Mercurial > octave-nkf
annotate scripts/special-matrix/gallery.m @ 20330:557979395ca9
gallery: allow single element vectors for 'house'.
| author | Carnë Draug <carandraug@octave.org> |
|---|---|
| date | Fri, 03 Jul 2015 15:01:38 +0100 |
| parents | 2102c4582e32 |
| children | 0b9d23557506 |
| 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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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."); | |
|
20329
2102c4582e32
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20266
diff
changeset
|
1214 elseif (! isnumeric (i) || ! isscalar (i) || i == 0 || abs (i) > n) |
|
18851
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|
1215 error ("gallery: I must be a nonzero scalar, and abs (I) <= N for gearmat matrix."); |
|
20329
2102c4582e32
gallery: fix chebspec, cycol, gearmat, hanowa, lauchli, and pei (bug #45466)
Massimiliano Fasi <massimiliano.fasi@gmail.com>
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20266
diff
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|
1216 elseif (! isnumeric (j) || ! isscalar (j) || i == 0 || abs (j) > n) |
|
18851
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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 |
|
9fc020886ae9
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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."); | |
|
20329
2102c4582e32
gallery: fix chebspec, cycol, gearmat, hanowa, lauchli, and pei (bug #45466)
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20266
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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."); | |
|
20330
557979395ca9
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Carnë Draug <carandraug@octave.org>
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20329
diff
changeset
|
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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changeset
|
1348 |
|
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|
1349 if (! (isnumeric (jidx) && isscalar (jidx) |
|
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|
1350 && jidx == fix (jidx) |
|
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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] " ... |
|
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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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|
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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|
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 | |
|
19979
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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|
1877 randstate = randn ("state"); |
|
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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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|
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 | |
| 2153 function A = randsvd (n, kappa = sqrt (1/eps), mode = 3, kl = n-1, ku = kl) | |
| 2154 ## RANDSVD Random matrix with pre-assigned singular values. | |
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2155 ## RANDSVD(N, KAPPA, MODE, KL, KU) is a (banded) random matrix of order N |
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2156 ## with COND(A) = KAPPA and singular values from the distribution MODE. |
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2157 ## N may be a 2-vector, in which case the matrix is N(1)-by-N(2). |
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2158 ## Available types: |
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2159 ## MODE = 1: one large singular value, |
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2160 ## MODE = 2: one small singular value, |
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2161 ## MODE = 3: geometrically distributed singular values, |
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2162 ## MODE = 4: arithmetically distributed singular values, |
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2163 ## MODE = 5: random singular values with unif. dist. logarithm. |
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2164 ## If omitted, MODE defaults to 3, and KAPPA defaults to SQRT(1/EPS). |
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2165 ## If MODE < 0 then the effect is as for ABS(MODE) except that in the |
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2166 ## original matrix of singular values the order of the diagonal entries |
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2167 ## is reversed: small to large instead of large to small. |
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2168 ## KL and KU are the lower and upper bandwidths respectively; if they |
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2169 ## are omitted a full matrix is produced. |
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2170 ## If only KL is present, KU defaults to KL. |
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2171 ## Special case: if KAPPA < 0 then a random full symmetric positive |
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2172 ## definite matrix is produced with COND(A) = -KAPPA and |
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2173 ## eigenvalues distributed according to MODE. |
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2174 ## KL and KU, if present, are ignored. |
| 16634 | 2175 ## |
|
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2176 ## Reference: |
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2177 ## N.J. Higham, Accuracy and Stability of Numerical Algorithms, |
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2178 ## Society for Industrial and Applied Mathematics, Philadelphia, PA, |
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2179 ## USA, 1996; sec. 26.3. |
| 16634 | 2180 ## |
|
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2181 ## This routine is similar to the more comprehensive Fortran routine xLATMS |
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2182 ## in the following reference: |
