Mercurial > hg > octave-jordi
annotate doc/interpreter/diagperm.txi @ 19595:0e1f5a750d00
maint: Periodic merge of gui-release to default.
| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Tue, 20 Jan 2015 10:24:46 -0500 |
| parents | 9e5b64b3c1fe 446c46af4b42 |
| children | 4197fc428c7d |
| rev | line source |
|---|---|
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1 @c Copyright (C) 2009-2013 Jaroslav Hajek |
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2 @c |
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3 @c This file is part of Octave. |
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4 @c |
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5 @c Octave is free software; you can redistribute it and/or modify it |
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6 @c under the terms of the GNU General Public License as published by the |
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7 @c Free Software Foundation; either version 3 of the License, or (at |
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8 @c your option) any later version. |
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9 @c |
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10 @c Octave is distributed in the hope that it will be useful, but WITHOUT |
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11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
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12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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13 @c for more details. |
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14 @c |
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15 @c You should have received a copy of the GNU General Public License |
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16 @c along with Octave; see the file COPYING. If not, see |
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17 @c <http://www.gnu.org/licenses/>. |
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18 |
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19 @node Diagonal and Permutation Matrices |
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20 @chapter Diagonal and Permutation Matrices |
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21 @cindex diagonal and permutation matrices |
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22 @cindex matrices, diagonal and permutation |
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23 |
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24 @menu |
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25 * Basic Usage:: Creation and Manipulation of Diagonal/Permutation Matrices |
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26 * Matrix Algebra:: Linear Algebra with Diagonal/Permutation Matrices |
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27 * Function Support:: Functions That Are Aware of These Matrices |
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28 * Example Code:: Examples of Usage |
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29 * Zeros Treatment:: Differences in Treatment of Zero Elements |
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30 @end menu |
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31 |
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32 @node Basic Usage |
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33 @section Creating and Manipulating Diagonal/Permutation Matrices |
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34 |
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35 A diagonal matrix is defined as a matrix that has zero entries outside the main |
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36 diagonal; that is, |
| 8917 | 37 @tex |
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38 $D_{ij} = 0$ if $i \neq j$ |
| 8917 | 39 @end tex |
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40 @ifnottex |
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41 @code{D(i,j) == 0} if @code{i != j}. |
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42 @end ifnottex |
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43 Most often, square diagonal matrices are considered; however, the definition can |
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44 equally be applied to non-square matrices, in which case we usually speak of a |
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45 rectangular diagonal matrix. |
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46 |
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47 A permutation matrix is defined as a square matrix that has a single element |
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48 equal to unity in each row and each column; all other elements are zero. That |
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49 is, there exists a permutation (vector) |
| 8917 | 50 @tex |
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51 $p$ such that $P_{ij}=1$ if $j = p_i$ and |
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52 $P_{ij}=0$ otherwise. |
| 8917 | 53 @end tex |
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54 @ifnottex |
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55 @code{p} such that @code{P(i,j) == 1} if @code{j == p(i)} and |
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56 @code{P(i,j) == 0} otherwise. |
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57 @end ifnottex |
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58 |
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59 Octave provides special treatment of real and complex rectangular diagonal |
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60 matrices, as well as permutation matrices. They are stored as special objects, |
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61 using efficient storage and algorithms, facilitating writing both readable and |
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62 efficient matrix algebra expressions in the Octave language. The special |
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63 treatment may be disabled by using the functions @dfn{disable_diagonal_matrix} |
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64 and @dfn{disable_permutation_matrix}. |
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65 |
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66 @DOCSTRING(disable_diagonal_matrix) |
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67 |
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68 @DOCSTRING(disable_permutation_matrix) |
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69 |
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70 The space savings are significant as demonstrated by the following code. |
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71 |
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72 @example |
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73 @group |
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74 x = diag (rand (10, 1)); |
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75 xf = full (x); |
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76 sizeof (x) |
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77 @result{} 80 |
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78 sizeof (xf) |
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79 @result{} 800 |
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80 @end group |
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81 @end example |
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82 |
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83 @menu |
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84 * Creating Diagonal Matrices:: |
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85 * Creating Permutation Matrices:: |
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86 * Explicit and Implicit Conversions:: |
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87 @end menu |
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88 |
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89 @node Creating Diagonal Matrices |
