131 ## F has all positive eigenvalues and they occur in reciprocal pairs |
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1132 ## (so that 1 is an eigenvalue if N is odd). |
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1133 ## The eigenvalues of F may be obtained in terms of the zeros of the |
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1134 ## Hermite polynomials. |
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1135 ## The FLOOR(N/2) smallest eigenvalues of F are ill conditioned, |
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1136 ## the more so for bigger N. |
| 16634 | 1137 ## |
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1138 ## DET(FRANK(N)') comes out far from 1 for large N---see Frank (1958) |
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1139 ## and Wilkinson (1960) for discussions. |
| 16634 | 1140 ## |
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1141 ## This version incorporates improvements suggested by W. Kahan. |
| 16634 | 1142 ## |
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1143 ## References: |
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1144 ## W.L. Frank, Computing eigenvalues of complex matrices by determinant |
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1145 ## evaluation and by methods of Danilewski and Wielandt, J. Soc. |
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1146 ## Indust. Appl. Math., 6 (1958), pp. 378-392 (see pp. 385, 388). |
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1147 ## G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the |
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1148 ## computation of the Jordan canonical form, SIAM Review, 18 (1976), |
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1149 ## pp. 578-619 (Section 13). |
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1150 ## H. Rutishauser, On test matrices, Programmation en Mathematiques |
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1151 ## Numeriques, Editions Centre Nat. Recherche Sci., Paris, 165, |
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1152 ## 1966, pp. 349-365. Section 9. |
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1153 ## J.H. Wilkinson, Error analysis of floating-point computation, |
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1154 ## Numer. Math., 2 (1960), pp. 319-340 (Section 8). |
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1155 ## J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University |
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1156 ## Press, 1965 (pp. 92-93). |
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1157 ## The next two references give details of the eigensystem, as does |
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1158 ## Rutishauser (see above). |
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1159 ## P.J. Eberlein, A note on the matrices denoted by B_n, SIAM J. Appl. |
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1160 ## Math., 20 (1971), pp. 87-92. |
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1161 ## J.M. Varah, A generalization of the Frank matrix, SIAM J. Sci. Stat. |
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diff
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|
1162 ## Comput., 7 (1986), pp. 835-839. |
| 16634 | 1163 |
| 1164 if (nargin < 1 || nargin > 2) | |
| 1165 error ("gallery: 1 to 2 arguments are required for frank matrix."); | |
| 1166 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1167 error ("gallery: N must be an integer for frank matrix."); | |
| 1168 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 1169 error ("gallery: K must be a numeric scalar for frank matrix."); | |
| 1170 endif | |
| 1171 | |
| 1172 p = n:-1:1; | |
| 1173 F = triu (p(ones (n, 1), :) - diag (ones (n-1, 1), -1), -1); | |
| 1174 | |
| 1175 switch (k) | |
| 1176 case (0), # do nothing | |
| 1177 case (1), F = F(p,p)'; | |
| 1178 otherwise | |
| 1179 error ("gallery: K must have a value of 0 or 1 for frank matrix."); | |
| 1180 endswitch | |
| 1181 endfunction | |
| 1182 | |
| 1183 function c = gcdmat (n) | |
| 1184 if (nargin != 1) | |
| 1185 error ("gallery: 1 argument is required for gcdmat matrix."); | |
| 1186 elseif (! isscalar (n) || ! isnumeric (n) || fix (n) != n) | |
| 1187 error ("gallery: N must be an integer for gcdmat matrix."); | |
| 1188 endif | |
| 1189 c = gcd (repmat ((1:n)', [1 n]), repmat (1:n, [n 1])); | |
| 1190 endfunction | |
| 1191 | |
| 1192 function A = gearmat (n, i = n, j = -n) | |
| 1193 ## NOTE: this function was named gearm in the original Test Matrix Toolbox | |
| 1194 ## GEARMAT Gear matrix. | |
|
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|
1195 ## A = GEARMAT(N,I,J) is the N-by-N matrix with ones on the sub- and |
|
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|
1196 ## super-diagonals, SIGN(I) in the (1,ABS(I)) position, SIGN(J) |
|
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|
1197 ## in the (N,N+1-ABS(J)) position, and zeros everywhere else. |
|
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|
1198 ## Defaults: I = N, j = -N. |
|
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|
1199 ## All eigenvalues are of the form 2*COS(a) and the eigenvectors |
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|
1200 ## are of the form [SIN(w+a), SIN(w+2a), ..., SIN(w+Na)]. |
|
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|
1201 ## The values of a and w are given in the reference below. |
|
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|
1202 ## A can have double and triple eigenvalues and can be defective. |
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|
1203 ## GEARMAT(N) is singular. |
| 16634 | 1204 ## |
|
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|
1205 ## (GEAR is a Simulink function, hence GEARMAT for Gear matrix.) |
|
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|
1206 ## Reference: |
|
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|
1207 ## C.W. Gear, A simple set of test matrices for eigenvalue programs, |
|
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|
1208 ## Math. Comp., 23 (1969), pp. 119-125. |
| 16634 | 1209 |
| 1210 if (nargin < 1 || nargin > 3) | |
| 1211 error ("gallery: 1 to 3 arguments are required for gearmat matrix."); | |
| 1212 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1213 error ("gallery: N must be an integer for gearmat matrix."); | |
|
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|
1214 elseif (! isnumeric (i) || ! isscalar (i) || i == 0 || abs (i) > n) |
|
18812
9ac2357f19bc
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|
1215 error ("gallery: I must be a nonzero scalar, and abs (I) <= N for gearmat matrix."); |
|
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2102c4582e32
gallery: fix chebspec, cycol, gearmat, hanowa, lauchli, and pei (bug #45466)
Massimiliano Fasi <massimiliano.fasi@gmail.com>
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|
1216 elseif (! isnumeric (j) || ! isscalar (j) || i == 0 || abs (j) > n) |
|
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|
1217 error ("gallery: J must be a nonzero scalar, and abs (J) <= N for gearmat matrix."); |
| 16634 | 1218 endif |
| 1219 | |
| 1220 A = diag (ones (n-1, 1), -1) + diag (ones (n-1, 1), 1); | |
| 1221 A(1, abs (i)) = sign (i); | |
| 1222 A(n, n+1 - abs (j)) = sign (j); | |
| 1223 endfunction | |
| 1224 | |
| 1225 function G = grcar (n, k = 3) | |
| 1226 ## GRCAR Grcar matrix - a Toeplitz matrix with sensitive eigenvalues. | |
|
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|
1227 ## GRCAR(N, K) is an N-by-N matrix with -1s on the |
|
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|
1228 ## subdiagonal, 1s on the diagonal, and K superdiagonals of 1s. |
|
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|
1229 ## The default is K = 3. The eigenvalues of this matrix form an |
|
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|
1230 ## interesting pattern in the complex plane (try PS(GRCAR(32))). |
| 16634 | 1231 ## |
|
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|
1232 ## References: |
|
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|
1233 ## J.F. Grcar, Operator coefficient methods for linear equations, |
|
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|
1234 ## Report SAND89-8691, Sandia National Laboratories, Albuquerque, |
|
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|
1235 ## New Mexico, 1989 (Appendix 2). |
|
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|
1236 ## N.M. Nachtigal, L. Reichel and L.N. Trefethen, A hybrid GMRES |
|
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|
1237 ## algorithm for nonsymmetric linear systems, SIAM J. Matrix Anal. |
|
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|
1238 ## Appl., 13 (1992), pp. 796-825. |
| 16634 | 1239 |
| 1240 if (nargin < 1 || nargin > 2) | |
| 1241 error ("gallery: 1 to 2 arguments are required for grcar matrix."); | |
| 1242 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1243 error ("gallery: N must be an integer for grcar matrix."); | |
| 1244 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 1245 error ("gallery: K must be a numeric scalar for grcar matrix."); | |
| 1246 endif | |
| 1247 | |
| 1248 G = tril (triu (ones (n)), k) - diag (ones (n-1, 1), -1); | |
| 1249 endfunction | |
| 1250 | |
| 1251 function A = hanowa (n, d = -1) | |
| 1252 ## HANOWA A matrix whose eigenvalues lie on a vertical line in the complex plane. | |
|
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|
1253 ## HANOWA(N, d) is the N-by-N block 2x2 matrix (thus N = 2M must be even) |
|
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|
1254 ## [d*EYE(M) -DIAG(1:M) |
|
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|
1255 ## DIAG(1:M) d*EYE(M)] |
|
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|
1256 ## It has complex eigenvalues lambda(k) = d +/- k*i (1 <= k <= M). |
|
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|
1257 ## Parameter d defaults to -1. |
| 16634 | 1258 ## |
|
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|
1259 ## Reference: |
|
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|
1260 ## E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary |
|
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|
1261 ## Differential Equations I: Nonstiff Problems, Springer-Verlag, |
|
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|
1262 ## Berlin, 1987. (pp. 86-87) |
| 16634 | 1263 |
| 1264 if (nargin < 1 || nargin > 2) | |
| 1265 error ("gallery: 1 to 2 arguments are required for hanowa matrix."); | |
| 1266 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1267 error ("gallery: N must be an integer for hanowa matrix."); | |
| 1268 elseif (rem (n, 2) != 0) | |
| 1269 error ("gallery: N must be even for hanowa matrix."); | |
|
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2102c4582e32
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|
1270 elseif (! isnumeric (d) || ! isscalar (d)) |
| 16634 | 1271 error ("gallery: D must be a numeric scalar for hanowa matrix."); |
| 1272 endif | |
| 1273 | |
| 1274 m = n/2; | |
| 1275 A = [ d*eye(m) -diag(1:m) | |
| 1276 diag(1:m) d*eye(m) ]; | |
| 1277 endfunction | |
| 1278 | |
| 1279 function [v, beta] = house (x) | |
| 1280 ## HOUSE Householder matrix. | |
|
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|
1281 ## If [v, beta] = HOUSE(x) then H = EYE - beta*v*v' is a Householder |
|
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|
1282 ## matrix such that Hx = -sign(x(1))*norm(x)*e_1. |
|
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|
1283 ## NB: If x = 0 then v = 0, beta = 1 is returned. |
|
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|
1284 ## x can be real or complex. |
|
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|
1285 ## sign(x) := exp(i*arg(x)) ( = x./abs(x) when x ~= 0). |
| 16634 | 1286 ## |
|
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|
1287 ## Theory: (textbook references Golub & Van Loan 1989, 38-43; |
|
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|
1288 ## Stewart 1973, 231-234, 262; Wilkinson 1965, 48-50). |
|
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|
1289 ## Hx = y: (I - beta*v*v')x = -s*e_1. |
|
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|
1290 ## Must have |s| = norm(x), v = x+s*e_1, and |
|
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|
1291 ## x'y = x'Hx =(x'Hx)' real => arg(s) = arg(x(1)). |
|
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|
1292 ## So take s = sign(x(1))*norm(x) (which avoids cancellation). |
|
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|
1293 ## v'v = (x(1)+s)^2 + x(2)^2 + ... + x(n)^2 |
|
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|
1294 ## = 2*norm(x)*(norm(x) + |x(1)|). |
|
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|
1295 ## |
|
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1296 ## References: |
|
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|
1297 ## G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, |
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|
1298 ## Johns Hopkins University Press, Baltimore, Maryland, 1989. |
|
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|
1299 ## G.W. Stewart, Introduction to Matrix Computations, Academic Press, |
|
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|
1300 ## New York, 1973, |
|
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|
1301 ## J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University |
|
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|
1302 ## Press, 1965. |
| 16634 | 1303 |
| 1304 if (nargin != 1) | |
| 1305 error ("gallery: 1 argument is required for house matrix."); | |
|
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Carnë Draug <carandraug@octave.org>
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20294
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|
1306 elseif (! isnumeric (x) || ! isvector (x)) |
| 16634 | 1307 error ("gallery: X must be a vector for house matrix."); |
| 1308 endif | |
| 1309 | |
| 1310 ## must be a column vector | |
| 1311 x = x(:); | |
| 1312 | |
| 1313 s = norm (x) * (sign (x(1)) + (x(1) == 0)); # Modification for sign (0) == 1. | |
| 1314 v = x; | |
| 1315 if (s == 0) | |
| 1316 ## Quit if x is the zero vector. | |
| 1317 beta = 1; | |
| 1318 else | |
| 1319 v(1) = v(1) + s; | |
| 1320 beta = 1/(s'*v(1)); # NB the conjugated s. | |
| 1321 ## beta = 1/(abs (s) * (abs (s) +abs(x(1)) would guarantee beta real. | |
| 1322 ## But beta as above can be non-real (due to rounding) only when x is complex. | |
| 1323 endif | |
| 1324 endfunction | |
| 1325 | |
|
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|
1326 function A = integerdata (varargin) |
|
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|
1327 |
|
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|
1328 if (nargin < 3) |
