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annotate doc/interpreter/matrix.txi @ 12498:294c0927c762
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| author | Rik <octave@nomad.inbox5.com> |
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| date | Thu, 03 Mar 2011 09:46:00 -0800 |
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| 11523 | 1 @c Copyright (C) 1996-2011 John W. Eaton |
| 7018 | 2 @c |
| 3 @c This file is part of Octave. | |
| 4 @c | |
| 5 @c Octave is free software; you can redistribute it and/or modify it | |
| 6 @c under the terms of the GNU General Public License as published by the | |
| 7 @c Free Software Foundation; either version 3 of the License, or (at | |
| 8 @c your option) any later version. | |
| 9 @c | |
| 10 @c Octave is distributed in the hope that it will be useful, but WITHOUT | |
| 11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
| 12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
| 13 @c for more details. | |
| 14 @c | |
| 15 @c You should have received a copy of the GNU General Public License | |
| 16 @c along with Octave; see the file COPYING. If not, see | |
| 17 @c <http://www.gnu.org/licenses/>. | |
| 3294 | 18 |
| 4167 | 19 @node Matrix Manipulation |
| 3294 | 20 @chapter Matrix Manipulation |
| 21 | |
| 22 There are a number of functions available for checking to see if the | |
| 23 elements of a matrix meet some condition, and for rearranging the | |
| 24 elements of a matrix. For example, Octave can easily tell you if all | |
| 25 the elements of a matrix are finite, or are less than some specified | |
| 26 value. Octave can also rotate the elements, extract the upper- or | |
| 27 lower-triangular parts, or sort the columns of a matrix. | |
| 28 | |
| 29 @menu | |
| 30 * Finding Elements and Checking Conditions:: | |
| 31 * Rearranging Matrices:: | |
| 6550 | 32 * Applying a Function to an Array:: |
| 3294 | 33 * Special Utility Matrices:: |
| 34 * Famous Matrices:: | |
| 35 @end menu | |
| 36 | |
| 4167 | 37 @node Finding Elements and Checking Conditions |
| 3294 | 38 @section Finding Elements and Checking Conditions |
| 39 | |
| 40 The functions @code{any} and @code{all} are useful for determining | |
| 41 whether any or all of the elements of a matrix satisfy some condition. | |
| 42 The @code{find} function is also useful in determining which elements of | |
| 43 a matrix meet a specified condition. | |
| 44 | |
| 3369 | 45 @DOCSTRING(any) |
| 3294 | 46 |
| 3369 | 47 @DOCSTRING(all) |
| 3294 | 48 |
| 49 Since the comparison operators (@pxref{Comparison Ops}) return matrices | |
| 50 of ones and zeros, it is easy to test a matrix for many things, not just | |
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51 whether the elements are nonzero. For example, |
| 3294 | 52 |
| 53 @example | |
| 54 @group | |
| 55 all (all (rand (5) < 0.9)) | |
| 56 @result{} 0 | |
| 57 @end group | |
| 58 @end example | |
| 59 | |
| 60 @noindent | |
| 61 tests a random 5 by 5 matrix to see if all of its elements are less | |
| 62 than 0.9. | |
| 63 | |
| 64 Note that in conditional contexts (like the test clause of @code{if} and | |
| 65 @code{while} statements) Octave treats the test as if you had typed | |
| 66 @code{all (all (condition))}. | |
| 67 | |
| 3428 | 68 @DOCSTRING(xor) |
| 69 | |
| 3369 | 70 @DOCSTRING(diff) |
| 3294 | 71 |
| 3369 | 72 @DOCSTRING(isinf) |
| 3294 | 73 |
| 3369 | 74 @DOCSTRING(isnan) |
| 3294 | 75 |
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76 @DOCSTRING(isfinite) |
| 3294 | 77 |
| 3369 | 78 @DOCSTRING(find) |
| 3294 | 79 |
| 9367 | 80 @DOCSTRING(lookup) |
| 81 | |
| 3428 | 82 @DOCSTRING(common_size) |
| 83 | |
| 4167 | 84 @node Rearranging Matrices |
| 3294 | 85 @section Rearranging Matrices |
| 86 | |
| 3369 | 87 @DOCSTRING(fliplr) |
| 3294 | 88 |
| 3369 | 89 @DOCSTRING(flipud) |
| 3294 | 90 |
| 4869 | 91 @DOCSTRING(flipdim) |
| 92 | |
| 3369 | 93 @DOCSTRING(rot90) |