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2183 ## J.W. Demmel and A. McKenney, A test matrix generation suite, |
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2184 ## LAPACK Working Note #9, Courant Institute of Mathematical Sciences, |
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2185 ## New York, 1989. |
| 16634 | 2186 |
| 2187 if (nargin < 1 || nargin > 5) | |
| 2188 error ("gallery: 1 to 5 arguments are required for randsvd matrix."); | |
| 2189 elseif (! isnumeric (n) || all (numel (n) != [1 2]) || fix (n) != n) | |
| 2190 error ("gallery: N must be a 1 or 2 element integer vector for randsvd matrix."); | |
| 2191 elseif (! isnumeric (kappa) || ! isscalar (kappa)) | |
| 2192 error ("gallery: KAPPA must be a numeric scalar for randsvd matrix."); | |
| 2193 elseif (abs (kappa) < 1) | |
| 2194 error ("gallery: KAPPA must larger than or equal to 1 for randsvd matrix."); | |
| 2195 elseif (! isnumeric (mode) || ! isscalar (mode)) | |
| 2196 error ("gallery: MODE must be a numeric scalar for randsvd matrix."); | |
| 2197 elseif (! isnumeric (kl) || ! isscalar (kl)) | |
| 2198 error ("gallery: KL must be a numeric scalar for randsvd matrix."); | |
| 2199 elseif (! isnumeric (ku) || ! isscalar (ku)) | |
| 2200 error ("gallery: KU must be a numeric scalar for randsvd matrix."); | |
| 2201 endif | |
| 2202 | |
| 2203 posdef = 0; | |
| 2204 if (kappa < 0) | |
| 2205 posdef = 1; | |
| 2206 kappa = -kappa; | |
| 2207 endif | |
| 2208 | |
| 2209 ## Parameter n specifies dimension: m-by-n. | |
| 2210 m = n(1); | |
| 2211 n = n(end); | |
| 2212 p = min ([m n]); | |
| 2213 | |
| 2214 ## If A will be a vector | |
| 2215 if (p == 1) | |
| 2216 A = randn (m, n); | |
|
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2217 A /= norm (A); |
|
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2218 return; |
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2219 endif |
| 16634 | 2220 |
| 2221 ## Set up vector sigma of singular values. | |
| 2222 switch (abs (mode)) | |
| 2223 case (1) | |
| 2224 sigma = ones (p, 1) ./ kappa; | |
| 2225 sigma(1) = 1; | |
| 2226 case (2) | |
| 2227 sigma = ones (p, 1); | |
| 2228 sigma(p) = 1 / kappa; | |
| 2229 case (3) | |
| 2230 factor = kappa^(-1/(p-1)); | |
| 2231 sigma = factor.^[0:p-1]; | |
| 2232 case (4) | |
| 2233 sigma = ones (p, 1) - (0:p-1)'/(p-1)*(1-1/kappa); | |
| 2234 case (5) | |
| 2235 ## In this case cond (A) <= kappa. | |
| 2236 rand ("uniform"); | |
| 2237 sigma = exp (-rand (p, 1) * log (kappa)); | |
| 2238 otherwise | |
|
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|
2239 error ("gallery: unknown MODE '%d' for randsvd matrix.", mode); |
| 16634 | 2240 endswitch |
| 2241 | |
| 2242 ## Convert to diagonal matrix of singular values. | |
| 2243 if (mode < 0) | |
|
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2244 sigma = sigma(p:-1:1); |
|
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2245 endif |
| 16634 | 2246 sigma = diag (sigma); |
| 2247 | |
| 2248 if (posdef) | |
| 2249 ## handle case where KAPPA was negative | |
| 2250 Q = qmult (p); | |
| 2251 A = Q' * sigma * Q; | |
| 2252 A = (A + A') / 2; # Ensure matrix is symmetric. | |
|
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2253 return; |
| 16634 | 2254 endif |
| 2255 | |
| 2256 if (m != n) | |
| 2257 ## Expand to m-by-n diagonal matrix | |
| 2258 sigma(m, n) = 0; | |
|
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2259 endif |
| 16634 | 2260 |
| 2261 if (kl == 0 && ku == 0) | |
| 2262 ## Diagonal matrix requested - nothing more to do. | |
| 2263 A = sigma; | |
| 2264 else | |
| 2265 ## A = U*sigma*V, where U, V are random orthogonal matrices from the | |
| 2266 ## Haar distribution. | |
| 2267 A = qmult (sigma'); | |
| 2268 A = qmult (A'); | |
| 2269 | |
| 2270 if (kl < n-1 || ku < n-1) | |
| 2271 ## Bandwidth reduction | |
| 2272 A = bandred (A, kl, ku); | |
| 2273 endif | |
| 2274 endif | |
| 2275 endfunction | |
| 2276 | |
| 2277 function A = redheff (n) | |
| 2278 ## REDHEFF A (0,1) matrix of Redheffer associated with the Riemann hypothesis. | |
|
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2279 ## A = REDHEFF(N) is an N-by-N matrix of 0s and 1s defined by |
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2280 ## A(i,j) = 1 if j = 1 or if i divides j, |
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2281 ## A(i,j) = 0 otherwise. |
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2282 ## It has N - FLOOR(LOG2(N)) - 1 eigenvalues equal to 1, |
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2283 ## a real eigenvalue (the spectral radius) approximately SQRT(N), |
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2284 ## a negative eigenvalue approximately -SQRT(N), |
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2285 ## and the remaining eigenvalues are provably ``small''. |
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2286 ## Barrett and Jarvis (1992) conjecture that |
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2287 ## ``the small eigenvalues all lie inside the unit circle |
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2288 ## ABS(Z) = 1'', |
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2289 ## and a proof of this conjecture, together with a proof that some |
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2290 ## eigenvalue tends to zero as N tends to infinity, would yield |
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2291 ## a new proof of the prime number theorem. |
|
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2292 ## The Riemann hypothesis is true if and only if |
|
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2293 ## DET(A) = O( N^(1/2+epsilon) ) for every epsilon > 0 |
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2294 ## (`!' denotes factorial). |
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2295 ## See also RIEMANN. |
| 16634 | 2296 ## |
|
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2297 ## Reference: |
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2298 ## W.W. Barrett and T.J. Jarvis, |