|
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|
1329 error ("gallery: At least 3 arguments required for integerdata matrix."); |
|
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|
1330 endif |
|
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|
1331 |
|
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|
1332 if (isnumeric (varargin{end})) |
|
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|
1333 jidx = varargin{end}; |
|
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|
1334 svec = [varargin{:}]; |
|
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|
1335 varargin(end) = []; |
|
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|
1336 elseif (ischar (varargin{end})) |
|
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|
1337 if (nargin < 4) |
|
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|
1338 error (["gallery: CLASS argument requires 4 inputs " ... |
|
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|
1339 "for integerdata matrix."]); |
|
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|
1340 endif |
|
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|
1341 jidx = varargin{end-1}; |
|
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|
1342 svec = [varargin{1:end-1}]; |
|
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|
1343 varargin(end-1) = []; |
|
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|
1344 else |
|
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|
1345 error (["gallery: J must be an integer in the range [0, 2^32-1] " ... |
|
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|
1346 "for integerdata matrix"]); |
|
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|
1347 endif |
|
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|
1348 |
|
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|
1349 if (! (isnumeric (jidx) && isscalar (jidx) |
|
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|
1350 && jidx == fix (jidx) |
|
1909e1ed63e6
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|
1351 && jidx >= 0 && jidx <= 0xFFFFFFFF)) |
|
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|
1352 error (["gallery: J must be an integer in the range [0, 2^32-1] " ... |
|
1909e1ed63e6
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|
1353 "for integerdata matrix"]); |
|
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|
1354 endif |
|
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|
1355 |
|
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|
1356 ## Save and restore random state. Initialization done so that reproducible |
|
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|
1357 ## data is available from gallery depending on the jidx and size vector. |
|
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|
1358 randstate = rand ("state"); |
|
16980
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|
1359 unwind_protect |
|
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|
1360 rand ("state", svec); |
|
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|
1361 A = randi (varargin{:}); |
|
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|
1362 unwind_protect_cleanup |
|
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|
1363 rand ("state", randstate); |
|
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|
1364 end_unwind_protect |
|
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diff
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|
1365 |
|
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|
1366 endfunction |
|
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|
1367 |
| 16634 | 1368 function A = invhess (x, y) |
| 1369 ## INVHESS Inverse of an upper Hessenberg matrix. | |
|
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|
1370 ## INVHESS(X, Y), where X is an N-vector and Y an N-1 vector, |
|
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|
1371 ## is the matrix whose lower triangle agrees with that of |
|
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|
1372 ## ONES(N,1)*X' and whose strict upper triangle agrees with |
|
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|
1373 ## that of [1 Y]*ONES(1,N). |
|
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|
1374 ## The matrix is nonsingular if X(1) ~= 0 and X(i+1) ~= Y(i) |
|
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|
1375 ## for all i, and its inverse is an upper Hessenberg matrix. |
|
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|
1376 ## If Y is omitted it defaults to -X(1:N-1). |
|
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|
1377 ## Special case: if X is a scalar INVHESS(X) is the same as |
|
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|
1378 ## INVHESS(1:X). |
| 16634 | 1379 ## |
|
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|
1380 ## References: |
|
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|
1381 ## F.N. Valvi and V.S. Geroyannis, Analytic inverses and |
|
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|
1382 ## determinants for a class of matrices, IMA Journal of Numerical |
|
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|
1383 ## Analysis, 7 (1987), pp. 123-128. |
|
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|
1384 ## W.-L. Cao and W.J. Stewart, A note on inverses of Hessenberg-like |
|
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|
1385 ## matrices, Linear Algebra and Appl., 76 (1986), pp. 233-240. |
|
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|
1386 ## Y. Ikebe, On inverses of Hessenberg matrices, Linear Algebra and |
|
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|
1387 ## Appl., 24 (1979), pp. 93-97. |
|
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|
1388 ## P. Rozsa, On the inverse of band matrices, Integral Equations and |
|
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|
1389 ## Operator Theory, 10 (1987), pp. 82-95. |
| 16634 | 1390 |
| 1391 if (nargin < 1 || nargin > 2) | |
| 1392 error ("gallery: 1 to 2 arguments are required for invhess matrix."); | |
| 1393 elseif (! isnumeric (x)) | |
| 1394 error ("gallery: X must be numeric for invhess matrix."); | |
| 1395 endif | |
| 1396 | |
| 1397 if (isscalar (x) && fix (x) == x) | |
| 1398 n = x; | |
| 1399 x = 1:n; | |
| 1400 elseif (! isscalar (x) && isvector (x)) | |
| 1401 n = numel (n); | |
| 1402 else | |
| 1403 error ("gallery: X must be an integer scalar, or a vector for invhess matrix."); | |
| 1404 endif | |
| 1405 | |
| 1406 if (nargin < 2) | |
| 1407 y = -x(1:end-1); | |
| 1408 elseif (! isvector (y) || numel (y) != numel (x) -1) | |
| 1409 error ("gallery: Y must be a vector of length -1 than X for invhess matrix."); | |
| 1410 endif | |
| 1411 | |
| 1412 x = x(:); | |
| 1413 y = y(:); | |
| 1414 | |
|
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|
1415 ## FIXME: On next line, z = x'; A = z(ones(n,1),:) would be more efficient. |
| 16634 | 1416 A = ones (n, 1) * x'; |
| 1417 for j = 2:n | |
| 1418 A(1:j-1,j) = y(1:j-1); | |
| 1419 endfor | |
| 1420 endfunction | |
| 1421 | |
| 1422 function A = invol (n) | |
| 1423 ## INVOL An involutory matrix. | |
|
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|
1424 ## A = INVOL(N) is an N-by-N involutory (A*A = EYE(N)) and |
|
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|
1425 ## ill-conditioned matrix. |
|
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|
1426 ## It is a diagonally scaled version of HILB(N). |
|
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|
1427 ## NB: B = (EYE(N)-A)/2 and B = (EYE(N)+A)/2 are idempotent (B*B = B). |
| 16634 | 1428 ## |
|
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|
1429 ## Reference: |
|
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|
1430 ## A.S. Householder and J.A. Carpenter, The singular values |
|
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|
1431 ## of involutory and of idempotent matrices, Numer. Math. 5 (1963), |
|
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|
1432 ## pp. 234-237. |
| 16634 | 1433 |
| 1434 if (nargin != 1) | |
| 1435 error ("gallery: 1 argument is required for invol matrix."); | |
| 1436 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1437 error ("gallery: N must be an integer for invol matrix."); | |
| 1438 endif | |
| 1439 | |
| 1440 A = hilb (n); | |
| 1441 | |
| 1442 d = -n; | |
| 1443 A(:, 1) = d * A(:, 1); | |
| 1444 | |
| 1445 for i = 1:n-1 | |
| 1446 d = -(n+i)*(n-i)*d/(i*i); | |
| 1447 A(i+1,:) = d * A(i+1,:); | |
| 1448 endfor | |
| 1449 endfunction | |
| 1450 | |
| 1451 function [A, detA] = ipjfact (n, k = 0) | |
| 1452 ## IPJFACT A Hankel matrix with factorial elements. | |
|
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|
1453 ## A = IPJFACT(N, K) is the matrix with |
|
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|
1454 ## A(i,j) = (i+j)! (K = 0, default) |
|
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|
1455 ## A(i,j) = 1/(i+j)! (K = 1) |
|
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|
1456 ## Both are Hankel matrices. |
|
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|
1457 ## The determinant and inverse are known explicitly. |
|
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|
1458 ## If a second output argument is present, d = DET(A) is returned: |
|
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diff
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|
1459 ## [A, d] = IPJFACT(N, K); |
| 16634 | 1460 ## |
|
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|
1461 ## Suggested by P. R. Graves-Morris. |
|
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|
1462 ## |
|
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|
1463 ## Reference: |
|
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|
1464 ## M.J.C. Gover, The explicit inverse of factorial Hankel matrices, |
|
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|
1465 ## Dept. of Mathematics, University of Bradford, 1993. |
| 16634 | 1466 |
| 1467 if (nargin < 1 || nargin > 2) | |
| 1468 error ("gallery: 1 to 2 arguments are required for ipjfact matrix."); | |
| 1469 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1470 error ("gallery: N must be an integer for ipjfact matrix."); | |
| 1471 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 1472 error ("gallery: K must be a numeric scalar for ipjfact matrix."); | |
| 1473 endif | |
| 1474 | |
| 1475 c = cumprod (2:n+1); | |
| 1476 d = cumprod (n+1:2*n) * c(n-1); | |
| 1477 | |
| 1478 A = hankel (c, d); | |
| 1479 | |
| 1480 switch (k) | |
| 1481 case (0), # do nothing | |
| 1482 case (1), A = ones (n) ./ A; | |
| 1483 otherwise | |
| 1484 error ("gallery: K must have a value of 0 or 1 for ipjfact matrix."); | |
| 1485 endswitch | |
| 1486 | |
| 1487 if (nargout == 2) | |
| 1488 d = 1; | |
| 1489 | |
| 1490 if (k == 0) | |
| 1491 for i = 1:n-1 | |
|
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|
1492 d *= prod (1:i+1) * prod (1:n-i); |
| 16634 | 1493 endfor |
|
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|
1494 d *= prod (1:n+1); |
| 16634 | 1495 |
| 1496 elseif (k == 1) | |
| 1497 for i = 0:n-1 | |
|
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|
1498 d *= prod (1:i) / prod (1:n+1+i); |
| 16634 | 1499 endfor |
| 1500 if (rem (n*(n-1)/2, 2)) | |
| 1501 d = -d; | |
| 1502 endif | |
| 1503 | |
| 1504 else | |
| 1505 error ("gallery: K must have a value of 0 or 1 for ipjfact matrix."); | |
| 1506 endif | |
| 1507 | |
| 1508 detA = d; | |
| 1509 endif | |
| 1510 endfunction | |
| 1511 | |
| 1512 function J = jordbloc (n, lambda = 1) | |
| 1513 ## JORDBLOC Jordan block. | |
|
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|
1514 ## JORDBLOC(N, LAMBDA) is the N-by-N Jordan block with eigenvalue |
|
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|
1515 ## LAMBDA. LAMBDA = 1 is the default. |
| 16634 | 1516 |
| 1517 if (nargin < 1 || nargin > 2) | |
| 1518 error ("gallery: 1 to 2 arguments are required for jordbloc matrix."); | |
| 1519 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1520 error ("gallery: N must be an integer for jordbloc matrix."); | |
| 1521 elseif (! isnumeric (lambda) || ! isscalar (lambda)) | |
| 1522 error ("gallery: LAMBDA must be a numeric scalar for jordbloc matrix."); | |
| 1523 endif | |
| 1524 | |
| 1525 J = lambda * eye (n) + diag (ones (n-1, 1), 1); | |
| 1526 endfunction | |
| 1527 | |
| 1528 function U = kahan (n, theta = 1.2, pert = 25) | |
| 1529 ## KAHAN Kahan matrix - upper trapezoidal. | |
|
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|
1530 ## KAHAN(N, THETA) is an upper trapezoidal matrix |
|
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|
1531 ## that has some interesting properties regarding estimation of |
|
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|
1532 ## condition and rank. |
|
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|
1533 ## The matrix is N-by-N unless N is a 2-vector, in which case it |
|
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|
1534 ## is N(1)-by-N(2). |
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|
1535 ## The parameter THETA defaults to 1.2. |
|
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|
1536 ## The useful range of THETA is 0 < THETA < PI. |
| 16634 | 1537 ## |
|
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|
1538 ## To ensure that the QR factorization with column pivoting does not |
|
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|
1539 ## interchange columns in the presence of rounding errors, the diagonal |
|
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|
1540 ## is perturbed by PERT*EPS*diag( [N:-1:1] ). |
|
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|
1541 ## The default is PERT = 25, which ensures no interchanges for KAHAN(N) |
|
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|
1542 ## up to at least N = 90 in IEEE arithmetic. |
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|
1543 ## KAHAN(N, THETA, PERT) uses the given value of PERT. |
| 16634 | 1544 ## |
|
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|
1545 ## The inverse of KAHAN(N, THETA) is known explicitly: see |
|
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|
1546 ## Higham (1987, p. 588), for example. |
|
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|
1547 ## The diagonal perturbation was suggested by Christian Bischof. |
| 16634 | 1548 ## |
|
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|
1549 ## References: |
|
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|
1550 ## W. Kahan, Numerical linear algebra, Canadian Math. Bulletin, |
|
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|
1551 ## 9 (1966), pp. 757-801. |
|
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|
1552 ## N.J. Higham, A survey of condition number estimation for |
|
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|
1553 ## triangular matrices, SIAM Review, 29 (1987), pp. 575-596. |
| 16634 | 1554 |
| 1555 if (nargin < 1 || nargin > 3) | |