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90 @subsection Creating Diagonal Matrices |
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91 |
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92 The most common and easiest way to create a diagonal matrix is using the |
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93 built-in function @dfn{diag}. The expression @code{diag (v)}, with @var{v} a |
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94 vector, will create a square diagonal matrix with elements on the main diagonal |
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95 given by the elements of @var{v}, and size equal to the length of @var{v}. |
| 8917 | 96 @code{diag (v, m, n)} can be used to construct a rectangular diagonal matrix. |
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97 The result of these expressions will be a special diagonal matrix object, rather |
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98 than a general matrix object. |
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99 |
| 8917 | 100 Diagonal matrix with unit elements can be created using @dfn{eye}. |
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101 Some other built-in functions can also return diagonal matrices. Examples |
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102 include |
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103 @dfn{balance} or @dfn{inv}. |
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104 |
| 8917 | 105 Example: |
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106 |
| 8917 | 107 @example |
| 108 diag (1:4) | |
| 109 @result{} | |
| 110 Diagonal Matrix | |
| 111 | |
| 112 1 0 0 0 | |
| 113 0 2 0 0 | |
| 114 0 0 3 0 | |
| 115 0 0 0 4 | |
| 116 | |
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117 diag (1:3,5,3) |
| 8917 | 118 |
| 119 @result{} | |
| 120 Diagonal Matrix | |
| 121 | |
| 122 1 0 0 | |
| 123 0 2 0 | |
| 124 0 0 3 | |
| 125 0 0 0 | |
| 126 0 0 0 | |
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127 @end example |
| 8917 | 128 |
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129 @node Creating Permutation Matrices |
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130 @subsection Creating Permutation Matrices |
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131 |
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132 For creating permutation matrices, Octave does not introduce a new function, but |
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133 rather overrides an existing syntax: permutation matrices can be conveniently |
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134 created by indexing an identity matrix by permutation vectors. |
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135 That is, if @var{q} is a permutation vector of length @var{n}, the expression |
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136 |
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137 @example |
| 8917 | 138 P = eye (n) (:, q); |
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139 @end example |
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140 |
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141 @noindent |
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142 will create a permutation matrix - a special matrix object. |
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143 |
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144 @example |
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145 eye (n) (q, :) |
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146 @end example |
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147 |
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148 @noindent |
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149 will also work (and create a row permutation matrix), as well as |
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150 |
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151 @example |
| 8917 | 152 eye (n) (q1, q2). |
| 153 @end example | |
| 154 | |
| 155 For example: | |
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156 |
| 8917 | 157 @example |
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158 @group |
| 8917 | 159 eye (4) ([1,3,2,4],:) |
| 160 @result{} | |
| 161 Permutation Matrix | |
| 162 | |
| 163 1 0 0 0 | |
| 164 0 0 1 0 | |
| 165 0 1 0 0 | |
| 166 0 0 0 1 | |
| 167 | |
| 168 eye (4) (:,[1,3,2,4]) | |
| 169 @result{} | |
| 170 Permutation Matrix | |
| 171 | |
| 172 1 0 0 0 | |
| 173 0 0 1 0 | |
| 174 0 1 0 0 | |
| 175 0 0 0 1 | |
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176 @end group |
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177 @end example |
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178 |
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179 Mathematically, an identity matrix is both diagonal and permutation matrix. |
| 8917 | 180 In Octave, @code{eye (n)} returns a diagonal matrix, because a matrix |
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181 can only have one class. You can convert this diagonal matrix to a permutation |
| 8917 | 182 matrix by indexing it by an identity permutation, as shown below. |
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183 This is a special property of the identity matrix; indexing other diagonal |
| 8917 | 184 matrices generally produces a full matrix. |
| 185 | |
| 186 @example | |
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187 @group |
| 8917 | 188 eye (3) |
| 189 @result{} | |
| 190 Diagonal Matrix | |
| 191 | |
| 192 1 0 0 | |
| 193 0 1 0 | |
| 194 0 0 1 | |
| 195 | |
| 196 eye(3)(1:3,:) | |
| 197 @result{} | |
| 198 Permutation Matrix | |
| 199 | |
| 200 1 0 0 | |
| 201 0 1 0 | |
| 202 0 0 1 | |
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203 @end group |
| 8917 | 204 @end example |
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205 |
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206 Some other built-in functions can also return permutation matrices. Examples |
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207 include |
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208 @dfn{inv} or @dfn{lu}. |
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209 |
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210 @node Explicit and Implicit Conversions |
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211 @subsection Explicit and Implicit Conversions |
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212 |
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213 The diagonal and permutation matrices are special objects in their own right. A |
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214 number of operations and built-in functions are defined for these matrices to |
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215 use special, more efficient code than would be used for a full matrix in the |
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216 same place. Examples are given in further sections. |
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217 |
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218 To facilitate smooth mixing with full matrices, backward compatibility, and |
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219 compatibility with @sc{matlab}, the diagonal and permutation matrices should |
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220 allow any operation that works on full matrices, and will either treat it |
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221 specially, or implicitly convert themselves to full matrices. |
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222 |
| 8917 | 223 Instances include matrix indexing, except for extracting a single element or |
| 224 a leading submatrix, indexed assignment, or applying most mapper functions, | |
| 225 such as @dfn{exp}. | |
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226 |
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227 An explicit conversion to a full matrix can be requested using the built-in |