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2299 ## Spectral Properties of a Matrix of Redheffer, |
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2300 ## Linear Algebra and Appl., 162 (1992), pp. 673-683. |
| 16634 | 2301 |
| 2302 if (nargin != 1) | |
| 2303 error ("gallery: 1 argument is required for redheff matrix."); | |
| 2304 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2305 error ("gallery: N must be an integer for redheff matrix."); | |
| 2306 endif | |
| 2307 | |
| 2308 i = (1:n)' * ones (1, n); | |
| 2309 A = ! rem (i', i); | |
| 2310 A(:,1) = ones (n, 1); | |
| 2311 endfunction | |
| 2312 | |
| 2313 function A = riemann (n) | |
| 2314 ## RIEMANN A matrix associated with the Riemann hypothesis. | |
|
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2315 ## A = RIEMANN(N) is an N-by-N matrix for which the |
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2316 ## Riemann hypothesis is true if and only if |
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2317 ## DET(A) = O( N! N^(-1/2+epsilon) ) for every epsilon > 0 |
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2318 ## (`!' denotes factorial). |
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2319 ## A = B(2:N+1, 2:N+1), where |
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2320 ## B(i,j) = i-1 if i divides j and -1 otherwise. |
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2321 ## Properties include, with M = N+1: |
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2322 ## Each eigenvalue E(i) satisfies ABS(E(i)) <= M - 1/M. |
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2323 ## i <= E(i) <= i+1 with at most M-SQRT(M) exceptions. |
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2324 ## All integers in the interval (M/3, M/2] are eigenvalues. |
| 16634 | 2325 ## |
|
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2326 ## See also REDHEFF. |
| 16634 | 2327 ## |
|
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2328 ## Reference: |
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2329 ## F. Roesler, Riemann's hypothesis as an eigenvalue problem, |
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|
2330 ## Linear Algebra and Appl., 81 (1986), pp. 153-198. |
| 16634 | 2331 |
| 2332 if (nargin != 1) | |
| 2333 error ("gallery: 1 argument is required for riemann matrix."); | |
| 2334 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2335 error ("gallery: N must be an integer for riemann matrix."); | |
| 2336 endif | |
| 2337 | |
|
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|
2338 n += 1; |
| 16634 | 2339 i = (2:n)' * ones (1, n-1); |
| 2340 j = i'; | |
| 2341 A = i .* (! rem (j, i)) - ones (n-1); | |
| 2342 endfunction | |
| 2343 | |
| 2344 function A = ris (n) | |
| 2345 ## NOTE: this function was named dingdong in the original Test Matrix Toolbox | |
| 2346 ## RIS Dingdong matrix - a symmetric Hankel matrix. | |
|
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2347 ## A = RIS(N) is the symmetric N-by-N Hankel matrix with |
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|
2348 ## A(i,j) = 0.5/(N-i-j+1.5). |
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2349 ## The eigenvalues of A cluster around PI/2 and -PI/2. |
| 16634 | 2350 ## |
|
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|
2351 ## Invented by F.N. Ris. |
| 16634 | 2352 ## |
|
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2353 ## Reference: |
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2354 ## J.C. Nash, Compact Numerical Methods for Computers: Linear |
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|
2355 ## Algebra and Function Minimisation, second edition, Adam Hilger, |
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|
2356 ## Bristol, 1990 (Appendix 1). |
| 16634 | 2357 |
| 2358 if (nargin != 1) | |
| 2359 error ("gallery: 1 argument is required for ris matrix."); | |
| 2360 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2361 error ("gallery: N must be an integer for ris matrix."); | |
| 2362 endif | |
| 2363 | |
| 2364 p = -2*(1:n) + (n+1.5); | |
| 2365 A = cauchy (p); | |
| 2366 endfunction | |
| 2367 | |
| 2368 function A = smoke (n, k = 0) | |
| 2369 ## SMOKE Smoke matrix - complex, with a `smoke ring' pseudospectrum. | |
|
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2370 ## SMOKE(N) is an N-by-N matrix with 1s on the |
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|
2371 ## superdiagonal, 1 in the (N,1) position, and powers of |
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|
2372 ## roots of unity along the diagonal. |
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2373 ## SMOKE(N, 1) is the same except for a zero (N,1) element. |
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|
2374 ## The eigenvalues of SMOKE(N, 1) are the N'th roots of unity; |
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|
2375 ## those of SMOKE(N) are the N'th roots of unity times 2^(1/N). |
| 16634 | 2376 ## |
|
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|
2377 ## Try PS(SMOKE(32)). For SMOKE(N, 1) the pseudospectrum looks |
|
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2378 ## like a sausage folded back on itself. |
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|
2379 ## GERSH(SMOKE(N, 1)) is interesting. |
| 16634 | 2380 ## |
|
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2381 ## Reference: |
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2382 ## L. Reichel and L.N. Trefethen, Eigenvalues and pseudo-eigenvalues of |
|
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|
2383 ## Toeplitz matrices, Linear Algebra and Appl., 162-164:153-185, 1992. |
| 16634 | 2384 |
| 2385 if (nargin < 1 || nargin > 2) | |
| 2386 error ("gallery: 1 to 2 arguments are required for smoke matrix."); | |
| 2387 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2388 error ("gallery: N must be an integer for smoke matrix."); | |
| 2389 elseif (! isnumeric (n) || ! isscalar (n)) | |
| 2390 error ("gallery: K must be a numeric scalar for smoke matrix."); | |
| 2391 endif | |
| 2392 | |
|