| 1556 error ("gallery: 1 to 3 arguments are required for kahan matrix."); | |
| 1557 elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n) | |
| 1558 error ("gallery: N must be a 1 or 2 element integer for kahan matrix."); | |
| 1559 elseif (! isnumeric (theta) || ! isscalar (theta)) | |
| 1560 error ("gallery: THETA must be a numeric scalar for kahan matrix."); | |
| 1561 elseif (! isnumeric (pert) || ! isscalar (pert)) | |
| 1562 error ("gallery: PERT must be a numeric scalar for kahan matrix."); | |
| 1563 endif | |
| 1564 | |
| 1565 ## Parameter n specifies dimension: r-by-n | |
| 1566 r = n(1); | |
| 1567 n = n(end); | |
| 1568 | |
| 1569 s = sin (theta); | |
| 1570 c = cos (theta); | |
| 1571 | |
| 1572 U = eye (n) - c * triu (ones (n), 1); | |
| 1573 U = diag (s.^[0:n-1]) * U + pert*eps* diag ([n:-1:1]); | |
| 1574 if (r > n) | |
| 1575 U(r,n) = 0; # Extend to an r-by-n matrix | |
| 1576 else | |
| 1577 U = U(1:r,:); # Reduce to an r-by-n matrix | |
| 1578 endif | |
| 1579 endfunction | |
| 1580 | |
| 1581 function A = kms (n, rho = 0.5) | |
| 1582 ## KMS Kac-Murdock-Szego Toeplitz matrix. | |
|
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|
1583 ## A = KMS(N, RHO) is the N-by-N Kac-Murdock-Szego Toeplitz matrix with |
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|
1584 ## A(i,j) = RHO^(ABS((i-j))) (for real RHO). |
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|
1585 ## If RHO is complex, then the same formula holds except that elements |
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|
1586 ## below the diagonal are conjugated. |
|
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|
1587 ## RHO defaults to 0.5. |
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|
1588 ## Properties: |
|
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|
1589 ## A has an LDL' factorization with |
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|
1590 ## L = INV(TRIW(N,-RHO,1)'), |
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|
1591 ## D(i,i) = (1-ABS(RHO)^2)*EYE(N) except D(1,1) = 1. |
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|
1592 ## A is positive definite if and only if 0 < ABS(RHO) < 1. |
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|
1593 ## INV(A) is tridiagonal. |
| 16634 | 1594 ## |
|
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|
1595 ## Reference: |
|
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|
1596 ## W.F. Trench, Numerical solution of the eigenvalue problem |
|
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|
1597 ## for Hermitian Toeplitz matrices, SIAM J. Matrix Analysis and Appl., |
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|
1598 ## 10 (1989), pp. 135-146 (and see the references therein). |
| 16634 | 1599 |
| 1600 if (nargin < 1 || nargin > 2) | |
| 1601 error ("gallery: 1 to 2 arguments are required for lauchli matrix."); | |
| 1602 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
|
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|
1603 error ("gallery: N must be an integer for lauchli matrix.") |
|
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|
1604 elseif (! isscalar (rho)) |
|
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|
1605 error ("gallery: RHO must be a scalar for lauchli matrix.") |
| 16634 | 1606 endif |
| 1607 | |
|
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|
1608 A = (1:n)'*ones (1,n); |
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|
1609 A = abs (A - A'); |
| 16634 | 1610 A = rho .^ A; |
|
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|
1611 if (imag (rho)) |
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|
1612 A = conj (tril (A,-1)) + triu (A); |
| 16634 | 1613 endif |
| 1614 endfunction | |
| 1615 | |
| 1616 function B = krylov (A, x, j) | |
| 1617 ## KRYLOV Krylov matrix. | |
|
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|
1618 ## KRYLOV(A, x, j) is the Krylov matrix |
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|
1619 ## [x, Ax, A^2x, ..., A^(j-1)x], |
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|
1620 ## where A is an n-by-n matrix and x is an n-vector. |
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|
1621 ## Defaults: x = ONES(n,1), j = n. |
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|
1622 ## KRYLOV(n) is the same as KRYLOV(RANDN(n)). |
| 16634 | 1623 ## |
|
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|
1624 ## Reference: |
|
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|
1625 ## G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, |
|
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|
1626 ## Johns Hopkins University Press, Baltimore, Maryland, 1989, p. 369. |
| 16634 | 1627 |
| 1628 if (nargin < 1 || nargin > 3) | |
| 1629 error ("gallery: 1 to 3 arguments are required for krylov matrix."); | |
| 1630 elseif (! isnumeric (A) || ! issquare (A) || ndims (A) != 2) | |
| 17199 | 1631 error ("gallery: A must be a square 2-D matrix for krylov matrix."); |
| 16634 | 1632 endif |
| 1633 | |
| 1634 n = length (A); | |
| 1635 if (isscalar (A)) | |
| 1636 n = A; | |
| 1637 A = randn (n); | |
| 1638 endif | |
| 1639 | |
| 1640 if (nargin < 2) | |
| 1641 x = ones (n, 1); | |
| 1642 elseif (! isvector (x) || numel (x) != n) | |
| 1643 error ("gallery: X must be a vector of length equal to A for krylov matrix."); | |
| 1644 endif | |
| 1645 | |
| 1646 if (nargin < 3) | |
| 1647 j = n; | |
| 1648 elseif (! isnumeric (j) || ! isscalar (j) || fix (j) != j) | |
| 1649 error ("gallery: J must be an integer for krylov matrix."); | |
| 1650 endif | |
| 1651 | |
| 1652 B = ones (n, j); | |
| 1653 B(:,1) = x(:); | |
| 1654 for i = 2:j | |
| 1655 B(:,i) = A*B(:,i-1); | |
| 1656 endfor | |
| 1657 endfunction | |
| 1658 | |
| 1659 function A = lauchli (n, mu = sqrt (eps)) | |
| 1660 ## LAUCHLI Lauchli matrix - rectangular. | |
|
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|
1661 ## LAUCHLI(N, MU) is the (N+1)-by-N matrix [ONES(1,N); MU*EYE(N))]. |
|
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|
1662 ## It is a well-known example in least squares and other problems |
|
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|
1663 ## that indicates the dangers of forming A'*A. |
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|
1664 ## MU defaults to SQRT(EPS). |
| 16634 | 1665 ## |
|
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|
1666 ## Reference: |
|
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|
1667 ## P. Lauchli, Jordan-Elimination und Ausgleichung nach |
|
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|
1668 ## kleinsten Quadraten, Numer. Math, 3 (1961), pp. 226-240. |
| 16634 | 1669 |
| 1670 if (nargin < 1 || nargin > 2) | |
| 1671 error ("gallery: 1 to 2 arguments are required for lauchli matrix."); | |
| 1672 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1673 error ("gallery: N must be an integer for lauchli matrix."); | |
| 1674 elseif (! isscalar (mu)) | |
| 1675 error ("gallery: MU must be a scalar for lauchli matrix."); | |
| 1676 endif | |
| 1677 | |
| 1678 A = [ones(1, n) | |
| 1679 mu*eye(n) ]; | |
| 1680 endfunction | |
| 1681 | |
| 1682 function A = lehmer (n) | |
| 1683 ## LEHMER Lehmer matrix - symmetric positive definite. | |
|
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|
1684 ## A = LEHMER(N) is the symmetric positive definite N-by-N matrix with |
|
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|
1685 ## A(i,j) = i/j for j >= i. |
|
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|
1686 ## A is totally nonnegative. INV(A) is tridiagonal, and explicit |
|
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|
1687 ## formulas are known for its entries. |
|
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|
1688 ## N <= COND(A) <= 4*N*N. |
| 16634 | 1689 ## |
|
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|
1690 ## References: |
|
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|
1691 ## M. Newman and J. Todd, The evaluation of matrix inversion |
|
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|
1692 ## programs, J. Soc. Indust. Appl. Math., 6 (1958), pp. 466-476. |
|
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|
1693 ## Solutions to problem E710 (proposed by D.H. Lehmer): The inverse |
|
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|
1694 ## of a matrix, Amer. Math. Monthly, 53 (1946), pp. 534-535. |
|
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|
1695 ## J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, |
|
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|
1696 ## Birkhauser, Basel, and Academic Press, New York, 1977, p. 154. |
| 16634 | 1697 |
| 1698 if (nargin != 1) | |
| 1699 error ("gallery: 1 argument is required for lehmer matrix."); | |
| 1700 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1701 error ("gallery: N must be an integer for lehmer matrix."); | |
| 1702 endif | |
| 1703 | |
| 1704 A = ones (n, 1) * (1:n); | |
| 1705 A = A./A'; | |
| 1706 A = tril (A) + tril (A, -1)'; | |
| 1707 endfunction | |
| 1708 | |
| 1709 function T = lesp (n) | |
| 1710 ## LESP A tridiagonal matrix with real, sensitive eigenvalues. | |
|
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|
1711 ## LESP(N) is an N-by-N matrix whose eigenvalues are real and smoothly |
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|
1712 ## distributed in the interval approximately [-2*N-3.5, -4.5]. |
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|
1713 ## The sensitivities of the eigenvalues increase exponentially as |
|
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|
1714 ## the eigenvalues grow more negative. |
|
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|
1715 ## The matrix is similar to the symmetric tridiagonal matrix with |
|
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|
1716 ## the same diagonal entries and with off-diagonal entries 1, |
|
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|
1717 ## via a similarity transformation with D = diag(1!,2!,...,N!). |
| 16634 | 1718 ## |
|
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|
1719 ## References: |
|
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|
1720 ## H.W.J. Lenferink and M.N. Spijker, On the use of stability regions in |
|
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|
1721 ## the numerical analysis of initial value problems, |
|
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|
1722 ## Math. Comp., 57 (1991), pp. 221-237. |
|
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|
1723 ## L.N. Trefethen, Pseudospectra of matrices, in Numerical Analysis 1991, |
|
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|
1724 ## Proceedings of the 14th Dundee Conference, |
|
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|
1725 ## D.F. Griffiths and G.A. Watson, eds, Pitman Research Notes in |
|
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|
1726 ## Mathematics, volume 260, Longman Scientific and Technical, Essex, |
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|
1727 ## UK, 1992, pp. 234-266. |
| 16634 | 1728 |
| 1729 if (nargin != 1) | |
| 1730 error ("gallery: 1 argument is required for lesp matrix."); | |
| 1731 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1732 error ("gallery: N must be an integer for lesp matrix."); | |
| 1733 endif | |
| 1734 | |
| 1735 x = 2:n; | |
| 1736 T = full (tridiag (ones (size (x)) ./x, -(2*[x n+1]+1), x)); | |
| 1737 endfunction | |
| 1738 | |
| 1739 function A = lotkin (n) | |
| 1740 ## LOTKIN Lotkin matrix. | |
|
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|
1741 ## A = LOTKIN(N) is the Hilbert matrix with its first row altered to |
|
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|
1742 ## all ones. A is unsymmetric, ill-conditioned, and has many negative |
|
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|
1743 ## eigenvalues of small magnitude. |
|
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|
1744 ## The inverse has integer entries and is known explicitly. |
| 16634 | 1745 ## |
|
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|
1746 ## Reference: |
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|
1747 ## M. Lotkin, A set of test matrices, MTAC, 9 (1955), pp. 153-161. |
| 16634 | 1748 |
| 1749 if (nargin != 1) | |
| 1750 error ("gallery: 1 argument is required for lotkin matrix."); | |
| 1751 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1752 error ("gallery: N must be an integer for lotkin matrix."); | |
| 1753 endif | |
| 1754 | |
| 1755 A = hilb (n); | |
| 1756 A(1,:) = ones (1, n); | |
| 1757 endfunction | |
| 1758 | |
| 1759 function A = minij (n) | |
| 1760 ## MINIJ Symmetric positive definite matrix MIN(i,j). | |
|
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|
1761 ## A = MINIJ(N) is the N-by-N symmetric positive definite matrix with |
|
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|
1762 ## A(i,j) = MIN(i,j). |
|
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|
1763 ## Properties, variations: |
|
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|
1764 ## INV(A) is tridiagonal: it is minus the second difference matrix |
|
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|
1765 ## except its (N,N) element is 1. |
|
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|
1766 ## 2*A-ONES(N) (Givens' matrix) has tridiagonal inverse and |
|
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|
1767 ## eigenvalues .5*sec^2([2r-1)PI/4N], r=1:N. |
|
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|
1768 ## (N+1)*ONES(N)-A also has a tridiagonal inverse. |
| 16634 | 1769 ## |
|
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|
1770 ## References: |
|
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|
1771 ## J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, |
|
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|
1772 ## Birkhauser, Basel, and Academic Press, New York, 1977, p. 158. |
|
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|
1773 ## D.E. Rutherford, Some continuant determinants arising in physics and |
|
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|
1774 ## chemistry---II, Proc. Royal Soc. Edin., 63, A (1952), pp. 232-241. |
|
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|
1775 ## (For the eigenvalues of Givens' matrix.) |
| 16634 | 1776 |
| 1777 if (nargin != 1) | |
| 1778 error ("gallery: 1 argument is required for minij matrix."); | |
| 1779 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1780 error ("gallery: N must be an integer for minij matrix."); | |
| 1781 endif | |
| 1782 | |
|
19944
941e782d0429
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|
1783 A = bsxfun (@min, 1:n, (1:n)'); |
| 16634 | 1784 endfunction |
| 1785 | |
| 1786 function A = moler (n, alpha = -1) | |
| 1787 ## MOLER Moler matrix - symmetric positive definite. | |
|
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|
1788 ## A = MOLER(N, ALPHA) is the symmetric positive definite N-by-N matrix |