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228 function @dfn{full}. It should also be noted that the diagonal and permutation |
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229 matrix objects will cache the result of the conversion after it is first |
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230 requested (explicitly or implicitly), so that subsequent conversions will |
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231 be very cheap. |
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232 |
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233 @node Matrix Algebra |
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234 @section Linear Algebra with Diagonal/Permutation Matrices |
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235 |
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236 As has been already said, diagonal and permutation matrices make it |
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237 possible to use efficient algorithms while preserving natural linear |
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238 algebra syntax. This section describes in detail the operations that |
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239 are treated specially when performed on these special matrix objects. |
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240 |
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241 @menu |
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242 * Expressions Involving Diagonal Matrices:: |
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243 * Expressions Involving Permutation Matrices:: |
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244 @end menu |
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245 |
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246 @node Expressions Involving Diagonal Matrices |
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247 @subsection Expressions Involving Diagonal Matrices |
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248 @cindex diagonal matrix expressions |
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249 |
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250 Assume @var{D} is a diagonal matrix. If @var{M} is a full matrix, |
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251 then @code{D*M} will scale the rows of @var{M}. That means, |
| 8917 | 252 if @code{S = D*M}, then for each pair of indices |
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253 i,j it holds |
| 8917 | 254 @tex |
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255 $$S_{ij} = D_{ii} M_{ij}$$ |
| 8917 | 256 @end tex |
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257 @ifnottex |
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258 |
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259 @example |
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260 S(i,j) = D(i,i) * M(i,j). |
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261 @end example |
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262 |
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263 @end ifnottex |
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264 Similarly, @code{M*D} will do a column scaling. |
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265 |
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266 The matrix @var{D} may also be rectangular, m-by-n where @code{m != n}. |
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267 If @code{m < n}, then the expression @code{D*M} is equivalent to |
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268 |
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269 @example |
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270 D(:,1:m) * M(1:m,:), |
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271 @end example |
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272 |
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273 @noindent |
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274 i.e., trailing @code{n-m} rows of @var{M} are ignored. If @code{m > n}, |
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275 then @code{D*M} is equivalent to |
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276 |
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277 @example |
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278 [D(1:n,n) * M; zeros(m-n, columns (M))], |
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279 @end example |
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280 |
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281 @noindent |
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282 i.e., null rows are appended to the result. |
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283 The situation for right-multiplication @code{M*D} is analogous. |
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284 |
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285 @cindex pseudoinverse |
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286 The expressions @code{D \ M} and @code{M / D} perform inverse scaling. |
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287 They are equivalent to solving a diagonal (or rectangular diagonal) |
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288 in a least-squares minimum-norm sense. In exact arithmetic, this is |
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289 equivalent to multiplying by a pseudoinverse. The pseudoinverse of |
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290 a rectangular diagonal matrix is again a rectangular diagonal matrix |
| 8917 | 291 with swapped dimensions, where each nonzero diagonal element is replaced |
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292 by its reciprocal. |
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293 The matrix division algorithms do, in fact, use division rather than |
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294 multiplication by reciprocals for better numerical accuracy; otherwise, they |
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295 honor the above definition. Note that a diagonal matrix is never truncated due |
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296 to ill-conditioning; otherwise, it would not be of much use for scaling. This |
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297 is typically consistent with linear algebra needs. A full matrix that only |
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298 happens to be diagonal (and is thus not a special object) is of course treated |
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299 normally. |
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300 |
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301 Multiplication and division by diagonal matrices work efficiently also when |
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302 combined with sparse matrices, i.e., @code{D*S}, where @var{D} is a diagonal |
| 9047 | 303 matrix and @var{S} is a sparse matrix scales the rows of the sparse matrix and |
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304 returns a sparse matrix. The expressions @code{S*D}, @code{D\S}, @code{S/D} |
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305 work analogically. |
| 9047 | 306 |
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307 If @var{D1} and @var{D2} are both diagonal matrices, then the expressions |
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308 |
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309 @example |
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310 @group |
| 8917 | 311 D1 + D2 |
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312 D1 - D2 |
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313 D1 * D2 |
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314 D1 / D2 |
| 8917 | 315 D1 \ D2 |
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316 @end group |
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317 @end example |
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318 |
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319 @noindent |
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320 again produce diagonal matrices, provided that normal |
