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linsolve.m: Use Octave coding conventions. * doc/interpreter/linalg.txi: Add linsolve to manual. * scripts/linear-algebra/linsolve.m: Redo docstring. Use Octave coding conventions. Add %!error input validation tests.
author Rik <rik@octave.org>
date Thu, 26 Sep 2013 09:38:51 -0700
parents f2a8592b8fbd
children
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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72c96de7a403 maint: update copyright notices for 2012
John W. Eaton <jwe@octave.org>
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1 @c Copyright (C) 1996-2012 John W. Eaton
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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 Linear Algebra
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20 @chapter Linear Algebra
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21
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22 This chapter documents the linear algebra functions of Octave.
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23 Reference material for many of these functions may be found in
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24 Golub and Van Loan, @cite{Matrix Computations, 2nd Ed.}, Johns Hopkins,
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25 1989, and in the @cite{@sc{lapack} Users' Guide}, SIAM, 1992.
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26
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27 @menu
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28 * Techniques Used for Linear Algebra::
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29 * Basic Matrix Functions::
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30 * Matrix Factorizations::
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31 * Functions of a Matrix::
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32 * Specialized Solvers::
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33 @end menu
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34
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35 @node Techniques Used for Linear Algebra
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36 @section Techniques Used for Linear Algebra
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37
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38 Octave includes a polymorphic solver, that selects an appropriate
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39 matrix factorization depending on the properties of the matrix itself.
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40 Generally, the cost of determining the matrix type is small relative to
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41 the cost of factorizing the matrix itself, but in any case the matrix
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42 type is cached once it is calculated, so that it is not re-determined
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43 each time it is used in a linear equation.
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44
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45 The selection tree for how the linear equation is solve or a matrix
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46 inverse is form is given by
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47
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48 @enumerate 1
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49 @item If the matrix is upper or lower triangular sparse a forward or
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50 backward substitution using the @sc{lapack} xTRTRS function, and goto 4.
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51
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52 @c Permuted triangular matrices currently disabled in the code
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53 @c
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54 @c @item If the matrix is a upper triangular matrix with column permutations
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55 @c or lower triangular matrix with row permutations, perform a forward or
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56 @c backward substitution, and goto 5.
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57
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58 @item If the matrix is square, Hermitian with a real positive diagonal,
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59 attempt Cholesky@tie{}factorization using the @sc{lapack} xPOTRF function.
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60
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61 @item If the Cholesky@tie{}factorization failed or the matrix is not
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62 Hermitian with a real positive diagonal, and the matrix is square, factorize
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63 using the @sc{lapack} xGETRF function.
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64
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65 @item If the matrix is not square, or any of the previous solvers flags
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66 a singular or near singular matrix, find a least squares solution using
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67 the @sc{lapack} xGELSD function.
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68 @end enumerate
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69
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70 The user can force the type of the matrix with the @code{matrix_type}
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71 function. This overcomes the cost of discovering the type of the matrix.
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72 However, it should be noted that identifying the type of the matrix incorrectly
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73 will lead to unpredictable results, and so @code{matrix_type} should be
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74 used with care.
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75
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76 It should be noted that the test for whether a matrix is a candidate for
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77 Cholesky@tie{}factorization, performed above and by the @code{matrix_type}
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78 function, does not give a certainty that the matrix is
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79 Hermitian. However, the attempt to factorize the matrix will quickly
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80 flag a non-Hermitian matrix.
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81
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82 @node Basic Matrix Functions
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83 @section Basic Matrix Functions
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84
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85 @DOCSTRING(balance)
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86
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87 @DOCSTRING(cond)
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88
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89 @DOCSTRING(det)
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90
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91 @DOCSTRING(eig)
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92
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93 @DOCSTRING(givens)
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94
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95 @DOCSTRING(planerot)
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96
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97 @DOCSTRING(inv)
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98
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99 @DOCSTRING(linsolve)
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100
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101 @DOCSTRING(matrix_type)
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102
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103 @DOCSTRING(norm)
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104
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105 @DOCSTRING(null)
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106
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107 @DOCSTRING(orth)
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108
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109 @DOCSTRING(mgorth)
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110
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111 @DOCSTRING(pinv)
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112
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113 @DOCSTRING(rank)
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114
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115 @DOCSTRING(rcond)
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116
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117 @DOCSTRING(trace)
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118
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119 @DOCSTRING(rref)
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120
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121 @node Matrix Factorizations
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122 @section Matrix Factorizations
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123
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124 @DOCSTRING(chol)
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125
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126 @DOCSTRING(cholinv)
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127
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128 @DOCSTRING(chol2inv)
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129
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130 @DOCSTRING(cholupdate)
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131
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132 @DOCSTRING(cholinsert)
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133
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134 @DOCSTRING(choldelete)
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135
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136 @DOCSTRING(cholshift)
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137
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138 @DOCSTRING(hess)
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139
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140 @DOCSTRING(lu)
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141
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142 @DOCSTRING(luupdate)
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143
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144 @DOCSTRING(qr)
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145
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146 @DOCSTRING(qrupdate)
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147
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Jaroslav Hajek <highegg@gmail.com>
parents: 7072
diff changeset
148 @DOCSTRING(qrinsert)
2df457529cfa implement expm1 and log1p functions
Jaroslav Hajek <highegg@gmail.com>
parents: 7072
diff changeset
149
2df457529cfa implement expm1 and log1p functions
Jaroslav Hajek <highegg@gmail.com>
parents: 7072
diff changeset
150 @DOCSTRING(qrdelete)
2df457529cfa implement expm1 and log1p functions
Jaroslav Hajek <highegg@gmail.com>
parents: 7072
diff changeset
151
8817
03b7f618ab3d include docstrings for new functions in the manual
John W. Eaton <jwe@octave.org>
parents: 8488
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152 @DOCSTRING(qrshift)
03b7f618ab3d include docstrings for new functions in the manual
John W. Eaton <jwe@octave.org>
parents: 8488
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153
3373
36405da8e173 [project @ 1999-11-23 20:54:17 by jwe]
jwe
parents: 3372
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154 @DOCSTRING(qz)
3372
f16c2ce14886 [project @ 1999-11-23 19:07:09 by jwe]
jwe
parents: 3294
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155
3373
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jwe
parents: 3372
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156 @DOCSTRING(qzhess)
3294
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jwe
parents:
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157
3373
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jwe
parents: 3372
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158 @DOCSTRING(schur)
3294
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jwe
parents:
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159
12188
1e206c8713a7 Add function rsf2csf to documentation.
