Mercurial > octave-antonio
annotate doc/interpreter/linalg.txi @ 8817:03b7f618ab3d
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| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Thu, 19 Feb 2009 15:39:19 -0500 |
| parents | cdb4788879b3 |
| children | eb63fbe60fab |
| rev | line source |
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| 6778 | 1 @c Copyright (C) 1996, 1997, 2007 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 Linear Algebra |
| 3294 | 20 @chapter Linear Algebra |
| 21 | |
| 22 This chapter documents the linear algebra functions of Octave. | |
| 23 Reference material for many of these functions may be found in | |
| 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. |
| 3294 | 26 |
| 27 @menu | |
| 6531 | 28 * Techniques used for Linear Algebra:: |
| 3294 | 29 * Basic Matrix Functions:: |
| 30 * Matrix Factorizations:: | |
| 31 * Functions of a Matrix:: | |
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32 * Specialized Solvers:: |
| 3294 | 33 @end menu |
| 34 | |
| 6531 | 35 @node Techniques used for Linear Algebra |
| 36 @section Techniques used for Linear Algebra | |
| 37 | |
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38 Octave includes a polymorphic solver, that selects an appropriate |
| 6531 | 39 matrix factorization depending on the properties of the matrix itself. |
| 40 Generally, the cost of determining the matrix type is small relative to | |
| 41 the cost of factorizing the matrix itself, but in any case the matrix | |
| 42 type is cached once it is calculated, so that it is not re-determined | |
| 43 each time it is used in a linear equation. | |
| 44 | |
| 45 The selection tree for how the linear equation is solve or a matrix | |
| 46 inverse is form is given by | |
| 47 | |
| 48 @enumerate 1 | |
| 49 @item If the matrix is upper or lower triangular sparse a forward or | |
| 6939 | 50 backward substitution using the @sc{Lapack} xTRTRS function, and goto 4. |
| 6531 | 51 |
| 52 @c Permuted triangular matrices currently disabled in the code | |
| 53 @c | |
| 54 @c @item If the matrix is a upper triangular matrix with column permutations | |
| 55 @c or lower triangular matrix with row permutations, perform a forward or | |
| 56 @c backward substitution, and goto 5. | |
| 57 | |
| 58 @item If the matrix is square, hermitian with a real positive diagonal, | |
| 59 attempt Cholesky factorization using the @sc{Lapack} xPOTRF function. | |
| 60 | |
| 61 @item If the Cholesky factorization failed or the matrix is not | |
| 62 hermitian with a real positive diagonal, and the matrix is square, factorize | |
| 63 using the @sc{Lapack} xGETRF function. | |
| 64 | |
| 65 @item If the matrix is not square, or any of the previous solvers flags | |
| 66 a singular or near singular matrix, find a least squares solution using | |
| 7072 | 67 the @sc{Lapack} xGELSD function. |
| 6531 | 68 @end enumerate |
| 69 | |
| 70 The user can force the type of the matrix with the @code{matrix_type} | |
| 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 |
| 6531 | 73 will lead to unpredictable results, and so @code{matrix_type} should be |
| 74 used with care. | |
| 75 | |
| 76 It should be noted that the test for whether a matrix is a candidate for | |
| 77 Cholesky factorization, performed above and by the @code{matrix_type} | |
| 78 function, does not give a certainty that the matrix is | |
| 79 Hermitian. However, the attempt to factorize the matrix will quickly | |
| 80 flag a non-Hermitian matrix. | |
| 81 | |
| 4167 | 82 @node Basic Matrix Functions |
| 3294 | 83 @section Basic Matrix Functions |
| 84 | |
| 3373 | 85 @DOCSTRING(balance) |
| 3294 | 86 |
| 3373 | 87 @DOCSTRING(cond) |
| 3294 | 88 |
| 3373 | 89 @DOCSTRING(det) |
| 3294 | 90 |
| 3428 | 91 @DOCSTRING(dmult) |
| 92 | |
| 93 @DOCSTRING(dot) | |
| 94 | |
| 3373 | 95 @DOCSTRING(eig) |
| 3294 | 96 |
| 3373 | 97 @DOCSTRING(givens) |
| 3294 | 98 |
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99 @DOCSTRING(planerot) |
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100 |
| 3373 | 101 @DOCSTRING(inv) |
| 3294 | 102 |
| 6531 | 103 @DOCSTRING(matrix_type) |
| 104 | |
| 3373 | 105 @DOCSTRING(norm) |
| 3294 | 106 |
| 3373 | 107 @DOCSTRING(null) |
| 3294 | 108 |
| 3373 | 109 @DOCSTRING(orth) |
| 3294 | 110 |
| 3373 | 111 @DOCSTRING(pinv) |
| 3294 | 112 |
| 3373 | 113 @DOCSTRING(rank) |
| 3294 | 114 |
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115 @DOCSTRING(rcond) |
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116 |
| 3373 | 117 @DOCSTRING(trace) |
| 3294 | 118 |
| 6502 | 119 @DOCSTRING(rref) |
| 120 | |
| 4167 | 121 @node Matrix Factorizations |
| 3294 | 122 @section Matrix Factorizations |
| 123 | |
| 3373 | 124 @DOCSTRING(chol) |
| 3294 | 125 |
| 6502 | 126 @DOCSTRING(cholinv) |
| 127 | |
| 128 @DOCSTRING(chol2inv) | |
| 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 |
| 3373 | 138 @DOCSTRING(hess) |
| 3294 | 139 |
| 3373 | 140 @DOCSTRING(lu) |
| 3294 | 141 |
| 3373 | 142 @DOCSTRING(qr) |
| 3294 | 143 |
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144 @DOCSTRING(qrupdate) |
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145 |
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146 @DOCSTRING(qrinsert) |
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147 |
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148 @DOCSTRING(qrdelete) |
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149 |
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150 @DOCSTRING(qrshift) |
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151 |
| 3373 | 152 @DOCSTRING(qz) |
| 3372 | 153 |
| 3373 | 154 @DOCSTRING(qzhess) |
| 3294 | 155 |
| 3373 | 156 @DOCSTRING(schur) |
| 3294 | 157 |
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158 @DOCSTRING(subspace) |
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159 |
| 3373 | 160 @DOCSTRING(svd) |
| 3294 | 161 |
| 5775 | 162 @c FIXME -- should there be a new section here? |
| 3428 | 163 |
| 164 @DOCSTRING(housh) | |
| 165 | |
| 166 @DOCSTRING(krylov) | |
| 167 | |
| 4167 | 168 @node Functions of a Matrix |
| 3294 | 169 @section Functions of a Matrix |
| 170 | |
| 3373 | 171 @DOCSTRING(expm) |
| 3294 | 172 |
| 3373 | 173 @DOCSTRING(logm) |
| 3294 | 174 |
| 3373 | 175 @DOCSTRING(sqrtm) |
| 3294 | 176 |
| 3373 | 177 @DOCSTRING(kron) |
| 3294 | 178 |
| 3373 | 179 @DOCSTRING(syl) |
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180 |
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181 @node Specialized Solvers |
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182 @section Specialized Solvers |
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183 |
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184 @DOCSTRING(bicgstab) |
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185 |
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186 @DOCSTRING(cgs) |