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|
2393 w = exp (2*pi*i/n); |
|
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|
2394 A = diag ( [w.^(1:n-1) 1] ) + diag (ones (n-1,1), 1); |
| 16634 | 2395 |
| 2396 switch (k) | |
| 2397 case (0), A(n,1) = 1; | |
| 2398 case (1), # do nothing | |
| 2399 otherwise, | |
| 2400 error ("gallery: K must have a value of 0 or 1 for smoke matrix."); | |
| 2401 endswitch | |
| 2402 endfunction | |
| 2403 | |
| 2404 function T = toeppd (n, m = n, w = rand (m,1), theta = rand (m,1)) | |
| 2405 ## NOTE: this function was named pdtoep in the original Test Matrix Toolbox | |
| 2406 ## TOEPPD Symmetric positive definite Toeplitz matrix. | |
|
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|
2407 ## TOEPPD(N, M, W, THETA) is an N-by-N symmetric positive (semi-) |
|
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|
2408 ## definite (SPD) Toeplitz matrix, comprised of the sum of M rank 2 |
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|
2409 ## (or, for certain THETA, rank 1) SPD Toeplitz matrices. |
|
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|
2410 ## Specifically, |
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|
2411 ## T = W(1)*T(THETA(1)) + ... + W(M)*T(THETA(M)), |
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|
2412 ## where T(THETA(k)) has (i,j) element COS(2*PI*THETA(k)*(i-j)). |
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|
2413 ## Defaults: M = N, W = RAND(M,1), THETA = RAND(M,1). |
| 16634 | 2414 ## |
|
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2415 ## Reference: |
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|
2416 ## G. Cybenko and C.F. Van Loan, Computing the minimum eigenvalue of |
|
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|
2417 ## a symmetric positive definite Toeplitz matrix, SIAM J. Sci. Stat. |
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|
2418 ## Comput., 7 (1986), pp. 123-131. |
| 16634 | 2419 |
| 2420 if (nargin < 1 || nargin > 4) | |
| 2421 error ("gallery: 1 to 4 arguments are required for toeppd matrix."); | |
| 2422 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2423 error ("gallery: N must be a numeric integer for toeppd matrix."); | |
| 2424 elseif (! isnumeric (m) || ! isscalar (m) || fix (m) != m) | |
| 2425 error ("gallery: M must be a numeric integer for toeppd matrix."); | |
| 2426 elseif (numel (w) != m || numel (theta) != m) | |
| 2427 error ("gallery: W and THETA must be vectors of length M for toeppd matrix."); | |
| 2428 endif | |
| 2429 | |
| 2430 T = zeros (n); | |
| 2431 E = 2*pi * ((1:n)' * ones (1, n) - ones (n, 1) * (1:n)); | |
| 2432 | |
| 2433 for i = 1:m | |
|
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|
2434 T += w(i) * cos (theta(i)*E); |
| 16634 | 2435 endfor |
| 2436 endfunction | |
| 2437 | |
| 2438 function P = toeppen (n, a = 1, b = -10, c = 0, d = 10, e = 1) | |
| 2439 ## NOTE: this function was named pentoep in the original Test Matrix Toolbox | |
| 2440 ## TOEPPEN Pentadiagonal Toeplitz matrix (sparse). | |
|
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|
2441 ## P = TOEPPEN(N, A, B, C, D, E) is the N-by-N pentadiagonal |
|
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|
2442 ## Toeplitz matrix with diagonals composed of the numbers |
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|
2443 ## A =: P(3,1), B =: P(2,1), C =: P(1,1), D =: P(1,2), E =: P(1,3). |
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|
2444 ## Default: (A,B,C,D,E) = (1,-10,0,10,1) (a matrix of Rutishauser). |
|
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|
2445 ## This matrix has eigenvalues lying approximately on |
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|
2446 ## the line segment 2*cos(2*t) + 20*i*sin(t). |
| 16634 | 2447 ## |
|
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|
2448 ## Interesting plots are |
|
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|
2449 ## PS(FULL(TOEPPEN(32,0,1,0,0,1/4))) - `triangle' |
|
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|
2450 ## PS(FULL(TOEPPEN(32,0,1/2,0,0,1))) - `propeller' |
|
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|
2451 ## PS(FULL(TOEPPEN(32,0,1/2,1,1,1))) - `fish' |
| 16634 | 2452 ## |
|
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|
2453 ## References: |
|
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|
2454 ## R.M. Beam and R.F. Warming, The asymptotic spectra of |
|
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|
2455 ## banded Toeplitz and quasi-Toeplitz matrices, SIAM J. Sci. |
|
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|
2456 ## Comput. 14 (4), 1993, pp. 971-1006. |
|
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|
2457 ## H. Rutishauser, On test matrices, Programmation en Mathematiques |
|
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|
2458 ## Numeriques, Editions Centre Nat. Recherche Sci., Paris, 165, |
|
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|
2459 ## 1966, pp. 349-365. |
| 16634 | 2460 |
| 2461 if (nargin < 1 || nargin > 6) | |
| 2462 error ("gallery: 1 to 6 arguments are required for toeppen matrix."); | |
| 2463 elseif (! isnumeric (n) || ! isscalar (n) || fix (n) != n) | |
| 2464 error ("gallery: N must be a numeric integer for toeppen matrix."); | |
|
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|
2465 elseif (any (! cellfun ("isnumeric", {a b c d e})) || any (cellfun ("numel", {a b c d e}) != 1)) |
| 16634 | 2466 error ("gallery: A, B, C, D and E must be numeric scalars for toeppen matrix."); |
| 2467 endif | |
| 2468 | |
| 2469 P = spdiags ([a*ones(n,1) b*ones(n,1) c*ones(n,1) d*ones(n,1) e*ones(n,1)], | |
| 2470 -2:2, n, n); | |
| 2471 endfunction | |
| 2472 | |
| 2473 function T = tridiag (n, x = -1, y = 2, z = -1) | |
| 2474 ## TRIDIAG Tridiagonal matrix (sparse). | |
|
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|
2475 ## TRIDIAG(X, Y, Z) is the tridiagonal matrix with subdiagonal X, |
|
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|
2476 ## diagonal Y, and superdiagonal Z. |
|
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|