Rik <octave@nomad.inbox5.com>
parents: 11593
diff changeset
160 @DOCSTRING(rsf2csf)
1e206c8713a7 Add function rsf2csf to documentation.
Rik <octave@nomad.inbox5.com>
parents: 11593
diff changeset
161
7984
bbaa5d7d0143 Some documentation updates
David Bateman <dbateman@free.fr>
parents: 7638
diff changeset
162 @DOCSTRING(subspace)
bbaa5d7d0143 Some documentation updates
David Bateman <dbateman@free.fr>
parents: 7638
diff changeset
163
3373
36405da8e173 [project @ 1999-11-23 20:54:17 by jwe]
jwe
parents: 3372
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164 @DOCSTRING(svd)
3294
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jwe
parents:
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165
11533
06c80c90a246 Add undocumented function svd_driver to documentation.
Rik <octave@nomad.inbox5.com>
parents: 11523
diff changeset
166 @DOCSTRING(svd_driver)
06c80c90a246 Add undocumented function svd_driver to documentation.
Rik <octave@nomad.inbox5.com>
parents: 11523
diff changeset
167
5775
ace8d8d26933 [project @ 2006-04-24 19:13:06 by jwe]
jwe
parents: 4167
diff changeset
168 @c FIXME -- should there be a new section here?
3428
5b77cf82393c [project @ 2000-01-14 02:54:53 by jwe]
jwe
parents: 3373
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169
5b77cf82393c [project @ 2000-01-14 02:54:53 by jwe]
jwe
parents: 3373
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170 @DOCSTRING(housh)
5b77cf82393c [project @ 2000-01-14 02:54:53 by jwe]
jwe
parents: 3373
diff changeset
171
5b77cf82393c [project @ 2000-01-14 02:54:53 by jwe]
jwe
parents: 3373
diff changeset
172 @DOCSTRING(krylov)
5b77cf82393c [project @ 2000-01-14 02:54:53 by jwe]
jwe
parents: 3373
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173
4167
aae05d51353c [project @ 2002-11-12 02:52:50 by jwe]
jwe
parents: 3428
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174 @node Functions of a Matrix
3294
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175 @section Functions of a Matrix
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe]
jwe
parents:
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176
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jwe
parents: 3372
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177 @DOCSTRING(expm)
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jwe
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178
3373
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jwe
parents: 3372
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179 @DOCSTRING(logm)
3294
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jwe
parents:
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180
3373
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jwe
parents: 3372
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181 @DOCSTRING(sqrtm)
3294
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jwe
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182
3373
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jwe
parents: 3372
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183 @DOCSTRING(kron)
3294
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jwe
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184
12572
dffd30f12752 Add blkmm function to documentation.
Rik <octave@nomad.inbox5.com>
parents: 12520
diff changeset
185 @DOCSTRING(blkmm)
dffd30f12752 Add blkmm function to documentation.
Rik <octave@nomad.inbox5.com>
parents: 12520
diff changeset
186
3373
36405da8e173 [project @ 1999-11-23 20:54:17 by jwe]
jwe
parents: 3372
diff changeset
187 @DOCSTRING(syl)
8817
03b7f618ab3d include docstrings for new functions in the manual
John W. Eaton <jwe@octave.org>
parents: 8488
diff changeset
188
03b7f618ab3d include docstrings for new functions in the manual
John W. Eaton <jwe@octave.org>
parents: 8488
diff changeset
189 @node Specialized Solvers
03b7f618ab3d include docstrings for new functions in the manual
John W. Eaton <jwe@octave.org>
parents: 8488
diff changeset
190 @section Specialized Solvers
03b7f618ab3d include docstrings for new functions in the manual
John W. Eaton <jwe@octave.org>
parents: 8488
diff changeset
191
13021
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents: 12572
diff changeset
192 @DOCSTRING(bicg)
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents: 12572
diff changeset
193
8817
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John W. Eaton <jwe@octave.org>
parents: 8488
diff changeset
194 @DOCSTRING(bicgstab)
03b7f618ab3d include docstrings for new functions in the manual
John W. Eaton <jwe@octave.org>
parents: 8488
diff changeset
195
03b7f618ab3d include docstrings for new functions in the manual
John W. Eaton <jwe@octave.org>
parents: 8488
diff changeset
196 @DOCSTRING(cgs)
12509
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents: 12188
diff changeset
197
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents: 12188
diff changeset
198 @DOCSTRING(gmres)