| 3294 | 94 |
| 4869 | 95 @DOCSTRING(rotdim) |
| 96 | |
| 4845 | 97 @DOCSTRING(cat) |
| 98 | |
| 99 @DOCSTRING(horzcat) | |
| 100 | |
| 101 @DOCSTRING(vertcat) | |
| 102 | |
| 103 @DOCSTRING(permute) | |
| 104 | |
| 105 @DOCSTRING(ipermute) | |
| 106 | |
| 3369 | 107 @DOCSTRING(reshape) |
| 3294 | 108 |
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109 @DOCSTRING(resize) |
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110 |
| 4894 | 111 @DOCSTRING(circshift) |
| 112 | |
| 113 @DOCSTRING(shiftdim) | |
| 114 | |
| 3369 | 115 @DOCSTRING(shift) |
| 3294 | 116 |
| 3369 | 117 @DOCSTRING(sort) |
| 3294 | 118 |
| 6550 | 119 @DOCSTRING(sortrows) |
| 120 | |
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121 @DOCSTRING(issorted) |
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122 |
| 3294 | 123 Since the @code{sort} function does not allow sort keys to be specified, |
| 124 it can't be used to order the rows of a matrix according to the values | |
| 125 of the elements in various columns@footnote{For example, to first sort | |
| 126 based on the values in column 1, and then, for any values that are | |
| 127 repeated in column 1, sort based on the values found in column 2, etc.} | |
| 128 in a single call. Using the second output, however, it is possible to | |
| 129 sort all rows based on the values in a given column. Here's an example | |
| 130 that sorts the rows of a matrix based on the values in the second | |
| 131 column. | |
| 132 | |
| 133 @example | |
| 134 @group | |
| 135 a = [1, 2; 2, 3; 3, 1]; | |
| 136 [s, i] = sort (a (:, 2)); | |
| 137 a (i, :) | |
| 138 @result{} 3 1 | |
| 139 1 2 | |
| 140 2 3 | |
| 141 @end group | |
| 142 @end example | |
| 143 | |
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144 @anchor{doc-triu} |
| 3369 | 145 @DOCSTRING(tril) |
| 3294 | 146 |
| 3369 | 147 @DOCSTRING(vec) |
| 3294 | 148 |
| 3369 | 149 @DOCSTRING(vech) |
| 3294 | 150 |
| 3428 | 151 @DOCSTRING(prepad) |
| 152 | |
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153 @DOCSTRING(postpad) |
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154 |
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155 @DOCSTRING(diag) |
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156 |
| 6550 | 157 @DOCSTRING(blkdiag) |
| 158 | |
| 159 @node Applying a Function to an Array | |
| 160 @section Applying a Function to an Array | |
| 161 | |
| 162 @DOCSTRING(arrayfun) | |
| 163 | |
| 6868 | 164 @DOCSTRING(bsxfun) |
| 165 | |
| 4167 | 166 @node Special Utility Matrices |
| 3294 | 167 @section Special Utility Matrices |
| 168 | |
| 3369 | 169 @DOCSTRING(eye) |
| 3294 | 170 |
| 3369 | 171 @DOCSTRING(ones) |
| 3294 | 172 |
| 3369 | 173 @DOCSTRING(zeros) |
| 3294 | 174 |
| 3920 | 175 @DOCSTRING(repmat) |
| 176 | |
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177 @DOCSTRING(repelems) |
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178 |
| 3369 | 179 @DOCSTRING(rand) |
| 3294 | 180 |
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181 @DOCSTRING(randi) |
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182 |
| 3369 | 183 @DOCSTRING(randn) |
| 3294 | 184 |
| 5730 | 185 @DOCSTRING(rande) |
| 186 | |
| 187 @DOCSTRING(randp) | |
| 188 | |
| 189 @DOCSTRING(randg) | |
| 190 | |
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191 The generators operate in the new or old style together, it is not |
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192 possible to mix the two. Initializing any generator with |
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193 @code{"state"} or @code{"seed"} causes the others to switch to the |
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194 same style for future calls. |
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195 |
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196 The state of each generator is independent and calls to different |