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321 dimension matching rules are obeyed. The relations used are same as described |
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322 above. |
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323 |
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324 Also, a diagonal matrix @var{D} can be multiplied or divided by a scalar, or |
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325 raised to a scalar power if it is square, producing diagonal matrix result in |
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326 all cases. |
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327 |
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328 A diagonal matrix can also be transposed or conjugate-transposed, giving the |
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329 expected result. Extracting a leading submatrix of a diagonal matrix, i.e., |
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330 @code{D(1:m,1:n)}, will produce a diagonal matrix, other indexing expressions |
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331 will implicitly convert to full matrix. |
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332 |
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333 Adding a diagonal matrix to a full matrix only operates on the diagonal |
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334 elements. Thus, |
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335 |
| 8917 | 336 @example |
| 337 A = A + eps * eye (n) | |
| 338 @end example | |
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339 |
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340 @noindent |
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341 is an efficient method of augmenting the diagonal of a matrix. Subtraction |
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342 works analogically. |
| 8917 | 343 |
| 344 When involved in expressions with other element-by-element operators, @code{.*}, | |
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345 @code{./}, @code{.\} or @code{.^}, an implicit conversion to full matrix will |
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346 take place. This is not always strictly necessary but chosen to facilitate |
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347 better consistency with @sc{matlab}. |
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348 |
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349 @node Expressions Involving Permutation Matrices |
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350 @subsection Expressions Involving Permutation Matrices |
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351 |
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352 If @var{P} is a permutation matrix and @var{M} a matrix, the expression |
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353 @code{P*M} will permute the rows of @var{M}. Similarly, @code{M*P} will |
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354 yield a column permutation. |
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355 Matrix division @code{P\M} and @code{M/P} can be used to do inverse permutation. |
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356 |
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357 The previously described syntax for creating permutation matrices can actually |
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358 help an user to understand the connection between a permutation matrix and |
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359 a permuting vector. Namely, the following holds, where @code{I = eye (n)} |
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360 is an identity matrix: |
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361 |
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362 @example |
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363 I(p,:) * M = (I*M) (p,:) = M(p,:) |
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364 @end example |
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365 |
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366 Similarly, |
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367 |
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368 @example |
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369 M * I(:,p) = (M*I) (:,p) = M(:,p) |
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370 @end example |
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371 |
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372 The expressions @code{I(p,:)} and @code{I(:,p)} are permutation matrices. |
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373 |
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374 A permutation matrix can be transposed (or conjugate-transposed, which is the |
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375 same, because a permutation matrix is never complex), inverting the |
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376 permutation, or equivalently, turning a row-permutation matrix into a |
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377 column-permutation one. For permutation matrices, transpose is equivalent to |
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378 inversion, thus @code{P\M} is equivalent to @code{P'*M}. Transpose of a |
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379 permutation matrix (or inverse) is a constant-time operation, flipping only a |
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380 flag internally, and thus the choice between the two above equivalent |
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381 expressions for inverse permuting is completely up to the user's taste. |
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382 |
| 9047 | 383 Multiplication and division by permutation matrices works efficiently also when |
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384 combined with sparse matrices, i.e., @code{P*S}, where @var{P} is a permutation |
| 9047 | 385 matrix and @var{S} is a sparse matrix permutes the rows of the sparse matrix and |
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386 returns a sparse matrix. The expressions @code{S*P}, @code{P\S}, @code{S/P} |
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387 work analogically. |
| 9047 | 388 |
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389 Two permutation matrices can be multiplied or divided (if their sizes match), |
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390 performing a composition of permutations. Also a permutation matrix can be |
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391 indexed by a permutation vector (or two vectors), giving again a permutation |
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392 matrix. Any other operations do not generally yield a permutation matrix and |
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393 will thus trigger the implicit conversion. |
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394 |
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395 @node Function Support |
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396 @section Functions That Are Aware of These Matrices |
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397 |
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398 This section lists the built-in functions that are aware of diagonal and |
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399 permutation matrices on input, or can return them as output. Passed to other |
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400 functions, these matrices will in general trigger an implicit conversion. |
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401 (Of course, user-defined dynamically linked functions may also work with |
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402 diagonal or permutation matrices). |
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403 |
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404 @menu |
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405 * Diagonal Matrix Functions:: |