|
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|
1789 ## U'*U where U = TRIW(N, ALPHA). |
|
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|
1790 ## For ALPHA = -1 (the default) A(i,j) = MIN(i,j)-2, A(i,i) = i. |
|
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|
1791 ## A has one small eigenvalue. |
| 16634 | 1792 ## |
|
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|
1793 ## Nash (1990) attributes the ALPHA = -1 matrix to Moler. |
| 16634 | 1794 ## |
|
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|
1795 ## Reference: |
|
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|
1796 ## J.C. Nash, Compact Numerical Methods for Computers: Linear |
|
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|
1797 ## Algebra and Function Minimisation, second edition, Adam Hilger, |
|
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|
1798 ## Bristol, 1990 (Appendix 1). |
| 16634 | 1799 |
| 1800 if (nargin < 1 || nargin > 2) | |
| 1801 error ("gallery: 1 to 2 arguments are required for moler matrix."); | |
| 1802 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1803 error ("gallery: N must be an integer for moler matrix."); | |
| 1804 elseif (! isscalar (alpha)) | |
| 1805 error ("gallery: ALPHA must be a scalar for moler matrix."); | |
| 1806 endif | |
| 1807 | |
| 1808 A = triw (n, alpha)' * triw (n, alpha); | |
| 1809 endfunction | |
| 1810 | |
| 1811 function [A, T] = neumann (n) | |
| 1812 ## NEUMANN Singular matrix from the discrete Neumann problem (sparse). | |
|
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|
1813 ## NEUMANN(N) is the singular, row diagonally dominant matrix resulting |
|
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|
1814 ## from discretizing the Neumann problem with the usual five point |
|
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|
1815 ## operator on a regular mesh. |
|
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|
1816 ## It has a one-dimensional null space with null vector ONES(N,1). |
|
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|
1817 ## The dimension N should be a perfect square, or else a 2-vector, |
|
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|
1818 ## in which case the dimension of the matrix is N(1)*N(2). |
| 16634 | 1819 ## |
|
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|
1820 ## Reference: |
|
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|
1821 ## R.J. Plemmons, Regular splittings and the discrete Neumann |
|
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|
1822 ## problem, Numer. Math., 25 (1976), pp. 153-161. |
| 16634 | 1823 |
| 1824 if (nargin != 1) | |
| 1825 error ("gallery: 1 argument is required for neumann matrix."); | |
| 1826 elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n) | |
| 1827 error ("gallery: N must be a 1 or 2 element integer for neumann matrix."); | |
| 1828 endif | |
| 1829 | |
| 1830 if (isscalar (n)) | |
| 1831 m = sqrt (n); | |
| 1832 if (m^2 != n) | |
| 1833 error ("gallery: N must be a perfect square for neumann matrix."); | |
| 1834 endif | |
| 1835 n(1) = m; | |
| 1836 n(2) = m; | |
| 1837 endif | |
| 1838 | |
| 1839 T = tridiag (n(1), -1, 2, -1); | |
| 1840 T(1,2) = -2; | |
| 1841 T(n(1),n(1)-1) = -2; | |
| 1842 | |
| 1843 A = kron (T, eye (n(2))) + kron (eye (n(2)), T); | |
| 1844 endfunction | |
| 1845 | |
|
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|
1846 function A = normaldata (varargin) |
|
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|
1847 |
|
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|
1848 if (nargin < 2) |
|
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|
1849 error ("gallery: At least 2 arguments required for normaldata matrix."); |
|
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|
1850 endif |
|
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|
1851 if (isnumeric (varargin{end})) |
|
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|
1852 jidx = varargin{end}; |
|
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|
1853 svec = [varargin{:}]; |
|
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|
1854 varargin(end) = []; |
|
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|
1855 elseif (ischar (varargin{end})) |
|
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|
1856 if (nargin < 3) |
|
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|
1857 error (["gallery: CLASS argument requires 3 inputs " ... |
|
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|
1858 "for normaldata matrix."]); |
|
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|
1859 endif |
|
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|
1860 jidx = varargin{end-1}; |
|
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|
1861 svec = [varargin{1:end-1}]; |
|
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|
1862 varargin(end-1) = []; |
|
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|
1863 else |
|
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|
1864 error (["gallery: J must be an integer in the range [0, 2^32-1] " ... |
|
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|
1865 "for normaldata matrix"]); |
|
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|
1866 endif |
|
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|
1867 |
|
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|
1868 if (! (isnumeric (jidx) && isscalar (jidx) |
|
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|
1869 && jidx == fix (jidx) |
|
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|
1870 && jidx >= 0 && jidx <= 0xFFFFFFFF)) |
|
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|
1871 error (["gallery: J must be an integer in the range [0, 2^32-1] " ... |
|
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|
1872 "for normaldata matrix"]); |
|
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|
1873 endif |
|
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|
1874 |
|
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|
1875 ## Save and restore random state. Initialization done so that reproducible |
|
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|
1876 ## data is available from gallery depending on the jidx and size vector. |
|
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17386
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|
1877 randstate = randn ("state"); |
|
16979
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diff
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|
1878 unwind_protect |
|
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|
1879 randn ("state", svec); |
|
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|
1880 A = randn (varargin{:}); |
|
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|
1881 unwind_protect_cleanup |
|
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|
1882 randn ("state", randstate); |
|
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|
1883 end_unwind_protect |
|
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diff
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|
1884 |
|
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|
1885 endfunction |
|
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|
1886 |
| 16634 | 1887 function Q = orthog (n, k = 1) |
| 1888 ## ORTHOG Orthogonal and nearly orthogonal matrices. | |
|
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|
1889 ## Q = ORTHOG(N, K) selects the K'th type of matrix of order N. |
|
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|
1890 ## K > 0 for exactly orthogonal matrices, K < 0 for diagonal scalings of |
|
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|
1891 ## orthogonal matrices. |
|
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|
1892 ## Available types: (K = 1 is the default) |
|
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|
1893 ## K = 1: Q(i,j) = SQRT(2/(n+1)) * SIN( i*j*PI/(n+1) ) |
|
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|
1894 ## Symmetric eigenvector matrix for second difference matrix. |
|
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|
1895 ## K = 2: Q(i,j) = 2/SQRT(2*n+1)) * SIN( 2*i*j*PI/(2*n+1) ) |
|
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|
1896 ## Symmetric. |
|
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|
1897 ## K = 3: Q(r,s) = EXP(2*PI*i*(r-1)*(s-1)/n) / SQRT(n) (i=SQRT(-1)) |
|
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|
1898 ## Unitary, the Fourier matrix. Q^4 is the identity. |
|
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|
1899 ## This is essentially the same matrix as FFT(EYE(N))/SQRT(N)! |
|
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|
1900 ## K = 4: Helmert matrix: a permutation of a lower Hessenberg matrix, |
|
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|
1901 ## whose first row is ONES(1:N)/SQRT(N). |
|
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|
1902 ## 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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|
1903 ## Symmetric matrix arising in the Hartley transform. |
|
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|
1904 ## K = -1: Q(i,j) = COS( (i-1)*(j-1)*PI/(n-1) ) |
|
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|
1905 ## Chebyshev Vandermonde-like matrix, based on extrema of T(n-1). |
|
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|
1906 ## K = -2: Q(i,j) = COS( (i-1)*(j-1/2)*PI/n) ) |
|
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|
1907 ## Chebyshev Vandermonde-like matrix, based on zeros of T(n). |
| 16634 | 1908 ## |
|
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|
1909 ## References: |
|
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|
1910 ## N.J. Higham and D.J. Higham, Large growth factors in Gaussian |
|
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|
1911 ## elimination with pivoting, SIAM J. Matrix Analysis and Appl., |
|
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|
1912 ## 10 (1989), pp. 155-164. |
|
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|
1913 ## P. Morton, On the eigenvectors of Schur's matrix, J. Number Theory, |
|
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1914 ## 12 (1980), pp. 122-127. (Re. ORTHOG(N, 3)) |
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1915 ## H.O. Lancaster, The Helmert Matrices, Amer. Math. Monthly, 72 (1965), |
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1916 ## pp. 4-12. |
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1917 ## D. Bini and P. Favati, On a matrix algebra related to the discrete |
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1918 ## Hartley transform, SIAM J. Matrix Anal. Appl., 14 (1993), |
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1919 ## pp. 500-507. |
| 16634 | 1920 |
| 1921 if (nargin < 1 || nargin > 2) | |
| 1922 error ("gallery: 1 to 2 arguments are required for orthog matrix."); | |
| 1923 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1924 error ("gallery: N must be an integer for orthog matrix."); | |
| 1925 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 1926 error ("gallery: K must be a numeric scalar for orthog matrix."); | |
| 1927 endif | |
| 1928 | |
| 1929 switch (k) | |
| 1930 case (1) | |
| 1931 ## E'vectors second difference matrix | |
| 1932 m = (1:n)'*(1:n) * (pi/(n+1)); | |
| 1933 Q = sin (m) * sqrt (2/(n+1)); | |
| 1934 | |
| 1935 case (2) | |
| 1936 m = (1:n)'*(1:n) * (2*pi/(2*n+1)); | |
| 1937 Q = sin (m) * (2/ sqrt (2*n+1)); | |
| 1938 | |
| 1939 case (3) | |
| 1940 ## Vandermonde based on roots of unity | |
| 1941 m = 0:n-1; | |
| 1942 Q = exp (m'*m*2*pi* sqrt (-1) / n) / sqrt (n); | |
| 1943 | |
| 1944 case (4) | |
| 1945 ## Helmert matrix | |
| 1946 Q = tril (ones (n)); | |
| 1947 Q(1,2:n) = ones (1, n-1); | |
| 1948 for i = 2:n | |
| 1949 Q(i,i) = -(i-1); | |
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1950 endfor |
| 16634 | 1951 Q = diag (sqrt ([n 1:n-1] .* [1:n])) \ Q; |
| 1952 | |
| 1953 case (5) | |
| 1954 ## Hartley matrix | |
| 1955 m = (0:n-1)'*(0:n-1) * (2*pi/n); | |
| 1956 Q = (cos (m) + sin (m)) / sqrt (n); | |
| 1957 | |
| 1958 case (-1) | |
| 1959 ## extrema of T(n-1) | |
| 1960 m = (0:n-1)'*(0:n-1) * (pi/(n-1)); | |
| 1961 Q = cos (m); | |
| 1962 | |
| 1963 case (-2) | |
| 1964 ## zeros of T(n) | |
| 1965 m = (0:n-1)'*(.5:n-.5) * (pi/n); | |
| 1966 Q = cos (m); | |
| 1967 | |
| 1968 otherwise | |
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1969 error ("gallery: unknown K '%d' for orthog matrix.", k); |
| 16634 | 1970 endswitch |
| 1971 endfunction | |
| 1972 | |
| 1973 function A = parter (n) | |
| 1974 ## PARTER Parter matrix - a Toeplitz matrix with singular values near PI. | |
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1975 ## PARTER(N) is the matrix with (i,j) element 1/(i-j+0.5). |
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1976 ## It is a Cauchy matrix and a Toeplitz matrix. |
| 16634 | 1977 ## |
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1978 ## At the Second SIAM Conference on Linear Algebra, Raleigh, N.C., |
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1979 ## 1985, Cleve Moler noted that most of the singular values of |
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1980 ## PARTER(N) are very close to PI. An explanation of the phenomenon |
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1981 ## was given by Parter; see also the paper by Tyrtyshnikov. |
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1982 ## |
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1983 ## References: |
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1984 ## The MathWorks Newsletter, Volume 1, Issue 1, March 1986, page 2. |
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1985 ## S.V. Parter, On the distribution of the singular values of Toeplitz |
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1986 ## matrices, Linear Algebra and Appl., 80 (1986), pp. 115-130. |
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1987 ## E.E. Tyrtyshnikov, Cauchy-Toeplitz matrices and some applications, |
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1988 ## Linear Algebra and Appl., 149 (1991), pp. 1-18. |
| 16634 | 1989 |
| 1990 if (nargin != 1) | |
| 1991 error ("gallery: 1 argument is required for parter matrix."); | |
| 1992 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 1993 error ("gallery: N must be an integer for parter matrix."); | |
| 1994 endif | |
| 1995 | |
| 1996 A = cauchy ((1:n) + 0.5, -(1:n)); | |
| 1997 endfunction | |
| 1998 | |
| 1999 function P = pei (n, alpha = 1) | |
| 2000 ## PEI Pei matrix. | |
|
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2001 ## PEI(N, ALPHA), where ALPHA is a scalar, is the symmetric matrix |
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2002 ## ALPHA*EYE(N) + ONES(N). |
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2003 ## If ALPHA is omitted then ALPHA = 1 is used. |