2477 ## X and Z must be vectors of dimension one less than Y. |
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|
2478 ## Alternatively TRIDIAG(N, C, D, E), where C, D, and E are all |
|
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|
2479 ## scalars, yields the Toeplitz tridiagonal matrix of order N |
|
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|
2480 ## with subdiagonal elements C, diagonal elements D, and superdiagonal |
|
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|
2481 ## elements E. This matrix has eigenvalues (Todd 1977) |
|
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|
2482 ## D + 2*SQRT(C*E)*COS(k*PI/(N+1)), k=1:N. |
|
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|
2483 ## TRIDIAG(N) is the same as TRIDIAG(N,-1,2,-1), which is |
|
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|
2484 ## a symmetric positive definite M-matrix (the negative of the |
|
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|
2485 ## second difference matrix). |
| 16634 | 2486 ## |
|
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|
2487 ## References: |
|
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|
2488 ## J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, |
|
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|
2489 ## Birkhauser, Basel, and Academic Press, New York, 1977, p. 155. |
|
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|
2490 ## D.E. Rutherford, Some continuant determinants arising in physics and |
|
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|
2491 ## chemistry---II, Proc. Royal Soc. Edin., 63, A (1952), pp. 232-241. |
| 16634 | 2492 |
| 2493 if (nargin != 1 && nargin != 3 && nargin != 4) | |
| 2494 error ("gallery: 1, 3, or 4 arguments are required for tridiag matrix."); | |
| 2495 elseif (nargin == 3) | |
| 2496 z = y; | |
| 2497 y = x; | |
| 2498 x = n; | |
| 2499 endif | |
| 2500 | |
| 2501 ## Force column vectors | |
| 2502 x = x(:); | |
| 2503 y = y(:); | |
| 2504 z = z(:); | |
| 2505 | |
| 2506 if (isscalar (x) && isscalar (y) && isscalar (z)) | |
| 2507 x *= ones (n-1, 1); | |
| 2508 z *= ones (n-1, 1); | |
| 2509 y *= ones (n, 1); | |
| 2510 elseif (numel (y) != numel (x) + 1) | |
| 2511 error ("gallery: X must have one element less than Y for tridiag matrix."); | |
| 2512 elseif (numel (y) != numel (z) + 1) | |
| 2513 error ("gallery: Z must have one element less than Y for tridiag matrix."); | |
| 2514 endif | |
| 2515 | |
| 2516 ## T = diag (x, -1) + diag (y) + diag (z, 1); # For non-sparse matrix. | |
| 2517 n = numel (y); | |
| 2518 T = spdiags ([[x;0] y [0;z]], -1:1, n, n); | |
| 2519 endfunction | |
| 2520 | |
|
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|
2521 function t = triw (n, alpha = -1, k = n(end) - 1) |
| 16634 | 2522 ## TRIW Upper triangular matrix discussed by Wilkinson and others. |
|
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|
2523 ## TRIW(N, ALPHA, K) is the upper triangular matrix with ones on |
|
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|
2524 ## the diagonal and ALPHAs on the first K >= 0 superdiagonals. |
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|
2525 ## N may be a 2-vector, in which case the matrix is N(1)-by-N(2) and |
|
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|
2526 ## upper trapezoidal. |
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|
2527 ## Defaults: ALPHA = -1, |
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|
2528 ## K = N - 1 (full upper triangle). |
|
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|
2529 ## TRIW(N) is a matrix discussed by Kahan, Golub and Wilkinson. |
| 16634 | 2530 ## |
|
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|
2531 ## Ostrowski (1954) shows that |
|
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|
2532 ## COND(TRIW(N,2)) = COT(PI/(4*N))^2, |
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|
2533 ## and for large ABS(ALPHA), |
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|
2534 ## COND(TRIW(N,ALPHA)) is approximately ABS(ALPHA)^N*SIN(PI/(4*N-2)). |
| 16634 | 2535 ## |
|
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|
2536 ## Adding -2^(2-N) to the (N,1) element makes TRIW(N) singular, |
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|
2537 ## as does adding -2^(1-N) to all elements in the first column. |
| 16634 | 2538 ## |
|
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|
2539 ## References: |
|
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|
2540 ## G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the |
|
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|
2541 ## computation of the Jordan canonical form, SIAM Review, |
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|
2542 ## 18(4), 1976, pp. 578-619. |
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|
2543 ## W. Kahan, Numerical linear algebra, Canadian Math. Bulletin, |
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|
2544 ## 9 (1966), pp. 757-801. |
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|
2545 ## A.M. Ostrowski, On the spectrum of a one-parametric family of |
|
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|
2546 ## matrices, J. Reine Angew. Math., 193 (3/4), 1954, pp. 143-160. |
|
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|
2547 ## J.H. Wilkinson, Singular-value decomposition---basic aspects, |
|
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|
2548 ## in D.A.H. Jacobs, ed., Numerical Software---Needs and Availability, |
|
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|
2549 ## Academic Press, London, 1978, pp. 109-135. |
| 16634 | 2550 |
| 2551 if (nargin < 1 || nargin > 3) | |
| 2552 error ("gallery: 1 to 3 arguments are required for triw matrix."); | |
| 2553 elseif (! isnumeric (n) || all (numel (n) != [1 2])) | |
| 2554 error ("gallery: N must be a 1 or 2 elements vector for triw matrix."); | |
| 2555 elseif (! isscalar (alpha)) | |
| 2556 error ("gallery: ALPHA must be a scalar for triw matrix."); | |
|
18880
3d33fe79816c
gallery.m: Return correct matrix if 3rd argument is not specified.