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197 generators can be interleaved without affecting the final result. For |
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198 example, |
| 3294 | 199 |
| 200 @example | |
| 201 @group | |
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202 rand ("state", [11, 22, 33]); |
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203 randn ("state", [44, 55, 66]); |
| 3294 | 204 u = rand (100, 1); |
| 205 n = randn (100, 1); | |
| 206 @end group | |
| 207 @end example | |
| 208 | |
| 209 @noindent | |
| 210 and | |
| 211 | |
| 212 @example | |
| 213 @group | |
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214 rand ("state", [11, 22, 33]); |
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215 randn ("state", [44, 55, 66]); |
| 3294 | 216 u = zeros (100, 1); |
| 217 n = zeros (100, 1); | |
| 218 for i = 1:100 | |
| 219 u(i) = rand (); | |
| 220 n(i) = randn (); | |
| 221 end | |
| 222 @end group | |
| 223 @end example | |
| 224 | |
| 225 @noindent | |
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226 produce equivalent results. When the generators are initialized in |
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227 the old style with @code{"seed"} only @code{rand} and @code{randn} are |
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228 independent, because the old @code{rande}, @code{randg} and |
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229 @code{randp} generators make calls to @code{rand} and @code{randn}. |
| 3294 | 230 |
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231 The generators are initialized with random states at start-up, so |
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232 that the sequences of random numbers are not the same each time you run |
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233 Octave.@footnote{The old versions of @code{rand} and @code{randn} |
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234 obtain their initial seeds from the system clock.} If you really do |
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235 need to reproduce a sequence of numbers exactly, you can set the state |
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236 or seed to a specific value. |
| 3294 | 237 |
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238 If invoked without arguments, @code{rand} and @code{randn} return a |
| 3294 | 239 single element of a random sequence. |
| 240 | |
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241 The original @code{rand} and @code{randn} functions use Fortran code from |
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242 @sc{ranlib}, a library of Fortran routines for random number generation, |
| 3294 | 243 compiled by Barry W. Brown and James Lovato of the Department of |
| 244 Biomathematics at The University of Texas, M.D. Anderson Cancer Center, | |
| 245 Houston, TX 77030. | |
| 246 | |
| 3428 | 247 @DOCSTRING(randperm) |
| 248 | |
| 3294 | 249 The functions @code{linspace} and @code{logspace} make it very easy to |
| 250 create vectors with evenly or logarithmically spaced elements. | |
| 251 @xref{Ranges}. | |
| 252 | |
| 3369 | 253 @DOCSTRING(linspace) |
| 3294 | 254 |
| 3369 | 255 @DOCSTRING(logspace) |
| 3294 | 256 |
| 4167 | 257 @node Famous Matrices |
| 3294 | 258 @section Famous Matrices |
| 259 | |
| 260 The following functions return famous matrix forms. | |
| 261 | |
| 6502 | 262 @DOCSTRING(hadamard) |
| 263 | |
| 3369 | 264 @DOCSTRING(hankel) |
| 3294 | 265 |
| 3369 | 266 @DOCSTRING(hilb) |
| 3294 | 267 |
| 3369 | 268 @DOCSTRING(invhilb) |
| 3294 | 269 |
| 6502 | 270 @DOCSTRING(magic) |
| 271 | |
| 272 @DOCSTRING(pascal) | |
| 273 | |
| 274 @DOCSTRING(rosser) | |
| 275 | |
| 3369 | 276 @DOCSTRING(sylvester_matrix) |
| 3294 | 277 |
| 3369 | 278 @DOCSTRING(toeplitz) |
| 3294 | 279 |
| 3369 | 280 @DOCSTRING(vander) |
| 6502 | 281 |
| 282 @DOCSTRING(wilkinson) |