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2004 ## The matrix is singular for ALPHA = 0, -N. |
| 16634 | 2005 ## |
|
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2006 ## Reference: |
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2007 ## M.L. Pei, A test matrix for inversion procedures, |
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2008 ## Comm. ACM, 5 (1962), p. 508. |
| 16634 | 2009 |
| 2010 if (nargin < 1 || nargin > 2) | |
| 2011 error ("gallery: 1 to 2 arguments are required for pei matrix."); | |
| 2012 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2013 error ("gallery: N must be an integer for pei matrix."); | |
|
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|
2014 elseif (! isnumeric (alpha) || ! isscalar (alpha)) |
| 16634 | 2015 error ("gallery: ALPHA must be a scalar for pei matrix."); |
| 2016 endif | |
| 2017 | |
| 2018 P = alpha * eye (n) + ones (n); | |
| 2019 endfunction | |
| 2020 | |
| 2021 function A = poisson (n) | |
| 2022 ## POISSON Block tridiagonal matrix from Poisson's equation (sparse). | |
|
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2023 ## POISSON(N) is the block tridiagonal matrix of order N^2 |
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2024 ## resulting from discretizing Poisson's equation with the |
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2025 ## 5-point operator on an N-by-N mesh. |
| 16634 | 2026 ## |
|
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2027 ## Reference: |
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2028 ## G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, |
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2029 ## Johns Hopkins University Press, Baltimore, Maryland, 1989 |
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2030 ## (Section 4.5.4). |
| 16634 | 2031 |
| 2032 if (nargin != 1) | |
| 2033 error ("gallery: 1 argument is required for poisson matrix."); | |
| 2034 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2035 error ("gallery: N must be an integer for poisson matrix."); | |
| 2036 endif | |
| 2037 | |
| 2038 S = tridiag (n, -1, 2, -1); | |
| 2039 I = speye (n); | |
| 2040 A = kron (I, S) + kron (S, I); | |
| 2041 endfunction | |
| 2042 | |
| 2043 function A = prolate (n, w = 0.25) | |
| 2044 ## PROLATE Prolate matrix - symmetric, ill-conditioned Toeplitz matrix. | |
|
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2045 ## A = PROLATE(N, W) is the N-by-N prolate matrix with parameter W. |
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2046 ## It is a symmetric Toeplitz matrix. |
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2047 ## If 0 < W < 0.5 then |
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2048 ## - A is positive definite |
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2049 ## - the eigenvalues of A are distinct, lie in (0, 1), and |
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2050 ## tend to cluster around 0 and 1. |
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2051 ## W defaults to 0.25. |
| 16634 | 2052 ## |
|
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2053 ## Reference: |
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2054 ## J.M. Varah. The Prolate matrix. Linear Algebra and Appl., |
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2055 ## 187:269--278, 1993. |
| 16634 | 2056 |
| 2057 if (nargin < 1 || nargin > 2) | |
| 2058 error ("gallery: 1 to 2 arguments are required for prolate matrix."); | |
| 2059 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2060 error ("gallery: N must be an integer for prolate matrix."); | |
| 2061 elseif (! isnumeric (w) || ! isscalar (w)) | |
| 2062 error ("gallery: W must be a scalar for prolate matrix."); | |
| 2063 endif | |
| 2064 | |
| 2065 a = zeros (n, 1); | |
| 2066 a(1) = 2*w; | |
| 2067 a(2:n) = sin (2*pi*w*(1:n-1)) ./ (pi*(1:n-1)); | |
| 2068 | |
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2069 A = toeplitz (a); |
| 16634 | 2070 endfunction |
| 2071 | |
| 2072 function H = randhess (x) | |
| 2073 ## NOTE: this function was named ohess in the original Test Matrix Toolbox | |
| 2074 ## RANDHESS Random, orthogonal upper Hessenberg matrix. | |
|
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2075 ## H = RANDHESS(N) is an N-by-N real, random, orthogonal |
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2076 ## upper Hessenberg matrix. |
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2077 ## Alternatively, H = RANDHESS(X), where X is an arbitrary real |
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2078 ## N-vector (N > 1) constructs H non-randomly using the elements |
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2079 ## of X as parameters. |
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2080 ## In both cases H is constructed via a product of N-1 Givens rotations. |
| 16634 | 2081 ## |
|
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2082 ## Note: See Gragg (1986) for how to represent an N-by-N (complex) |
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2083 ## unitary Hessenberg matrix with positive subdiagonal elements in terms |
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2084 ## of 2N-1 real parameters (the Schur parametrization). |
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2085 ## This M-file handles the real case only and is intended simply as a |
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2086 ## convenient way to generate random or non-random orthogonal Hessenberg |
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2087 ## matrices. |
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2088 ## |
|
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2089 ## Reference: |
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2090 ## W.B. Gragg, The QR algorithm for unitary Hessenberg matrices, |
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2091 ## J. Comp. Appl. Math., 16 (1986), pp. 1-8. |
| 16634 | 2092 |
| 2093 if (nargin != 1) | |
| 2094 error ("gallery: 1 argument is required for randhess matrix."); | |
| 2095 elseif (! isnumeric (x) || ! isreal (x)) | |
| 2096 error ("gallery: N or X must be numeric real values for randhess matrix."); | |
| 2097 endif | |
| 2098 | |
| 2099 if (isscalar (x)) | |
| 2100 n = x; | |
| 2101 x = rand (n-1, 1) * 2*pi; | |
| 2102 H = eye (n); | |
| 2103 H(n,n) = sign (randn); | |
| 2104 elseif (isvector (x)) | |
| 2105 n = numel (x); | |
| 2106 H = eye (n); | |
| 2107 H(n,n) = sign (x(n)) + (x(n) == 0); # Second term ensures H(n,n) nonzero. | |
| 2108 else | |
| 2109 error ("gallery: N or X must be a scalar or a vector for randhess matrix."); | |
| 2110 endif | |
| 2111 | |
| 2112 for i = n:-1:2 | |
| 2113 ## Apply Givens rotation through angle x(i-1). | |
| 2114 theta = x(i-1); | |
| 2115 c = cos (theta); | |
| 2116 s = sin (theta); | |
| 2117 H([i-1 i], :) = [ c*H(i-1,:)+s*H(i,:) | |
| 2118 -s*H(i-1,:)+c*H(i,:) ]; | |
| 2119 endfor | |
| 2120 endfunction | |
| 2121 | |
| 2122 function A = rando (n, k = 1) | |
| 2123 ## RANDO Random matrix with elements -1, 0 or 1. | |
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2124 ## A = RANDO(N, K) is a random N-by-N matrix with elements from |
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2125 ## one of the following discrete distributions (default K = 1): |
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2126 ## K = 1: A(i,j) = 0 or 1 with equal probability, |
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2127 ## K = 2: A(i,j) = -1 or 1 with equal probability, |
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2128 ## K = 3: A(i,j) = -1, 0 or 1 with equal probability. |
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2129 ## N may be a 2-vector, in which case the matrix is N(1)-by-N(2). |
| 16634 | 2130 |
| 2131 if (nargin < 1 || nargin > 2) | |
| 2132 error ("gallery: 1 to 2 arguments are required for rando matrix."); | |
| 2133 elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n) | |
| 2134 error ("gallery: N must be an integer for rando matrix."); | |
| 2135 elseif (! isnumeric (k) || ! isscalar (k)) | |
| 2136 error ("gallery: K must be a numeric scalar for smoke matrix."); | |
| 2137 endif | |
| 2138 | |
| 2139 ## Parameter n specifies dimension: m-by-n. | |
| 2140 m = n(1); | |
| 2141 n = n(end); | |
| 2142 | |
| 2143 switch (k) | |
| 2144 case (1), A = floor ( rand(m, n) + 0.5); # {0, 1} | |
| 2145 case (2), A = 2*floor ( rand(m, n) + 0.5) -1; # {-1, 1} | |
| 2146 case (3), A = round (3*rand(m, n) - 1.5); # {-1, 0, 1} | |
| 2147 otherwise | |
|
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2148 error ("gallery: unknown K '%d' for smoke matrix.", k); |
| 16634 | 2149 endswitch |
| 2150 | |
| 2151 endfunction | |
| 2152 | |
|
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2153 function A = randsvd (n, kappa = sqrt (1/eps), mode = 3, kl = max (n) -1, |
|
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|
2154 ku = kl) |
| 16634 | 2155 ## RANDSVD Random matrix with pre-assigned singular values. |
|
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|
2156 ## RANDSVD(N, KAPPA, MODE, KL, KU) is a (banded) random matrix of order N |
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|
2157 ## with COND(A) = KAPPA and singular values from the distribution MODE. |
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|
2158 ## N may be a 2-vector, in which case the matrix is N(1)-by-N(2). |
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2159 ## Available types: |
|
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|
2160 ## MODE = 1: one large singular value, |
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2161 ## MODE = 2: one small singular value, |
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2162 ## MODE = 3: geometrically distributed singular values, |
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|
2163 ## MODE = 4: arithmetically distributed singular values, |
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2164 ## MODE = 5: random singular values with unif. dist. logarithm. |
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|
2165 ## If omitted, MODE defaults to 3, and KAPPA defaults to SQRT(1/EPS). |
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|
2166 ## If MODE < 0 then the effect is as for ABS(MODE) except that in the |
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|
2167 ## original matrix of singular values the order of the diagonal entries |
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|
2168 ## is reversed: small to large instead of large to small. |
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|
2169 ## KL and KU are the lower and upper bandwidths respectively; if they |
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2170 ## are omitted a full matrix is produced. |
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|
2171 ## If only KL is present, KU defaults to KL. |
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|
2172 ## Special case: if KAPPA < 0 then a random full symmetric positive |
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|
2173 ## definite matrix is produced with COND(A) = -KAPPA and |
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|
2174 ## eigenvalues distributed according to MODE. |
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2175 ## KL and KU, if present, are ignored. |
| 16634 | 2176 ## |
|
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|
2177 ## Reference: |
|
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|
2178 ## N.J. Higham, Accuracy and Stability of Numerical Algorithms, |
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|
2179 ## Society for Industrial and Applied Mathematics, Philadelphia, PA, |
|
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|
2180 ## USA, 1996; sec. 26.3. |
| 16634 | 2181 ## |
|
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|
2182 ## This routine is similar to the more comprehensive Fortran routine xLATMS |
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|
2183 ## in the following reference: |
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|
2184 ## J.W. Demmel and A. McKenney, A test matrix generation suite, |
|
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|
2185 ## LAPACK Working Note #9, Courant Institute of Mathematical Sciences, |
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|
2186 ## New York, 1989. |
| 16634 | 2187 |
| 2188 if (nargin < 1 || nargin > 5) | |
| 2189 error ("gallery: 1 to 5 arguments are required for randsvd matrix."); | |
| 2190 elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n) | |
| 2191 error ("gallery: N must be a 1 or 2 element integer vector for randsvd matrix."); | |
| 2192 elseif (! isnumeric (kappa) || ! isscalar (kappa)) | |
| 2193 error ("gallery: KAPPA must be a numeric scalar for randsvd matrix."); | |
| 2194 elseif (abs (kappa) < 1) | |
| 2195 error ("gallery: KAPPA must larger than or equal to 1 for randsvd matrix."); | |
| 2196 elseif (! isnumeric (mode) || ! isscalar (mode)) | |
| 2197 error ("gallery: MODE must be a numeric scalar for randsvd matrix."); | |
| 2198 elseif (! isnumeric (kl) || ! isscalar (kl)) | |
| 2199 error ("gallery: KL must be a numeric scalar for randsvd matrix."); | |
| 2200 elseif (! isnumeric (ku) || ! isscalar (ku)) | |
| 2201 error ("gallery: KU must be a numeric scalar for randsvd matrix."); | |
| 2202 endif | |
| 2203 | |
| 2204 posdef = 0; | |
| 2205 if (kappa < 0) | |
| 2206 posdef = 1; | |
| 2207 kappa = -kappa; | |
| 2208 endif | |
| 2209 | |
| 2210 ## Parameter n specifies dimension: m-by-n. | |
| 2211 m = n(1); | |
| 2212 n = n(end); | |
| 2213 p = min ([m n]); | |
| 2214 | |
| 2215 ## If A will be a vector | |
| 2216 if (p == 1) | |
| 2217 A = randn (m, n); | |
|
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|
2218 A /= norm (A); |
|
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|
2219 return; |
|
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|
2220 endif |
| 16634 | 2221 |
| 2222 ## Set up vector sigma of singular values. | |
| 2223 switch (abs (mode)) | |
| 2224 case (1) | |
| 2225 sigma = ones (p, 1) ./ kappa; | |
| 2226 sigma(1) = 1; | |
| 2227 case (2) | |
| 2228 sigma = ones (p, 1); | |
| 2229 sigma(p) = 1 / kappa; | |
| 2230 case (3) | |
| 2231 factor = kappa^(-1/(p-1)); | |
| 2232 sigma = factor.^[0:p-1]; | |
| 2233 case (4) | |
| 2234 sigma = ones (p, 1) - (0:p-1)'/(p-1)*(1-1/kappa); | |
| 2235 case (5) | |
| 2236 ## In this case cond (A) <= kappa. | |
| 2237 rand ("uniform"); | |
| 2238 sigma = exp (-rand (p, 1) * log (kappa)); | |
| 2239 otherwise | |
|
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John W. Eaton <jwe@octave.org>
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|
2240 error ("gallery: unknown MODE '%d' for randsvd matrix.", mode); |
| 16634 | 2241 endswitch |
| 2242 | |
| 2243 ## Convert to diagonal matrix of singular values. | |
| 2244 if (mode < 0) | |
|