Rik <rik@octave.org>
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18851
diff
changeset
|
2557 elseif (! isscalar (k) || ! isnumeric (k) || fix (k) != k || k < 0) |
|
3d33fe79816c
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Rik <rik@octave.org>
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18851
diff
changeset
|
2558 error ("gallery: K must be a numeric integer >= 0 for triw matrix."); |
| 16634 | 2559 endif |
| 2560 | |
| 2561 m = n(1); # Parameter n specifies dimension: m-by-n. | |
| 2562 n = n(end); | |
| 2563 | |
| 2564 t = tril (eye (m, n) + alpha * triu (ones (m, n), 1), k); | |
| 2565 endfunction | |
| 2566 | |
|
16978
00379f9f8773
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|
2567 function A = uniformdata (varargin) |
|
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|
2568 |
|
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|
2569 if (nargin < 2) |
|
16979
9aa293e00475
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Rik <rik@octave.org>
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diff
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|
2570 error ("gallery: At least 2 arguments required for uniformdata matrix."); |
|
16978
00379f9f8773
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|
2571 endif |
|
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|
2572 if (isnumeric (varargin{end})) |
|
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|
2573 jidx = varargin{end}; |
|
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|
2574 svec = [varargin{:}]; |
|
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|
2575 varargin(end) = []; |
|
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|
2576 elseif (ischar (varargin{end})) |
|
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Rik <rik@octave.org>
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diff
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|
2577 if (nargin < 3) |
|
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9aa293e00475
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diff
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|
2578 error (["gallery: CLASS argument requires 3 inputs " ... |
|
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diff
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|
2579 "for uniformdata matrix."]); |
|
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|
2580 endif |
|
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diff
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|
2581 jidx = varargin{end-1}; |
|
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Rik <rik@octave.org>
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diff
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|
2582 svec = [varargin{1:end-1}]; |
|
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Rik <rik@octave.org>
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|
2583 varargin(end-1) = []; |
|
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|
2584 else |
|
16979
9aa293e00475
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|
2585 error (["gallery: J must be an integer in the range [0, 2^32-1] " ... |
|
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|
2586 "for uniformdata matrix"]); |
|
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|
2587 endif |
|
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diff
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|
2588 |
|
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|
2589 if (! (isnumeric (jidx) && isscalar (jidx) |
|
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|
2590 && jidx == fix (jidx) |
|
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|
2591 && jidx >= 0 && jidx <= 0xFFFFFFFF)) |
|
16979
9aa293e00475
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|
2592 error (["gallery: J must be an integer in the range [0, 2^32-1] " ... |
|
9aa293e00475
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diff
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|
2593 "for uniformdata matrix"]); |
|
16978
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|
2594 endif |
|
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|
2595 |
|
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|
2596 ## Save and restore random state. Initialization done so that reproducible |
|
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|
2597 ## data is available from gallery depending on the jidx and size vector. |
|
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|
2598 randstate = rand ("state"); |
|
16978
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|
2599 unwind_protect |
|
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|
2600 rand ("state", svec); |
|
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|
2601 A = rand (varargin{:}); |
|
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|
2602 unwind_protect_cleanup |
|
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|
2603 rand ("state", randstate); |
|
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|
2604 end_unwind_protect |
|
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diff
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|
2605 |
|
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|
2606 endfunction |
|
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|
2607 |
| 16634 | 2608 function A = wathen (nx, ny, k = 0) |
|
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|
2609 ## WATHEN returns the Wathen matrix. |
| 16634 | 2610 ## |
|
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|
2611 ## Discussion: |
| 16634 | 2612 ## |
|
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|
2613 ## The Wathen matrix is a finite element matrix which is sparse. |
| 16634 | 2614 ## |
|
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|
2615 ## The entries of the matrix depend in part on a physical quantity |
|
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|
2616 ## related to density. That density is here assigned random values between |
|
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|
2617 ## 0 and 100. |
| 16634 | 2618 ## |
|
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|
2619 ## A = WATHEN ( NX, NY ) is a sparse random N-by-N finite element matrix |
|
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|
2620 ## where N = 3*NX*NY + 2*NX + 2*NY + 1. |
| 16634 | 2621 ## |
|
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|
2622 ## A is the consistent mass matrix for a regular NX-by-NY |
|
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|
2623 ## grid of 8-node (serendipity) elements in 2 space dimensions. |
| 16634 | 2624 ## |
|
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|
2625 ## Here is an illustration for NX = 3, NX = 2: |