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406 * Permutation Matrix Functions:: |
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407 @end menu |
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408 |
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409 @node Diagonal Matrix Functions |
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410 @subsection Diagonal Matrix Functions |
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411 |
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412 @dfn{inv} and @dfn{pinv} can be applied to a diagonal matrix, yielding again |
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413 a diagonal matrix. @dfn{det} will use an efficient straightforward calculation |
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414 when given a diagonal matrix, as well as @dfn{cond}. |
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415 The following mapper functions can be applied to a diagonal matrix |
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416 without converting it to a full one: |
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417 @dfn{abs}, @dfn{real}, @dfn{imag}, @dfn{conj}, @dfn{sqrt}. |
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418 A diagonal matrix can also be returned from the @dfn{balance} |
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419 and @dfn{svd} functions. |
| 9047 | 420 The @dfn{sparse} function will convert a diagonal matrix efficiently to a |
| 421 sparse matrix. | |
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422 |
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423 @node Permutation Matrix Functions |
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424 @subsection Permutation Matrix Functions |
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425 @cindex matrix, permutation functions |
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426 @cindex permutation matrix functions |
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427 |
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428 @dfn{inv} and @dfn{pinv} will invert a permutation matrix, preserving its |
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429 specialness. @dfn{det} can be applied to a permutation matrix, efficiently |
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430 calculating the sign of the permutation (which is equal to the determinant). |
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431 |
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432 A permutation matrix can also be returned from the built-in functions |
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433 @dfn{lu} and @dfn{qr}, if a pivoted factorization is requested. |
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434 |
| 9047 | 435 The @dfn{sparse} function will convert a permutation matrix efficiently to a |
| 436 sparse matrix. | |
| 437 The @dfn{find} function will also work efficiently with a permutation matrix, | |
| 438 making it possible to conveniently obtain the permutation indices. | |
| 439 | |
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440 @node Example Code |
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441 @section Examples of Usage |
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442 |
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443 The following can be used to solve a linear system @code{A*x = b} |
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444 using the pivoted LU@tie{}factorization: |
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445 |
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446 @example |
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447 @group |
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448 [L, U, P] = lu (A); ## now L*U = P*A |
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449 x = U \ L \ P*b; |
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450 @end group |
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451 @end example |
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452 |
| 8917 | 453 @noindent |
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454 This is one way to normalize columns of a matrix @var{X} to unit norm: |
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455 |
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456 @example |
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457 @group |
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458 s = norm (X, "columns"); |
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459 X /= diag (s); |
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460 @end group |
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461 @end example |
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462 |
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463 @noindent |
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464 The same can also be accomplished with broadcasting (@pxref{Broadcasting}): |
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465 |
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466 @example |
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467 @group |
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468 s = norm (X, "columns"); |
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469 X ./= s; |
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470 @end group |
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471 @end example |
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472 |
| 8917 | 473 @noindent |
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474 The following expression is a way to efficiently calculate the sign of a |
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475 permutation, given by a permutation vector @var{p}. It will also work |
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476 in earlier versions of Octave, but slowly. |
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477 |
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478 @example |
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479 det (eye (length (p))(p, :)) |
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480 @end example |
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481 |
| 8917 | 482 @noindent |
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483 Finally, here's how to solve a linear system @code{A*x = b} |
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484 with @nospell{Tikhonov} regularization (ridge regression) using SVD (a skeleton |
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485 only): |
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486 |
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487 @example |
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488 @group |
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489 m = rows (A); n = columns (A); |
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490 [U, S, V] = svd (A); |
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491 ## determine the regularization factor alpha |
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492 ## alpha = @dots{} |
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493 ## transform to orthogonal basis |
| 8917 | 494 b = U'*b; |
| 495 ## Use the standard formula, replacing A with S. | |
| 496 ## S is diagonal, so the following will be very fast and accurate. | |
| 497 x = (S'*S + alpha^2 * eye (n)) \ (S' * b); | |
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498 ## transform to solution basis |
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499 x = V*x; |
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500 @end group |
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501 @end example |
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502 |
| 8917 | 503 |
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504 @node Zeros Treatment |
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505 @section Differences in Treatment of Zero Elements |
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506 @cindex matrix, zero elements |