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|
2245 sigma = sigma(p:-1:1); |
|
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|
2246 endif |
| 16634 | 2247 sigma = diag (sigma); |
| 2248 | |
| 2249 if (posdef) | |
| 2250 ## handle case where KAPPA was negative | |
| 2251 Q = qmult (p); | |
| 2252 A = Q' * sigma * Q; | |
| 2253 A = (A + A') / 2; # Ensure matrix is symmetric. | |
|
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|
2254 return; |
| 16634 | 2255 endif |
| 2256 | |
| 2257 if (m != n) | |
| 2258 ## Expand to m-by-n diagonal matrix | |
| 2259 sigma(m, n) = 0; | |
|
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|
2260 endif |
| 16634 | 2261 |
| 2262 if (kl == 0 && ku == 0) | |
| 2263 ## Diagonal matrix requested - nothing more to do. | |
| 2264 A = sigma; | |
| 2265 else | |
| 2266 ## A = U*sigma*V, where U, V are random orthogonal matrices from the | |
| 2267 ## Haar distribution. | |
| 2268 A = qmult (sigma'); | |
| 2269 A = qmult (A'); | |
| 2270 | |
| 2271 if (kl < n-1 || ku < n-1) | |
| 2272 ## Bandwidth reduction | |
| 2273 A = bandred (A, kl, ku); | |
| 2274 endif | |
| 2275 endif | |
| 2276 endfunction | |
| 2277 | |
| 2278 function A = redheff (n) | |
| 2279 ## REDHEFF A (0,1) matrix of Redheffer associated with the Riemann hypothesis. | |
|
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|
2280 ## A = REDHEFF(N) is an N-by-N matrix of 0s and 1s defined by |
|
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|
2281 ## A(i,j) = 1 if j = 1 or if i divides j, |
|
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|
2282 ## A(i,j) = 0 otherwise. |
|
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|
2283 ## It has N - FLOOR(LOG2(N)) - 1 eigenvalues equal to 1, |
|
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|
2284 ## a real eigenvalue (the spectral radius) approximately SQRT(N), |
|
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|
2285 ## a negative eigenvalue approximately -SQRT(N), |
|
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|
2286 ## and the remaining eigenvalues are provably ``small''. |
|
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|
2287 ## Barrett and Jarvis (1992) conjecture that |
|
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|
2288 ## ``the small eigenvalues all lie inside the unit circle |
|
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|
2289 ## ABS(Z) = 1'', |
|
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|
2290 ## and a proof of this conjecture, together with a proof that some |
|
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|
2291 ## eigenvalue tends to zero as N tends to infinity, would yield |
|
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|
2292 ## a new proof of the prime number theorem. |
|
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|
2293 ## The Riemann hypothesis is true if and only if |
|
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|
2294 ## DET(A) = O( N^(1/2+epsilon) ) for every epsilon > 0 |
|
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|
2295 ## (`!' denotes factorial). |
|
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|
2296 ## See also RIEMANN. |
| 16634 | 2297 ## |
|
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|
2298 ## Reference: |
|
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|
2299 ## W.W. Barrett and T.J. Jarvis, |
|
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|
2300 ## Spectral Properties of a Matrix of Redheffer, |
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|
2301 ## Linear Algebra and Appl., 162 (1992), pp. 673-683. |
| 16634 | 2302 |
| 2303 if (nargin != 1) | |
| 2304 error ("gallery: 1 argument is required for redheff matrix."); | |
| 2305 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2306 error ("gallery: N must be an integer for redheff matrix."); | |
| 2307 endif | |
| 2308 | |
| 2309 i = (1:n)' * ones (1, n); | |
| 2310 A = ! rem (i', i); | |
| 2311 A(:,1) = ones (n, 1); | |
| 2312 endfunction | |
| 2313 | |
| 2314 function A = riemann (n) | |
| 2315 ## RIEMANN A matrix associated with the Riemann hypothesis. | |
|
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|
2316 ## A = RIEMANN(N) is an N-by-N matrix for which the |
|
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|
2317 ## Riemann hypothesis is true if and only if |
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|
2318 ## DET(A) = O( N! N^(-1/2+epsilon) ) for every epsilon > 0 |
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|
2319 ## (`!' denotes factorial). |
|
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|
2320 ## A = B(2:N+1, 2:N+1), where |
|
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|
2321 ## B(i,j) = i-1 if i divides j and -1 otherwise. |
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|
2322 ## Properties include, with M = N+1: |
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|
2323 ## Each eigenvalue E(i) satisfies ABS(E(i)) <= M - 1/M. |
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|
2324 ## i <= E(i) <= i+1 with at most M-SQRT(M) exceptions. |
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|
2325 ## All integers in the interval (M/3, M/2] are eigenvalues. |
| 16634 | 2326 ## |
|
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|
2327 ## See also REDHEFF. |
| 16634 | 2328 ## |
|
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|
2329 ## Reference: |
|
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|
2330 ## F. Roesler, Riemann's hypothesis as an eigenvalue problem, |
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|
2331 ## Linear Algebra and Appl., 81 (1986), pp. 153-198. |
| 16634 | 2332 |
| 2333 if (nargin != 1) | |
| 2334 error ("gallery: 1 argument is required for riemann matrix."); | |
| 2335 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2336 error ("gallery: N must be an integer for riemann matrix."); | |
| 2337 endif | |
| 2338 | |
|
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|
2339 n += 1; |
| 16634 | 2340 i = (2:n)' * ones (1, n-1); |
| 2341 j = i'; | |
| 2342 A = i .* (! rem (j, i)) - ones (n-1); | |
| 2343 endfunction | |
| 2344 | |
| 2345 function A = ris (n) | |
| 2346 ## NOTE: this function was named dingdong in the original Test Matrix Toolbox | |
| 2347 ## RIS Dingdong matrix - a symmetric Hankel matrix. | |
|
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|
2348 ## A = RIS(N) is the symmetric N-by-N Hankel matrix with |
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|
2349 ## A(i,j) = 0.5/(N-i-j+1.5). |
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|
2350 ## The eigenvalues of A cluster around PI/2 and -PI/2. |
| 16634 | 2351 ## |
|
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|
2352 ## Invented by F.N. Ris. |
| 16634 | 2353 ## |
|
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|
2354 ## Reference: |
|
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|
2355 ## J.C. Nash, Compact Numerical Methods for Computers: Linear |
|
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|
2356 ## Algebra and Function Minimisation, second edition, Adam Hilger, |
|
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|
2357 ## Bristol, 1990 (Appendix 1). |
| 16634 | 2358 |
| 2359 if (nargin != 1) | |
| 2360 error ("gallery: 1 argument is required for ris matrix."); | |
| 2361 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2362 error ("gallery: N must be an integer for ris matrix."); | |
| 2363 endif | |
| 2364 | |
| 2365 p = -2*(1:n) + (n+1.5); | |
| 2366 A = cauchy (p); | |
| 2367 endfunction | |
| 2368 | |
| 2369 function A = smoke (n, k = 0) | |
| 2370 ## SMOKE Smoke matrix - complex, with a `smoke ring' pseudospectrum. | |
|
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|
2371 ## SMOKE(N) is an N-by-N matrix with 1s on the |
|
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|
2372 ## superdiagonal, 1 in the (N,1) position, and powers of |
|
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|
2373 ## roots of unity along the diagonal. |
|
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|
2374 ## SMOKE(N, 1) is the same except for a zero (N,1) element. |
|
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|
2375 ## The eigenvalues of SMOKE(N, 1) are the N'th roots of unity; |
|
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|
2376 ## those of SMOKE(N) are the N'th roots of unity times 2^(1/N). |
| 16634 | 2377 ## |
|
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|
2378 ## Try PS(SMOKE(32)). For SMOKE(N, 1) the pseudospectrum looks |
|
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|
2379 ## like a sausage folded back on itself. |
|
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|
2380 ## GERSH(SMOKE(N, 1)) is interesting. |
| 16634 | 2381 ## |
|
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|
2382 ## Reference: |
|
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|
2383 ## L. Reichel and L.N. Trefethen, Eigenvalues and pseudo-eigenvalues of |
|
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|
2384 ## Toeplitz matrices, Linear Algebra and Appl., 162-164:153-185, 1992. |
| 16634 | 2385 |
| 2386 if (nargin < 1 || nargin > 2) | |
| 2387 error ("gallery: 1 to 2 arguments are required for smoke matrix."); | |
| 2388 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2389 error ("gallery: N must be an integer for smoke matrix."); | |
| 2390 elseif (! isnumeric (n) || ! isscalar (n)) | |
| 2391 error ("gallery: K must be a numeric scalar for smoke matrix."); | |
| 2392 endif | |
| 2393 | |
|
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|
2394 w = exp (2*pi*i/n); |
|
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|
2395 A = diag ( [w.^(1:n-1) 1] ) + diag (ones (n-1,1), 1); |
| 16634 | 2396 |
| 2397 switch (k) | |
| 2398 case (0), A(n,1) = 1; | |
| 2399 case (1), # do nothing | |
| 2400 otherwise, | |
| 2401 error ("gallery: K must have a value of 0 or 1 for smoke matrix."); | |
| 2402 endswitch | |
| 2403 endfunction | |
| 2404 | |
| 2405 function T = toeppd (n, m = n, w = rand (m,1), theta = rand (m,1)) | |
| 2406 ## NOTE: this function was named pdtoep in the original Test Matrix Toolbox | |
| 2407 ## TOEPPD Symmetric positive definite Toeplitz matrix. | |
|
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|
2408 ## TOEPPD(N, M, W, THETA) is an N-by-N symmetric positive (semi-) |
|
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|
2409 ## definite (SPD) Toeplitz matrix, comprised of the sum of M rank 2 |
|
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|
2410 ## (or, for certain THETA, rank 1) SPD Toeplitz matrices. |
|
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|
2411 ## Specifically, |
|
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|
2412 ## T = W(1)*T(THETA(1)) + ... + W(M)*T(THETA(M)), |
|
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|
2413 ## where T(THETA(k)) has (i,j) element COS(2*PI*THETA(k)*(i-j)). |
|
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|
2414 ## Defaults: M = N, W = RAND(M,1), THETA = RAND(M,1). |
| 16634 | 2415 ## |
|
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|
2416 ## Reference: |
|
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|
2417 ## G. Cybenko and C.F. Van Loan, Computing the minimum eigenvalue of |
|
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|
2418 ## a symmetric positive definite Toeplitz matrix, SIAM J. Sci. Stat. |
|
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|
2419 ## Comput., 7 (1986), pp. 123-131. |
| 16634 | 2420 |
| 2421 if (nargin < 1 || nargin > 4) | |
| 2422 error ("gallery: 1 to 4 arguments are required for toeppd matrix."); | |
| 2423 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2424 error ("gallery: N must be a numeric integer for toeppd matrix."); | |
| 2425 elseif (! isnumeric (m) || ! isscalar (m) || fix (m) != m) | |
| 2426 error ("gallery: M must be a numeric integer for toeppd matrix."); | |
| 2427 elseif (numel (w) != m || numel (theta) != m) | |
| 2428 error ("gallery: W and THETA must be vectors of length M for toeppd matrix."); | |
| 2429 endif | |
| 2430 | |
| 2431 T = zeros (n); | |
| 2432 E = 2*pi * ((1:n)' * ones (1, n) - ones (n, 1) * (1:n)); | |
| 2433 | |
| 2434 for i = 1:m | |
|
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|
2435 T += w(i) * cos (theta(i)*E); |
| 16634 | 2436 endfor |
| 2437 endfunction | |
| 2438 | |
| 2439 function P = toeppen (n, a = 1, b = -10, c = 0, d = 10, e = 1) | |
| 2440 ## NOTE: this function was named pentoep in the original Test Matrix Toolbox | |
| 2441 ## TOEPPEN Pentadiagonal Toeplitz matrix (sparse). | |
|
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|
2442 ## P = TOEPPEN(N, A, B, C, D, E) is the N-by-N pentadiagonal |
|
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|
2443 ## Toeplitz matrix with diagonals composed of the numbers |
|
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|
2444 ## A =: P(3,1), B =: P(2,1), C =: P(1,1), D =: P(1,2), E =: P(1,3). |
|
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|
2445 ## Default: (A,B,C,D,E) = (1,-10,0,10,1) (a matrix of Rutishauser). |
|
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|
2446 ## This matrix has eigenvalues lying approximately on |
|
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|
2447 ## the line segment 2*cos(2*t) + 20*i*sin(t). |
| 16634 | 2448 ## |
|
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|
2449 ## Interesting plots are |
|
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|
2450 ## PS(FULL(TOEPPEN(32,0,1,0,0,1/4))) - `triangle' |
|
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|
2451 ## PS(FULL(TOEPPEN(32,0,1/2,0,0,1))) - `propeller' |
|
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|
2452 ## PS(FULL(TOEPPEN(32,0,1/2,1,1,1))) - `fish' |
| 16634 | 2453 ## |
|
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|
2454 ## References: |
|
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|
2455 ## R.M. Beam and R.F. Warming, The asymptotic spectra of |
|
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|
2456 ## banded Toeplitz and quasi-Toeplitz matrices, SIAM J. Sci. |
|
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|
2457 ## Comput. 14 (4), 1993, pp. 971-1006. |
|
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|
2458 ## H. Rutishauser, On test matrices, Programmation en Mathematiques |
|
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|
2459 ## Numeriques, Editions Centre Nat. Recherche Sci., Paris, 165, |
|
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|
2460 ## 1966, pp. 349-365. |
| 16634 | 2461 |
| 2462 if (nargin < 1 || nargin > 6) | |
| 2463 error ("gallery: 1 to 6 arguments are required for toeppen matrix."); | |
| 2464 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2465 error ("gallery: N must be a numeric integer for toeppen matrix."); | |
|
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Replace cellfun() occurrences with faster code where possible.
