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1 // tc-expm.cc -*- C++ -*- |
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2 /* |
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3 |
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4 Copyright (C) 1993 John W. Eaton |
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5 |
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6 This file is part of Octave. |
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7 |
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8 Octave is free software; you can redistribute it and/or modify it |
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9 under the terms of the GNU General Public License as published by the |
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10 Free Software Foundation; either version 2, or (at your option) any |
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11 later version. |
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12 |
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13 Octave is distributed in the hope that it will be useful, but WITHOUT |
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14 ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
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15 FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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16 for more details. |
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17 |
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18 You should have received a copy of the GNU General Public License |
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19 along with Octave; see the file COPYING. If not, write to the Free |
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20 Software Foundation, 675 Mass Ave, Cambridge, MA 02139, USA. |
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21 |
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22 */ |
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23 |
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24 // Written by A. S. Hodel <scotte@eng.auburn.edu> |
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25 |
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26 #ifdef __GNUG__ |
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27 #pragma implementation |
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28 #endif |
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29 |
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30 #include <math.h> |
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31 |
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32 #include "Matrix.h" |
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33 |
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34 #include "tree-const.h" |
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35 #include "user-prefs.h" |
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36 #include "gripes.h" |
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37 #include "error.h" |
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38 #include "f-expm.h" |
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39 |
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40 #ifdef WITH_DLD |
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41 tree_constant * |
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42 builtin_matrix_exp_2 (tree_constant *args, int nargin, int nargout) |
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43 { |
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44 tree_constant *retval = new tree_constant [2]; |
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45 retval[0] = matrix_exp (args[1]); |
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46 return retval; |
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47 } |
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48 #endif |
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49 |
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50 extern "C" |
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51 { |
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52 double F77_FCN (dlange) (const char*, const int*, const int*, |
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53 const double*, const int*, double*); |
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54 |
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55 double F77_FCN (zlange) (const char*, const int*, const int*, |
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56 const Complex*, const int*, double*); |
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57 } |
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58 |
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59 tree_constant |
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52
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60 matrix_exp (tree_constant& a) |
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61 { |
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62 tree_constant retval; |
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63 tree_constant tmp = a.make_numeric (); |
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64 |
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65 // Constants for matrix exponential calculation. |
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66 |
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67 static double padec[] = |
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68 { |
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69 5.0000000000000000e-1, |
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70 1.1666666666666667e-1, |
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71 1.6666666666666667e-2, |
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72 1.6025641025641026e-3, |
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73 1.0683760683760684e-4, |
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74 4.8562548562548563e-6, |
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75 1.3875013875013875e-7, |
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76 1.9270852604185938e-9, |
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77 }; |
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78 |
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79 if (tmp.is_empty ()) |
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80 { |
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81 int flag = user_pref.propagate_empty_matrices; |
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82 if (flag != 0) |
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83 { |
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84 if (flag < 0) |
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85 gripe_empty_arg ("expm", 0); |
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86 Matrix m; |
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87 retval = tree_constant (m); |
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88 } |
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89 else gripe_empty_arg ("expm", 1); |
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90 } |
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91 else if (tmp.rows () != tmp.columns ()) |
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92 gripe_square_matrix_required ("expm"); |
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93 else |
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94 { |
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95 int i, j; |
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96 int n_cols = tmp.columns (); |
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97 |
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98 char* balance_job = "B"; // variables for balancing |
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99 |
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100 int sqpow; // power for scaling and squaring |
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101 double inf_norm; // norm of preconditioned matrix |
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102 int minus_one_j; // used in computing pade approx |
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103 |
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104 switch (tmp.const_type ()) |
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105 { |
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106 case tree_constant_rep::complex_matrix_constant: |
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107 { |
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108 ComplexMatrix m = tmp.complex_matrix_value (); |
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109 Complex trshift = 0.0; // trace shift value |
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110 |
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111 // Preconditioning step 1: trace normalization. |
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112 |
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113 for (i = 0; i < n_cols; i++) |
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114 trshift += m.elem (i, i); |
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115 trshift /= n_cols; |
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116 for (i = 0; i < n_cols; i++) |
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117 m.elem (i, i) -= trshift; |
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118 |
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119 // Preconditioning step 2: eigenvalue balancing. |
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120 |
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121 ComplexAEPBALANCE mbal (m, balance_job); |
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122 m = mbal.balanced_matrix (); |
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123 ComplexMatrix d = mbal.balancing_matrix (); |
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124 |
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125 // Preconditioning step 3: scaling. |
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126 |
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127 ColumnVector work (n_cols); |
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128 inf_norm = F77_FCN (zlange) ("I", &n_cols, &n_cols, m. |
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129 fortran_vec (), &n_cols, |
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130 work.fortran_vec ()); |
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131 |
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132 sqpow = (int) (1.0 + log (inf_norm) / log (2.0)); |
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133 |
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134 // Check whether we need to square at all. |
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135 |
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52
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136 if (sqpow < 0) |
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137 sqpow = 0; |
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138 else |
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139 { |
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140 for (inf_norm = 1.0, i = 0; i < sqpow; i++) |
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141 inf_norm *= 2.0; |
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142 |
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143 m = m / inf_norm; |
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144 } |
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145 |
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146 // npp, dpp: pade' approx polynomial matrices. |
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147 |
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148 ComplexMatrix npp (n_cols, n_cols, 0.0); |
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149 ComplexMatrix dpp = npp; |
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150 |
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151 // Now powers a^8 ... a^1. |
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152 |
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153 minus_one_j = -1; |
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154 for (j = 7; j >= 0; j--) |
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155 { |