| 16634 | 2626 ## |
|
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|
2627 ## 23-24-25-26-27-28-29 |
|
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|
2628 ## | | | | |
|
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|
2629 ## 19 20 21 22 |
|
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|
2630 ## | | | | |
|
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|
2631 ## 12-13-14-15-16-17-18 |
|
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|
2632 ## | | | | |
|
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|
2633 ## 8 9 10 11 |
|
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|
2634 ## | | | | |
|
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|
2635 ## 1--2--3--4--5--6--7 |
| 16634 | 2636 ## |
|
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|
2637 ## For this example, the total number of nodes is, as expected, |
| 16634 | 2638 ## |
|
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|
2639 ## N = 3 * 3 * 2 + 2 * 2 + 2 * 3 + 1 = 29. |
| 16634 | 2640 ## |
|
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|
2641 ## A is symmetric positive definite for any (positive) values of |
|
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|
2642 ## the density, RHO(NX,NY), which is chosen randomly in this routine. |
| 16634 | 2643 ## |
|
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|
2644 ## In particular, if D = DIAG(DIAG(A)), then |
|
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|
2645 ## 0.25 <= EIG(INV(D)*A) <= 4.5 |
|
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|
2646 ## for any positive integers NX and NY and any densities RHO(NX,NY). |
| 16634 | 2647 ## |
|
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|
2648 ## A = WATHEN ( NX, NY, 1 ) returns the diagonally scaled matrix. |
| 16634 | 2649 ## |
|
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|
2650 ## Modified: |
| 16634 | 2651 ## |
|
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|
2652 ## 17 September 2007 |
| 16634 | 2653 ## |
|
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|
2654 ## Author: |
| 16634 | 2655 ## |
|
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|
2656 ## Nicholas Higham |
| 16634 | 2657 ## |
|
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|
2658 ## Reference: |
| 16634 | 2659 ## |
|
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|
2660 ## Nicholas Higham, |
|
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|
2661 ## Algorithm 694: A Collection of Test Matrices in MATLAB, |
|
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|
2662 ## ACM Transactions on Mathematical Software, |
|
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|
2663 ## Volume 17, Number 3, September 1991, pages 289-305. |
| 16634 | 2664 ## |
|
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|
2665 ## Andrew Wathen, |
|
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|
2666 ## Realistic eigenvalue bounds for the Galerkin mass matrix, |
|
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|
2667 ## IMA Journal of Numerical Analysis, |
|
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|
2668 ## Volume 7, 1987, pages 449-457. |
| 16634 | 2669 ## |
|
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|
2670 ## Parameters: |
| 16634 | 2671 ## |
|
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|
2672 ## Input, integer NX, NY, the number of elements in the X and Y directions |
|
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|
2673 ## of the finite element grid. NX and NY must each be at least 1. |
| 16634 | 2674 ## |
|
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2675 ## Optional input, integer K, is used to request that the diagonally scaled |
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|
2676 ## version of the matrix be returned. This happens if K is specified with |
|
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|
2677 ## the value 1. |
| 16634 | 2678 ## |
|
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2679 ## Output, sparse real A(N,N), the matrix. The dimension N is determined by |
|
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|
2680 ## NX and NY, as described above. A is stored in the MATLAB sparse matrix |
|
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|
2681 ## format. |
| 16634 | 2682 |
| 2683 if (nargin < 2 || nargin > 3) | |
| 2684 error ("gallery: 2 or 3 arguments are required for wathen matrix."); | |
| 2685 elseif (! isnumeric (nx) || ! isscalar (nx) || nx < 1) | |
| 2686 error ("gallery: NX must be a positive scalar for wathen matrix."); | |
| 2687 elseif (! isnumeric (ny) || ! isscalar (ny) || ny < 1) | |
| 2688 error ("gallery: NY must be a positive scalar for wathen matrix."); | |
| 2689 elseif (! isscalar (k)) | |
| 2690 error ("gallery: K must be a scalar for wathen matrix."); | |
| 2691 endif | |
| 2692 | |
| 2693 e1 = [ 6 -6 2 -8 | |
| 2694 -6 32 -6 20 | |
| 2695 2 -6 6 -6 | |
| 2696 -8 20 -6 32 ]; | |
| 2697 | |
| 2698 e2 = [ 3 -8 2 -6 | |
| 2699 -8 16 -8 20 | |
| 2700 2 -8 3 -8 | |
| 2701 -6 20 -8 16 ]; | |
| 2702 | |
| 2703 e = [ e1 e2 | |
| 2704 e2' e1] / 45; | |
| 2705 | |
| 2706 n = 3*nx*ny + 2*nx + 2*ny + 1; | |
| 2707 | |
| 2708 A = sparse (n, n); | |
| 2709 | |
| 2710 rho = 100 * rand (nx, ny); | |
| 2711 | |
| 2712 for j = 1:ny | |
| 2713 for i = 1:nx | |
|
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2714 ## |
| 16634 | 2715 ## For the element (I,J), determine the indices of the 8 nodes. |
|
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|
2716 ## |
| 16634 | 2717 nn(1) = 3*j*nx + 2*i + 2*j + 1; |
| 2718 nn(2) = nn(1) - 1; | |
| 2719 nn(3) = nn(2) - 1; | |
| 2720 nn(4) = (3*j - 1) * nx + 2*j + i - 1; | |
| 2721 nn(5) = 3 * (j-1) * nx + 2*i + 2*j - 3; | |
| 2722 nn(6) = nn(5) + 1; | |
| 2723 nn(7) = nn(6) + 1; | |
| 2724 nn(8) = nn(4) + 1; | |
| 2725 | |
| 2726 em = e * rho(i,j); | |
| 2727 | |
| 2728 for krow = 1:8 | |
| 2729 for kcol = 1:8 | |
| 2730 A(nn(krow),nn(kcol)) = A(nn(krow),nn(kcol)) + em(krow,kcol); | |
| 2731 endfor | |
| 2732 endfor | |
| 2733 | |
| 2734 endfor | |
| 2735 endfor | |
| 2736 | |
| 2737 ## If requested, return A with diagonal scaling. | |
| 2738 if (k) | |
| 2739 A = diag (diag (A)) \ A; | |
| 2740 endif | |
| 2741 endfunction | |
| 2742 | |
| 2743 function [A, b] = wilk (n) | |
| 2744 ## WILK Various specific matrices devised/discussed by Wilkinson. | |
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2745 ## [A, b] = WILK(N) is the matrix or system of order N. |
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2746 ## N = 3: upper triangular system Ux=b illustrating inaccurate solution. |
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2747 ## N = 4: lower triangular system Lx=b, ill-conditioned. |