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507 |
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508 Making diagonal and permutation matrices special matrix objects in their own |
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509 right and the consequent usage of smarter algorithms for certain operations |
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510 implies, as a side effect, small differences in treating zeros. |
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511 The contents of this section apply also to sparse matrices, discussed in |
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512 the following chapter. (@pxref{Sparse Matrices}) |
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513 |
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514 The IEEE floating point standard defines the result of the expressions |
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515 @code{0*Inf} and @code{0*NaN} as @code{NaN}. This is widely agreed to be a |
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516 good compromise. Numerical software dealing with structured and sparse matrices |
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517 (including Octave) however, almost always makes a distinction between a |
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518 "numerical zero" and an "assumed zero". A "numerical zero" is a zero value |
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519 occurring in a place where any floating-point value could occur. It is |
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520 normally stored somewhere in memory as an explicit value. An "assumed zero", on |
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521 the contrary, is a zero matrix element implied by the matrix structure |
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522 (diagonal, triangular) or a sparsity pattern; its value is usually not stored |
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523 explicitly anywhere, but is implied by the underlying data structure. |
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524 |
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525 The primary distinction is that an assumed zero, when multiplied |
| 8917 | 526 by any number, or divided by any nonzero number, |
| 527 yields *always* a zero, even when, e.g., multiplied by @code{Inf} | |
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528 or divided by @code{NaN}. |
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529 The reason for this behavior is that the numerical multiplication is not |
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530 actually performed anywhere by the underlying algorithm; the result is |
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531 just assumed to be zero. Equivalently, one can say that the part of the |
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532 computation involving assumed zeros is performed symbolically, not numerically. |
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533 |
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534 This behavior not only facilitates the most straightforward and efficient |
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535 implementation of algorithms, but also preserves certain useful invariants, |
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536 like: |
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537 |
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538 @itemize |
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539 @item scalar * diagonal matrix is a diagonal matrix |
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540 |
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541 @item sparse matrix / scalar preserves the sparsity pattern |
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542 |
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543 @item permutation matrix * matrix is equivalent to permuting rows |
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544 @end itemize |
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545 |
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546 all of these natural mathematical truths would be invalidated by treating |
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547 assumed zeros as numerical ones. |
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548 |
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549 Note that @sc{matlab} does not strictly follow this principle and converts |
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550 assumed zeros to numerical zeros in certain cases, while not doing so in |
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551 other cases. As of today, there are no intentions to mimic such behavior |
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552 in Octave. |
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553 |
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554 Examples of effects of assumed zeros vs. numerical zeros: |
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555 |
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556 @example |
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557 Inf * eye (3) |
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558 @result{} |
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559 Inf 0 0 |
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560 0 Inf 0 |
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561 0 0 Inf |
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562 |
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563 Inf * speye (3) |
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564 @result{} |
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565 Compressed Column Sparse (rows = 3, cols = 3, nnz = 3 [33%]) |
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566 |
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567 (1, 1) -> Inf |
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568 (2, 2) -> Inf |
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569 (3, 3) -> Inf |
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570 |
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571 Inf * full (eye (3)) |
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572 @result{} |
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573 Inf NaN NaN |
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574 NaN Inf NaN |
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575 NaN NaN Inf |
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576 |
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577 @end example |
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578 |
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579 @example |
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580 @group |
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581 diag (1:3) * [NaN; 1; 1] |
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582 @result{} |
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583 NaN |
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584 2 |
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585 3 |
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586 |
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587 sparse (1:3,1:3,1:3) * [NaN; 1; 1] |
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588 @result{} |
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589 NaN |
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590 2 |
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591 3 |
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592 [1,0,0;0,2,0;0,0,3] * [NaN; 1; 1] |
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593 @result{} |
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594 NaN |
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595 NaN |
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596 NaN |
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597 @end group |
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598 @end example |
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599 |