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17338
diff
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|
2466 elseif (any (! cellfun ("isnumeric", {a b c d e})) || any (cellfun ("numel", {a b c d e}) != 1)) |
| 16634 | 2467 error ("gallery: A, B, C, D and E must be numeric scalars for toeppen matrix."); |
| 2468 endif | |
| 2469 | |
| 2470 P = spdiags ([a*ones(n,1) b*ones(n,1) c*ones(n,1) d*ones(n,1) e*ones(n,1)], | |
| 2471 -2:2, n, n); | |
| 2472 endfunction | |
| 2473 | |
| 2474 function T = tridiag (n, x = -1, y = 2, z = -1) | |
| 2475 ## TRIDIAG Tridiagonal matrix (sparse). | |
|
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|
2476 ## TRIDIAG(X, Y, Z) is the tridiagonal matrix with subdiagonal X, |
|
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|
2477 ## diagonal Y, and superdiagonal Z. |
|
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diff
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|
2478 ## X and Z must be vectors of dimension one less than Y. |
|
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|
2479 ## Alternatively TRIDIAG(N, C, D, E), where C, D, and E are all |
|
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diff
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|
2480 ## scalars, yields the Toeplitz tridiagonal matrix of order N |
|
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diff
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|
2481 ## with subdiagonal elements C, diagonal elements D, and superdiagonal |
|
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|
2482 ## elements E. This matrix has eigenvalues (Todd 1977) |
|
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|
2483 ## D + 2*SQRT(C*E)*COS(k*PI/(N+1)), k=1:N. |
|
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diff
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|
2484 ## TRIDIAG(N) is the same as TRIDIAG(N,-1,2,-1), which is |
|
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|
2485 ## a symmetric positive definite M-matrix (the negative of the |
|
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diff
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|
2486 ## second difference matrix). |
| 16634 | 2487 ## |
|
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|
2488 ## References: |
|
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|
2489 ## J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, |
|
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|
2490 ## Birkhauser, Basel, and Academic Press, New York, 1977, p. 155. |
|
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|
2491 ## D.E. Rutherford, Some continuant determinants arising in physics and |
|
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diff
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|
2492 ## chemistry---II, Proc. Royal Soc. Edin., 63, A (1952), pp. 232-241. |
| 16634 | 2493 |
| 2494 if (nargin != 1 && nargin != 3 && nargin != 4) | |
| 2495 error ("gallery: 1, 3, or 4 arguments are required for tridiag matrix."); | |
| 2496 elseif (nargin == 3) | |
| 2497 z = y; | |
| 2498 y = x; | |
| 2499 x = n; | |
| 2500 endif | |
| 2501 | |
| 2502 ## Force column vectors | |
| 2503 x = x(:); | |
| 2504 y = y(:); | |
| 2505 z = z(:); | |
| 2506 | |
| 2507 if (isscalar (x) && isscalar (y) && isscalar (z)) | |
| 2508 x *= ones (n-1, 1); | |
| 2509 z *= ones (n-1, 1); | |
| 2510 y *= ones (n, 1); | |
| 2511 elseif (numel (y) != numel (x) + 1) | |
| 2512 error ("gallery: X must have one element less than Y for tridiag matrix."); | |
| 2513 elseif (numel (y) != numel (z) + 1) | |
| 2514 error ("gallery: Z must have one element less than Y for tridiag matrix."); | |
| 2515 endif | |
| 2516 | |
| 2517 ## T = diag (x, -1) + diag (y) + diag (z, 1); # For non-sparse matrix. | |
| 2518 n = numel (y); | |
| 2519 T = spdiags ([[x;0] y [0;z]], -1:1, n, n); | |
| 2520 endfunction | |
| 2521 | |
|
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|
2522 function t = triw (n, alpha = -1, k = n(end) - 1) |
| 16634 | 2523 ## TRIW Upper triangular matrix discussed by Wilkinson and others. |
|
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|
2524 ## TRIW(N, ALPHA, K) is the upper triangular matrix with ones on |
|
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|
2525 ## the diagonal and ALPHAs on the first K >= 0 superdiagonals. |
|
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|
2526 ## N may be a 2-vector, in which case the matrix is N(1)-by-N(2) and |
|
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|
2527 ## upper trapezoidal. |
|
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|
2528 ## Defaults: ALPHA = -1, |
|
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|
2529 ## K = N - 1 (full upper triangle). |
|
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|
2530 ## TRIW(N) is a matrix discussed by Kahan, Golub and Wilkinson. |
| 16634 | 2531 ## |
|
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|
2532 ## Ostrowski (1954) shows that |
|
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|
2533 ## COND(TRIW(N,2)) = COT(PI/(4*N))^2, |
|
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|
2534 ## and for large ABS(ALPHA), |
|
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|
2535 ## COND(TRIW(N,ALPHA)) is approximately ABS(ALPHA)^N*SIN(PI/(4*N-2)). |
| 16634 | 2536 ## |
|
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|
2537 ## Adding -2^(2-N) to the (N,1) element makes TRIW(N) singular, |
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|
2538 ## as does adding -2^(1-N) to all elements in the first column. |
| 16634 | 2539 ## |
|
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|
2540 ## References: |
|
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|
2541 ## G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the |
|
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|
2542 ## computation of the Jordan canonical form, SIAM Review, |
|
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|
2543 ## 18(4), 1976, pp. 578-619. |
|
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Rik <rik@octave.org>
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diff
changeset
|
2544 ## W. Kahan, Numerical linear algebra, Canadian Math. Bulletin, |
|
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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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|
2545 ## 9 (1966), pp. 757-801. |
|
9fc020886ae9
maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
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diff
changeset
|
2546 ## 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>
parents:
19697
diff
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|
2547 ## matrices, J. Reine Angew. Math., 193 (3/4), 1954, pp. 143-160. |
|
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diff
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|
2548 ## J.H. Wilkinson, Singular-value decomposition---basic aspects, |
|
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Rik <rik@octave.org>
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19697
diff
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|
2549 ## in D.A.H. Jacobs, ed., Numerical Software---Needs and Availability, |
|
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Rik <rik@octave.org>
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19697
diff
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|
2550 ## Academic Press, London, 1978, pp. 109-135. |
| 16634 | 2551 |
| 2552 if (nargin < 1 || nargin > 3) | |
| 2553 error ("gallery: 1 to 3 arguments are required for triw matrix."); | |
| 2554 elseif (! isnumeric (n) || all (numel (n) != [1 2])) | |
| 2555 error ("gallery: N must be a 1 or 2 elements vector for triw matrix."); | |
| 2556 elseif (! isscalar (alpha)) | |
| 2557 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
|
2558 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
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|
2559 error ("gallery: K must be a numeric integer >= 0 for triw matrix."); |
| 16634 | 2560 endif |
| 2561 | |
| 2562 m = n(1); # Parameter n specifies dimension: m-by-n. | |
| 2563 n = n(end); | |
| 2564 | |
| 2565 t = tril (eye (m, n) + alpha * triu (ones (m, n), 1), k); | |
| 2566 endfunction | |
| 2567 | |
|
16978
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Rik <rik@octave.org>
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diff
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|
2568 function A = uniformdata (varargin) |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
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16933
diff
changeset
|
2569 |
|
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gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
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16933
diff
changeset
|
2570 if (nargin < 2) |
|
16979
9aa293e00475
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Rik <rik@octave.org>
parents:
16978
diff
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|
2571 error ("gallery: At least 2 arguments required 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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|
2572 endif |
|
00379f9f8773
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Rik <rik@octave.org>
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16933
diff
changeset
|
2573 if (isnumeric (varargin{end})) |
|
00379f9f8773
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Rik <rik@octave.org>
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16933
diff
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|
2574 jidx = varargin{end}; |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
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|
2575 svec = [varargin{:}]; |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
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16933
diff
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|
2576 varargin(end) = []; |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
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16933
diff
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|
2577 elseif (ischar (varargin{end})) |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2578 if (nargin < 3) |
|
16979
9aa293e00475
gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
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16978
diff
changeset
|
2579 error (["gallery: CLASS argument requires 3 inputs " ... |
|
9aa293e00475
gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents:
16978
diff
changeset
|
2580 "for uniformdata matrix."]); |
|
16978
00379f9f8773
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diff
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|
2581 endif |
|
00379f9f8773
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Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2582 jidx = varargin{end-1}; |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
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16933
diff
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|
2583 svec = [varargin{1:end-1}]; |
|
00379f9f8773
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Rik <rik@octave.org>
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16933
diff
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|
2584 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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|
2585 else |
|
16979
9aa293e00475
gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents:
16978
diff
changeset
|
2586 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
|
2587 "for uniformdata matrix"]); |
|
16978
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
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|
2588 endif |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2589 |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2590 if (! (isnumeric (jidx) && isscalar (jidx) |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
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|
2591 && jidx == fix (jidx) |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
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16933
diff
changeset
|
2592 && jidx >= 0 && jidx <= 0xFFFFFFFF)) |
|
16979
9aa293e00475
gallery.m: Add 'normaldata' matrix to function.
Rik <rik@octave.org>
parents:
16978
diff
changeset
|
2593 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
|
2594 "for uniformdata matrix"]); |
|
16978
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
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|
2595 endif |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2596 |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
parents:
16933
diff
changeset
|
2597 ## 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
|
2598 ## 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>
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17386
diff
changeset
|
2599 randstate = rand ("state"); |
|
16978
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
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16933
diff
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|
2600 unwind_protect |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
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16933
diff
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|
2601 rand ("state", svec); |
|
00379f9f8773
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Rik <rik@octave.org>
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16933
diff
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|
2602 A = rand (varargin{:}); |
|
00379f9f8773
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Rik <rik@octave.org>
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16933
diff
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|
2603 unwind_protect_cleanup |
|
00379f9f8773
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Rik <rik@octave.org>
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diff
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|
2604 rand ("state", randstate); |
|
00379f9f8773
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Rik <rik@octave.org>
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16933
diff
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|
2605 end_unwind_protect |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
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16933
diff
changeset
|
2606 |
|
00379f9f8773
gallery.m: Add 'uniformdata' matrix to function.
Rik <rik@octave.org>
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16933
diff
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|
2607 endfunction |
|
00379f9f8773
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Rik <rik@octave.org>
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16933
diff
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|
2608 |
| 16634 | 2609 function A = wathen (nx, ny, k = 0) |
|
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|
2610 ## WATHEN returns the Wathen matrix. |
| 16634 | 2611 ## |
|
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Rik <rik@octave.org>
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19697
diff
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|
2612 ## Discussion: |
| 16634 | 2613 ## |
|
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Rik <rik@octave.org>
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19697
diff
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|
2614 ## The Wathen matrix is a finite element matrix which is sparse. |
| 16634 | 2615 ## |
|
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maint: Clean up m-files to follow Octave coding conventions.