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2748 ## N = 5: HILB(6)(1:5,2:6)*1.8144. Symmetric positive definite. |
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2749 ## N = 21: W21+, tridiagonal. Eigenvalue problem. |
| 16634 | 2750 ## |
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2751 ## References: |
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2752 ## J.H. Wilkinson, Error analysis of direct methods of matrix inversion, |
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2753 ## J. Assoc. Comput. Mach., 8 (1961), pp. 281-330. |
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2754 ## J.H. Wilkinson, Rounding Errors in Algebraic Processes, Notes on Applied |
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2755 ## Science No. 32, Her Majesty's Stationery Office, London, 1963. |
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2756 ## J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University |
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2757 ## Press, 1965. |
| 16634 | 2758 |
| 2759 if (nargin != 1) | |
| 2760 error ("gallery: 1 argument is required for wilk matrix."); | |
| 2761 elseif (! isnumeric (n) || ! isscalar (n)) | |
| 2762 error ("gallery: N must be a numeric scalar for wilk matrix."); | |
| 2763 endif | |
| 2764 | |
| 2765 if (n == 3) | |
| 2766 ## Wilkinson (1961) p.323. | |
| 2767 A = [ 1e-10 0.9 -0.4 | |
| 2768 0 0.9 -0.4 | |
| 2769 0 0 1e-10 ]; | |
| 2770 | |
| 2771 b = [ 0 | |
| 2772 0 | |
| 2773 1]; | |
| 2774 | |
| 2775 elseif (n == 4) | |
| 2776 ## Wilkinson (1963) p.105. | |
| 2777 A = [0.9143e-4 0 0 0 | |
| 2778 0.8762 0.7156e-4 0 0 | |
| 2779 0.7943 0.8143 0.9504e-4 0 | |
| 2780 0.8017 0.6123 0.7165 0.7123e-4]; | |
| 2781 | |
| 2782 b = [0.6524 | |
| 2783 0.3127 | |
| 2784 0.4186 | |
| 2785 0.7853]; | |
| 2786 | |
| 2787 elseif (n == 5) | |
| 2788 ## Wilkinson (1965), p.234. | |
| 2789 A = hilb (6); | |
| 2790 A = A(1:5, 2:6) * 1.8144; | |
| 2791 | |
| 2792 elseif (n == 21) | |
| 2793 ## Wilkinson (1965), p.308. | |
| 2794 E = diag (ones (n-1, 1), 1); | |
| 2795 m = (n-1)/2; | |
| 2796 A = diag (abs (-m:m)) + E + E'; | |
| 2797 | |
| 2798 else | |
|
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2799 error ("gallery: unknown N '%d' for wilk matrix.", n); |
| 16634 | 2800 endif |
| 2801 endfunction | |
| 2802 | |
| 2803 ## NOTE: bandred is part of the Test Matrix Toolbox and is used by randsvd() | |
| 2804 function A = bandred (A, kl, ku) | |
| 2805 ## BANDRED Band reduction by two-sided unitary transformations. | |
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2806 ## B = BANDRED(A, KL, KU) is a matrix unitarily equivalent to A |
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2807 ## with lower bandwidth KL and upper bandwidth KU |
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2808 ## (i.e. B(i,j) = 0 if i > j+KL or j > i+KU). |
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2809 ## The reduction is performed using Householder transformations. |
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2810 ## If KU is omitted it defaults to KL. |
| 16634 | 2811 ## |
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2812 ## Called by RANDSVD. |
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2813 ## This is a `standard' reduction. Cf. reduction to bidiagonal form |
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2814 ## prior to computing the SVD. This code is a little wasteful in that |
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2815 ## it computes certain elements which are immediately set to zero! |
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2816 ## |
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2817 ## Reference: |
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2818 ## G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, |
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2819 ## Johns Hopkins University Press, Baltimore, Maryland, 1989. |
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2820 ## Section 5.4.3. |
| 16634 | 2821 |
| 2822 ## Check for special case where order of left/right transformations matters. | |
| 2823 ## Easiest approach is to work on the transpose, flipping back at the end. | |
| 2824 flip = false; | |
| 2825 if (ku == 0) | |
| 2826 flip = true; | |
| 2827 A = A'; | |
| 2828 [ku, kl] = deal (kl, ku); | |
| 2829 endif | |
| 2830 | |
| 2831 [m, n] = size (A); | |
| 2832 | |
| 2833 for j = 1:min (min (m, n), max (m-kl-1, n-ku-1)) | |
| 2834 if (j+kl+1 <= m) | |
| 2835 [v, beta] = house (A(j+kl:m,j)); | |
| 2836 temp = A(j+kl:m,j:n); | |
| 2837 A(j+kl:m,j:n) = temp - beta*v*(v'*temp); | |
| 2838 A(j+kl+1:m,j) = zeros (m-j-kl, 1); | |
| 2839 endif | |
| 2840 | |
| 2841 if (j+ku+1 <= n) | |
| 2842 [v, beta] = house (A(j,j+ku:n)'); | |
| 2843 temp = A(j:m,j+ku:n); | |
| 2844 A(j:m,j+ku:n) = temp - beta*(temp*v)*v'; | |
| 2845 A(j,j+ku+1:n) = zeros (1, n-j-ku); | |
| 2846 endif | |
| 2847 endfor | |
| 2848 | |
| 2849 if (flip) | |
| 2850 A = A'; | |
| 2851 endif | |
| 2852 endfunction | |
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2853 |
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2854 |
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2855 ## BIST testing for just a few functions to verify that the main gallery |
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2856 ## dispatch function works. |
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2857 %assert (gallery ("clement", 3), [0 1 0; 2 0 2; 0 1 0]) |
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2858 %assert (gallery ("invhess", 2), [1 -1; 1 2]) |
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2859 |
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2860 ## Test input validation of main dispatch function only |
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2861 %!error gallery () |
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2862 %!error <NAME must be a string> gallery (123) |
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2863 %!error <matrix binomial not implemented> gallery ("binomial") |
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2864 %!error <unknown matrix with NAME foobar> gallery ("foobar") |
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2865 |
|
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2866 %!assert (gallery ("minij", 4), [1 1 1 1; 1 2 2 2; 1 2 3 3; 1 2 3 4]) |
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2867 %!assert (gallery ("minij", 1), 1) |
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2868 %!assert (gallery ("minij", 0), []) |
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2869 %!assert (gallery ("minij", -1), []) |