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1 @c Copyright (C) 1996, 1997, 2007 John W. Eaton |
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2 @c This is part of the Octave manual. |
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3 @c For copying conditions, see the file gpl.texi. |
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4 |
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5 @node Linear Algebra |
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6 @chapter Linear Algebra |
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7 |
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8 This chapter documents the linear algebra functions of Octave. |
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9 Reference material for many of these functions may be found in |
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10 Golub and Van Loan, @cite{Matrix Computations, 2nd Ed.}, Johns Hopkins, |
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11 1989, and in @cite{@sc{Lapack} Users' Guide}, SIAM, 1992. |
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12 |
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13 @menu |
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14 * Techniques used for Linear Algebra:: |
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15 * Basic Matrix Functions:: |
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16 * Matrix Factorizations:: |
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17 * Functions of a Matrix:: |
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18 @end menu |
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19 |
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20 @node Techniques used for Linear Algebra |
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21 @section Techniques used for Linear Algebra |
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22 |
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23 Octave includes a poly-morphic solver, that selects an appropriate |
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24 matrix factorization depending on the properties of the matrix itself. |
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25 Generally, the cost of determining the matrix type is small relative to |
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26 the cost of factorizing the matrix itself, but in any case the matrix |
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27 type is cached once it is calculated, so that it is not re-determined |
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28 each time it is used in a linear equation. |
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29 |
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30 The selection tree for how the linear equation is solve or a matrix |
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31 inverse is form is given by |
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32 |
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33 @enumerate 1 |
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34 @item If the matrix is upper or lower triangular sparse a forward or |
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35 backward substitution using the @sc{Lapack} xTRTRS function, and goto 4. |
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36 |
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37 @c Permuted triangular matrices currently disabled in the code |
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38 @c |
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39 @c @item If the matrix is a upper triangular matrix with column permutations |
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40 @c or lower triangular matrix with row permutations, perform a forward or |
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41 @c backward substitution, and goto 5. |
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42 |
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43 @item If the matrix is square, hermitian with a real positive diagonal, |
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44 attempt Cholesky factorization using the @sc{Lapack} xPOTRF function. |
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45 |
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46 @item If the Cholesky factorization failed or the matrix is not |
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47 hermitian with a real positive diagonal, and the matrix is square, factorize |
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48 using the @sc{Lapack} xGETRF function. |
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49 |
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50 @item If the matrix is not square, or any of the previous solvers flags |
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51 a singular or near singular matrix, find a least squares solution using |
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52 the @sc{Lapack} xGELSY function. |
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53 @end enumerate |
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54 |
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55 The user can force the type of the matrix with the @code{matrix_type} |
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56 function. This overcomes the cost of discovering the type of the matrix. |
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57 However, it should be noted incorrectly identifying the type of the matrix |
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58 will lead to unpredictable results, and so @code{matrix_type} should be |
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59 used with care. |
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60 |
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61 It should be noted that the test for whether a matrix is a candidate for |
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62 Cholesky factorization, performed above and by the @code{matrix_type} |
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63 function, does not give a certainty that the matrix is |
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64 Hermitian. However, the attempt to factorize the matrix will quickly |
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65 flag a non-Hermitian matrix. |
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66 |
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67 @node Basic Matrix Functions |
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68 @section Basic Matrix Functions |
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69 |
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70 @DOCSTRING(balance) |
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71 |
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72 @DOCSTRING(cond) |
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73 |
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74 @DOCSTRING(det) |
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75 |
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76 @DOCSTRING(dmult) |
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77 |
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78 @DOCSTRING(dot) |
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79 |
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80 @DOCSTRING(eig) |
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81 |
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82 @DOCSTRING(givens) |
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83 |
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84 @DOCSTRING(inv) |
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85 |
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86 @DOCSTRING(matrix_type) |
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87 |
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88 @DOCSTRING(norm) |
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89 |
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90 @DOCSTRING(null) |
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91 |
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92 @DOCSTRING(orth) |
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93 |
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94 @DOCSTRING(pinv) |
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95 |
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96 @DOCSTRING(rank) |
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97 |
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98 @DOCSTRING(trace) |
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99 |
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100 @DOCSTRING(rref) |
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101 |
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102 @node Matrix Factorizations |
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103 @section Matrix Factorizations |
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104 |
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105 @DOCSTRING(chol) |
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106 |
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107 @DOCSTRING(cholinv) |
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108 |
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109 @DOCSTRING(chol2inv) |
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110 |
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111 @DOCSTRING(hess) |
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112 |
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113 @DOCSTRING(lu) |
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114 |
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115 @DOCSTRING(qr) |
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116 |
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117 @DOCSTRING(qz) |
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118 |
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119 @DOCSTRING(qzhess) |
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120 |
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121 @DOCSTRING(schur) |
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122 |
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123 @DOCSTRING(svd) |
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124 |
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125 @c FIXME -- should there be a new section here? |
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126 |
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127 @DOCSTRING(housh) |
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128 |
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129 @DOCSTRING(krylov) |
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130 |
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131 @node Functions of a Matrix |
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132 @section Functions of a Matrix |
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133 |
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134 @DOCSTRING(expm) |
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135 |
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136 @DOCSTRING(logm) |
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137 |
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138 @DOCSTRING(sqrtm) |
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139 |
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140 @DOCSTRING(kron) |
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141 |
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142 @DOCSTRING(syl) |