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19697
diff
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|
2616 ## The entries of the matrix depend in part on a physical quantity |
|
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Rik <rik@octave.org>
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19697
diff
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|
2617 ## related to density. That density is here assigned random values between |
|
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19697
diff
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|
2618 ## 0 and 100. |
| 16634 | 2619 ## |
|
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Rik <rik@octave.org>
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diff
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|
2620 ## A = WATHEN ( NX, NY ) is a sparse random N-by-N finite element matrix |
|
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Rik <rik@octave.org>
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19697
diff
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|
2621 ## where N = 3*NX*NY + 2*NX + 2*NY + 1. |
| 16634 | 2622 ## |
|
19833
9fc020886ae9
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Rik <rik@octave.org>
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diff
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|
2623 ## A is the consistent mass matrix for a regular NX-by-NY |
|
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Rik <rik@octave.org>
parents:
19697
diff
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|
2624 ## grid of 8-node (serendipity) elements in 2 space dimensions. |
| 16634 | 2625 ## |
|
19833
9fc020886ae9
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Rik <rik@octave.org>
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19697
diff
changeset
|
2626 ## Here is an illustration for NX = 3, NX = 2: |
| 16634 | 2627 ## |
|
19833
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Rik <rik@octave.org>
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19697
diff
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|
2628 ## 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>
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19697
diff
changeset
|
2629 ## | | | | |
|
9fc020886ae9
maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents:
19697
diff
changeset
|
2630 ## 19 20 21 22 |
|
9fc020886ae9
maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents:
19697
diff
changeset
|
2631 ## | | | | |
|
9fc020886ae9
maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents:
19697
diff
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|
2632 ## 12-13-14-15-16-17-18 |
|
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maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents:
19697
diff
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|
2633 ## | | | | |
|
9fc020886ae9
maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents:
19697
diff
changeset
|
2634 ## 8 9 10 11 |
|
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maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents:
19697
diff
changeset
|
2635 ## | | | | |
|
9fc020886ae9
maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents:
19697
diff
changeset
|
2636 ## 1--2--3--4--5--6--7 |
| 16634 | 2637 ## |
|
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9fc020886ae9
maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
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19697
diff
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|
2638 ## For this example, the total number of nodes is, as expected, |
| 16634 | 2639 ## |
|
19833
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Rik <rik@octave.org>
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19697
diff
changeset
|
2640 ## N = 3 * 3 * 2 + 2 * 2 + 2 * 3 + 1 = 29. |
| 16634 | 2641 ## |
|
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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|
2642 ## 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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|
2643 ## the density, RHO(NX,NY), which is chosen randomly in this routine. |
| 16634 | 2644 ## |
|
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diff
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|
2645 ## In particular, if D = DIAG(DIAG(A)), then |
|
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|
2646 ## 0.25 <= EIG(INV(D)*A) <= 4.5 |
|
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19697
diff
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|
2647 ## for any positive integers NX and NY and any densities RHO(NX,NY). |
| 16634 | 2648 ## |
|
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diff
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|
2649 ## A = WATHEN ( NX, NY, 1 ) returns the diagonally scaled matrix. |
| 16634 | 2650 ## |
|
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|
2651 ## Modified: |
| 16634 | 2652 ## |
|
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19697
diff
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|
2653 ## 17 September 2007 |
| 16634 | 2654 ## |
|
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|
2655 ## Author: |
| 16634 | 2656 ## |
|
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|
2657 ## Nicholas Higham |
| 16634 | 2658 ## |
|
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diff
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|
2659 ## Reference: |
| 16634 | 2660 ## |
|
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19697
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|
2661 ## Nicholas Higham, |
|
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diff
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|
2662 ## Algorithm 694: A Collection of Test Matrices in MATLAB, |
|
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19697
diff
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|
2663 ## ACM Transactions on Mathematical Software, |
|
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19697
diff
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|
2664 ## Volume 17, Number 3, September 1991, pages 289-305. |
| 16634 | 2665 ## |
|
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|
2666 ## Andrew Wathen, |
|
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19697
diff
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|
2667 ## Realistic eigenvalue bounds for the Galerkin mass matrix, |
|
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diff
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|
2668 ## IMA Journal of Numerical Analysis, |
|
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diff
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|
2669 ## Volume 7, 1987, pages 449-457. |
| 16634 | 2670 ## |
|
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|
2671 ## Parameters: |
| 16634 | 2672 ## |
|
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|
2673 ## Input, integer NX, NY, the number of elements in the X and Y directions |
|
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|
2674 ## of the finite element grid. NX and NY must each be at least 1. |
| 16634 | 2675 ## |
|
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|
2676 ## Optional input, integer K, is used to request that the diagonally scaled |
|
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|
2677 ## version of the matrix be returned. This happens if K is specified with |
|
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|
2678 ## the value 1. |
| 16634 | 2679 ## |
|
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|
2680 ## Output, sparse real A(N,N), the matrix. The dimension N is determined by |
|
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|
2681 ## NX and NY, as described above. A is stored in the MATLAB sparse matrix |
|
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|
2682 ## format. |
| 16634 | 2683 |
| 2684 if (nargin < 2 || nargin > 3) | |
| 2685 error ("gallery: 2 or 3 arguments are required for wathen matrix."); | |
| 2686 elseif (! isnumeric (nx) || ! isscalar (nx) || nx < 1) | |
| 2687 error ("gallery: NX must be a positive scalar for wathen matrix."); | |
| 2688 elseif (! isnumeric (ny) || ! isscalar (ny) || ny < 1) | |
| 2689 error ("gallery: NY must be a positive scalar for wathen matrix."); | |
| 2690 elseif (! isscalar (k)) | |
| 2691 error ("gallery: K must be a scalar for wathen matrix."); | |
| 2692 endif | |
| 2693 | |
| 2694 e1 = [ 6 -6 2 -8 | |
| 2695 -6 32 -6 20 | |
| 2696 2 -6 6 -6 | |
| 2697 -8 20 -6 32 ]; | |
| 2698 | |
| 2699 e2 = [ 3 -8 2 -6 | |
| 2700 -8 16 -8 20 | |
| 2701 2 -8 3 -8 | |
| 2702 -6 20 -8 16 ]; | |
| 2703 | |
| 2704 e = [ e1 e2 | |
| 2705 e2' e1] / 45; | |
| 2706 | |
| 2707 n = 3*nx*ny + 2*nx + 2*ny + 1; | |
| 2708 | |
| 2709 A = sparse (n, n); | |
| 2710 | |
| 2711 rho = 100 * rand (nx, ny); | |
| 2712 | |
| 2713 for j = 1:ny | |
| 2714 for i = 1:nx | |
|
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2715 ## |
| 16634 | 2716 ## For the element (I,J), determine the indices of the 8 nodes. |
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2717 ## |
| 16634 | 2718 nn(1) = 3*j*nx + 2*i + 2*j + 1; |
| 2719 nn(2) = nn(1) - 1; | |
| 2720 nn(3) = nn(2) - 1; | |
| 2721 nn(4) = (3*j - 1) * nx + 2*j + i - 1; | |
| 2722 nn(5) = 3 * (j-1) * nx + 2*i + 2*j - 3; | |
| 2723 nn(6) = nn(5) + 1; | |
| 2724 nn(7) = nn(6) + 1; | |
| 2725 nn(8) = nn(4) + 1; | |
| 2726 | |
| 2727 em = e * rho(i,j); | |
| 2728 | |
| 2729 for krow = 1:8 | |
| 2730 for kcol = 1:8 | |
| 2731 A(nn(krow),nn(kcol)) = A(nn(krow),nn(kcol)) + em(krow,kcol); | |
| 2732 endfor | |
| 2733 endfor | |
| 2734 | |
| 2735 endfor | |
| 2736 endfor | |
| 2737 | |
| 2738 ## If requested, return A with diagonal scaling. | |
| 2739 if (k) | |
| 2740 A = diag (diag (A)) \ A; | |
| 2741 endif | |
| 2742 endfunction | |
| 2743 | |
| 2744 function [A, b] = wilk (n) | |
| 2745 ## WILK Various specific matrices devised/discussed by Wilkinson. | |
|
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2746 ## [A, b] = WILK(N) is the matrix or system of order N. |
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2747 ## N = 3: upper triangular system Ux=b illustrating inaccurate solution. |
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2748 ## N = 4: lower triangular system Lx=b, ill-conditioned. |
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2749 ## N = 5: HILB(6)(1:5,2:6)*1.8144. Symmetric positive definite. |
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2750 ## N = 21: W21+, tridiagonal. Eigenvalue problem. |
| 16634 | 2751 ## |
|
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2752 ## References: |
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2753 ## J.H. Wilkinson, Error analysis of direct methods of matrix inversion, |
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2754 ## J. Assoc. Comput. Mach., 8 (1961), pp. 281-330. |
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2755 ## J.H. Wilkinson, Rounding Errors in Algebraic Processes, Notes on Applied |
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2756 ## Science No. 32, Her Majesty's Stationery Office, London, 1963. |
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2757 ## J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University |
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2758 ## Press, 1965. |
| 16634 | 2759 |
| 2760 if (nargin != 1) | |
| 2761 error ("gallery: 1 argument is required for wilk matrix."); | |
| 2762 elseif (! isnumeric (n) || ! isscalar (n)) | |
| 2763 error ("gallery: N must be a numeric scalar for wilk matrix."); | |
| 2764 endif | |
| 2765 | |
| 2766 if (n == 3) | |
| 2767 ## Wilkinson (1961) p.323. | |
| 2768 A = [ 1e-10 0.9 -0.4 | |
| 2769 0 0.9 -0.4 | |
| 2770 0 0 1e-10 ]; | |
| 2771 | |
| 2772 b = [ 0 | |
| 2773 0 | |
| 2774 1]; | |
| 2775 | |
| 2776 elseif (n == 4) | |
| 2777 ## Wilkinson (1963) p.105. | |
| 2778 A = [0.9143e-4 0 0 0 | |
| 2779 0.8762 0.7156e-4 0 0 | |
| 2780 0.7943 0.8143 0.9504e-4 0 | |
| 2781 0.8017 0.6123 0.7165 0.7123e-4]; | |
| 2782 | |
| 2783 b = [0.6524 | |
| 2784 0.3127 | |
| 2785 0.4186 | |
| 2786 0.7853]; | |
| 2787 | |
| 2788 elseif (n == 5) | |
| 2789 ## Wilkinson (1965), p.234. | |
| 2790 A = hilb (6); | |
| 2791 A = A(1:5, 2:6) * 1.8144; | |
| 2792 | |
| 2793 elseif (n == 21) | |
| 2794 ## Wilkinson (1965), p.308. | |
| 2795 E = diag (ones (n-1, 1), 1); | |
| 2796 m = (n-1)/2; | |
| 2797 A = diag (abs (-m:m)) + E + E'; | |
| 2798 | |
| 2799 else | |
|
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2800 error ("gallery: unknown N '%d' for wilk matrix.", n); |
| 16634 | 2801 endif |
| 2802 endfunction | |
| 2803 | |
| 2804 ## NOTE: bandred is part of the Test Matrix Toolbox and is used by randsvd() | |
| 2805 function A = bandred (A, kl, ku) | |
| 2806 ## BANDRED Band reduction by two-sided unitary transformations. | |
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2807 ## B = BANDRED(A, KL, KU) is a matrix unitarily equivalent to A |
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2808 ## with lower bandwidth KL and upper bandwidth KU |
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2809 ## (i.e. B(i,j) = 0 if i > j+KL or j > i+KU). |
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2810 ## The reduction is performed using Householder transformations. |
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2811 ## If KU is omitted it defaults to KL. |
| 16634 | 2812 ## |
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2813 ## Called by RANDSVD. |
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2814 ## This is a `standard' reduction. Cf. reduction to bidiagonal form |
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2815 ## prior to computing the SVD. This code is a little wasteful in that |
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2816 ## it computes certain elements which are immediately set to zero! |
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2817 ## |
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2818 ## Reference: |
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2819 ## G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, |
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2820 ## Johns Hopkins University Press, Baltimore, Maryland, 1989. |
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2821 ## Section 5.4.3. |
| 16634 | 2822 |
| 2823 ## Check for special case where order of left/right transformations matters. | |
| 2824 ## Easiest approach is to work on the transpose, flipping back at the end. | |
| 2825 flip = false; | |
| 2826 if (ku == 0) | |
| 2827 flip = true; | |
| 2828 A = A'; | |
| 2829 [ku, kl] = deal (kl, ku); | |
| 2830 endif | |
| 2831 | |
| 2832 [m, n] = size (A); | |
| 2833 | |
| 2834 for j = 1:min (min (m, n), max (m-kl-1, n-ku-1)) | |
| 2835 if (j+kl+1 <= m) | |
| 2836 [v, beta] = house (A(j+kl:m,j)); | |
| 2837 temp = A(j+kl:m,j:n); | |
| 2838 A(j+kl:m,j:n) = temp - beta*v*(v'*temp); | |
| 2839 A(j+kl+1:m,j) = zeros (m-j-kl, 1); | |
| 2840 endif | |
| 2841 | |
| 2842 if (j+ku+1 <= n) | |
| 2843 [v, beta] = house (A(j,j+ku:n)'); | |
| 2844 temp = A(j:m,j+ku:n); | |
| 2845 A(j:m,j+ku:n) = temp - beta*(temp*v)*v'; | |
| 2846 A(j,j+ku+1:n) = zeros (1, n-j-ku); | |
| 2847 endif | |
| 2848 endfor | |
| 2849 | |
| 2850 if (flip) | |
| 2851 A = A'; | |
| 2852 endif | |
| 2853 endfunction | |
|
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2854 |
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2855 ## NOTE: qmult is part of the Test Matrix Toolbox and is used by randsvd() |
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2856 function B = qmult (A) |
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2857 ## QMULT Pre-multiply by random orthogonal matrix. |
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2858 ## QMULT(A) is Q*A where Q is a random real orthogonal matrix from |
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2859 ## the Haar distribution, of dimension the number of rows in A. |
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2860 ## Special case: if A is a scalar then QMULT(A) is the same as |
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2861 ## QMULT(EYE(A)). |
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2862 ## |
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2863 ## Called by RANDSVD. |
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2864 ## |
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2865 ## Reference: |
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2866 ## G.W. Stewart, The efficient generation of random |
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2867 ## orthogonal matrices with an application to condition estimators, |
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2868 ## SIAM J. Numer. Anal., 17 (1980), 403-409. |
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2869 |
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2870 [n, m] = size (A); |
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2871 |
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2872 ## Handle scalar A |
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2873 if (isscalar (A)) |
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2874 n = A; |
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2875 A = eye (n); |
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2876 endif |
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2877 |
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2878 d = zeros (n); |
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2879 |
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2880 for k = n-1:-1:1 |
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2881 ## Generate random Householder transformation. |
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2882 x = randn (n-k+1, 1); |
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2883 s = norm (x); |
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2884 sgn = sign (x(1)) + (x(1) == 0); # Modification for sign(1)=1. |
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2885 s = sgn*s; |
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2886 d(k) = -sgn; |
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2887 x(1) = x(1) + s; |
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2888 beta = s*x(1); |
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2889 |
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2890 ## Apply the transformation to A. |
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2891 y = x'*A(k:n,:); |
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2892 A(k:n,:) = A(k:n,:) - x*(y/beta); |
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2893 endfor |
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2894 |
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2895 ## Tidy up signs |
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2896 for i = 1:n-1 |
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2897 A(i,:) = d(i)*A(i,:); |
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2898 endfor |
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2899 A(n,:) = A(n,:) * sign (randn); |
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2900 B = A; |
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2901 endfunction |
|
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2902 |
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2903 ## BIST testing for just a few functions to verify that the main gallery |
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2904 ## dispatch function works. |
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2905 %assert (gallery ("clement", 3), [0 1 0; 2 0 2; 0 1 0]) |
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2906 %assert (gallery ("invhess", 2), [1 -1; 1 2]) |
|
cdfc8bc9ab62
gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents:
19040
diff
changeset
|
2907 |
|
cdfc8bc9ab62
gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents:
19040
diff
changeset
|
2908 ## Test input validation of main dispatch function only |
|
cdfc8bc9ab62
gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents:
19040
diff
changeset
|
2909 %!error gallery () |
|
cdfc8bc9ab62
gallery.m: Add BIST tests for central dispatch function.
Rik <rik@octave.org>
parents:
19040
diff
changeset
|
2910 %!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
|
2911 %!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
|
2912 %!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
|
2913 |
|
19944
941e782d0429
gallery: performance increase for minij matrix.
Carnë Draug <carandraug@octave.org>
parents:
19833
diff
changeset
|
2914 %!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
|
2915 %!assert (gallery ("minij", 1), 1) |
|
941e782d0429
gallery: performance increase for minij matrix.
Carnë Draug <carandraug@octave.org>
parents:
19833
diff
changeset
|
2916 %!assert (gallery ("minij", 0), []) |
|
941e782d0429
gallery: performance increase for minij matrix.
Carnë Draug <carandraug@octave.org>
parents:
19833
diff
changeset
|
2917 %!assert (gallery ("minij", -1), []) |