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156 npp = m * npp + m * padec[j]; |
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157 dpp = m * dpp + m * (minus_one_j * padec[j]); |
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158 minus_one_j *= -1; |
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159 } |
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160 |
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161 // Zero power. |
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162 |
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163 dpp = -dpp; |
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164 for (j = 0; j < n_cols; j++) |
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165 { |
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166 npp.elem (j, j) += 1.0; |
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167 dpp.elem (j, j) += 1.0; |
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168 } |
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169 |
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170 // Compute pade approximation = inverse (dpp) * npp. |
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171 |
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172 ComplexMatrix result = dpp.solve (npp); |
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173 |
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174 // Reverse preconditioning step 3: repeated squaring. |
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175 |
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176 while (sqpow) |
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177 { |
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178 result = result * result; |
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179 sqpow--; |
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180 } |
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181 |
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182 // reverse preconditioning step 2: inverse balancing XXX FIXME XXX: |
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183 // should probably do this with lapack calls instead of a complete |
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184 // matrix inversion. |
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185 |
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186 result = result.transpose (); |
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187 d = d.transpose (); |
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188 result = result * d; |
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189 result = d.solve (result); |
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190 result = result.transpose (); |
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191 |
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192 // Reverse preconditioning step 1: fix trace normalization. |
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193 |
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194 result = result * exp (trshift); |
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195 |
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196 retval = tree_constant (result); |
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197 } |
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198 break; |
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199 case tree_constant_rep::complex_scalar_constant: |
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200 { |
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201 Complex c = tmp.complex_value (); |
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202 retval = tree_constant (exp (c)); |
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203 } |
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204 break; |
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205 case tree_constant_rep::matrix_constant: |
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206 { |
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207 |
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208 // Compute the exponential. |
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209 |
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210 Matrix m = tmp.matrix_value (); |
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211 |
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212 double trshift = 0; // trace shift value |
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213 |
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214 // Preconditioning step 1: trace normalization. |
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215 |
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216 for (i = 0; i < n_cols; i++) |
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217 trshift += m.elem (i, i); |
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218 trshift /= n_cols; |
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219 for (i = 0; i < n_cols; i++) |
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220 m.elem (i, i) -= trshift; |
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221 |
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222 // Preconditioning step 2: balancing. |
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223 |
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224 AEPBALANCE mbal (m, balance_job); |
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225 m = mbal.balanced_matrix (); |
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226 Matrix d = mbal.balancing_matrix (); |
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227 |
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228 // Preconditioning step 3: scaling. |
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229 |
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230 ColumnVector work(n_cols); |
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231 inf_norm = F77_FCN (dlange) ("I", &n_cols, &n_cols, |
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232 m.fortran_vec (), &n_cols, |
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233 work.fortran_vec ()); |
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234 |
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235 sqpow = (int) (1.0 + log (inf_norm) / log (2.0)); |
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236 |
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237 // Check whether we need to square at all. |
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238 |
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239 if (sqpow < 0) |
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240 sqpow = 0; |
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241 else |
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242 { |
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243 for (inf_norm = 1.0, i = 0; i < sqpow; i++) |
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244 inf_norm *= 2.0; |
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245 |
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246 m = m / inf_norm; |
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247 } |
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248 |
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249 // npp, dpp: pade' approx polynomial matrices. |
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250 |
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251 Matrix npp (n_cols, n_cols, 0.0); |
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252 Matrix dpp = npp; |
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253 |
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254 // now powers a^8 ... a^1. |
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255 |
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256 minus_one_j = -1; |
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257 for (j = 7; j >= 0; j--) |
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258 { |
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259 npp = m * npp + m * padec[j]; |
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260 dpp = m * dpp + m * (minus_one_j * padec[j]); |
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261 minus_one_j *= -1; |
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262 } |
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263 // Zero power. |
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264 |
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265 dpp = -dpp; |
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266 for(j = 0; j < n_cols; j++) |
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267 { |
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268 npp.elem (j, j) += 1.0; |
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269 dpp.elem (j, j) += 1.0; |
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270 } |
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271 |
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272 // Compute pade approximation = inverse (dpp) * npp. |
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273 |
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274 Matrix result = dpp.solve (npp); |
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275 |
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276 // Reverse preconditioning step 3: repeated squaring. |
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277 |
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278 while(sqpow) |
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279 { |
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280 result = result * result; |
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281 sqpow--; |
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282 } |
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283 |
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284 // Reverse preconditioning step 2: inverse balancing. |
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285 |
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286 result = result.transpose(); |
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287 d = d.transpose (); |
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288 result = result * d; |
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289 result = d.solve (result); |
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290 result = result.transpose (); |
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291 |
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292 // Reverse preconditioning step 1: fix trace normalization. |
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293 |
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294 result = result * exp (trshift); |
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295 |
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296 retval = tree_constant (result); |
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297 } |
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298 break; |
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299 case tree_constant_rep::scalar_constant: |
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300 { |
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301 double d = tmp.double_value (); |
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302 retval = tree_constant (exp (d)); |
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303 } |
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304 break; |
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305 default: |
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306 panic_impossible(); |
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307 break; |
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308 } |
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309 } |
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310 return retval; |
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311 } |
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312 |
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313 /* |
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314 ;;; Local Variables: *** |
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315 ;;; mode: C++ *** |
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316 ;;; page-delimiter: "^/\\*" *** |
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317 ;;; End: *** |
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318 */ |
