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1 // f-expm.cc -*- C++ -*- |
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2 /* |
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3 |
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4 Copyright (C) 1993, 1994 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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240
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26 #ifdef HAVE_CONFIG_H |
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27 #include "config.h" |
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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 "dMatrix.h" |
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33 #include "CMatrix.h" |
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34 #include "CColVector.h" |
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35 #include "dbleAEPBAL.h" |
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36 #include "CmplxAEPBAL.h" |
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234
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37 #include "f77-uscore.h" |
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38 |
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39 #include "tree-const.h" |
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40 #include "user-prefs.h" |
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41 #include "gripes.h" |
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42 #include "error.h" |
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43 #include "defun-dld.h" |
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44 |
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45 extern "C" |
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46 { |
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47 double F77_FCN (dlange) (const char*, const int*, const int*, |
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48 const double*, const int*, double*); |
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49 |
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50 double F77_FCN (zlange) (const char*, const int*, const int*, |
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51 const Complex*, const int*, double*); |
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52 } |
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53 |
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54 DEFUN_DLD ("expm", Fexpm, Sexpm, 2, 1, |
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55 "expm (X): matrix exponential, e^A") |
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56 { |
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57 Octave_object retval; |
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58 |
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59 int nargin = args.length (); |
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60 |
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61 if (nargin != 2) |
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62 { |
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63 print_usage ("expm"); |
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64 return retval; |
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65 } |
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66 |
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67 tree_constant tmp = args(1).make_numeric (); |
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68 |
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69 // Constants for matrix exponential calculation. |
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70 |
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71 static double padec [] = |
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72 { |
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73 5.0000000000000000e-1, |
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74 1.1666666666666667e-1, |
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75 1.6666666666666667e-2, |
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76 1.6025641025641026e-3, |
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77 1.0683760683760684e-4, |
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78 4.8562548562548563e-6, |
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79 1.3875013875013875e-7, |
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80 1.9270852604185938e-9, |
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81 }; |
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82 |
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83 if (tmp.is_empty ()) |
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84 { |
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85 int flag = user_pref.propagate_empty_matrices; |
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86 if (flag != 0) |
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87 { |
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88 if (flag < 0) |
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89 gripe_empty_arg ("expm", 0); |
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90 |
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91 retval.resize (1, Matrix ()); |
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92 } |
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93 else gripe_empty_arg ("expm", 1); |
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94 } |
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95 else if (tmp.rows () != tmp.columns ()) |
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96 gripe_square_matrix_required ("expm"); |
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97 else |
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98 { |
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99 int i, j; |
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100 int n_cols = tmp.columns (); |
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101 |
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102 char* balance_job = "B"; // variables for balancing |
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103 |
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104 int sqpow; // power for scaling and squaring |
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105 double inf_norm; // norm of preconditioned matrix |
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106 int minus_one_j; // used in computing pade approx |
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107 |
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108 switch (tmp.const_type ()) |
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109 { |
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110 case tree_constant_rep::complex_matrix_constant: |
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111 { |
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112 ComplexMatrix m = tmp.complex_matrix_value (); |
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113 Complex trshift = 0.0; // trace shift value |
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114 |
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115 // Preconditioning step 1: trace normalization. |
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116 |
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117 for (i = 0; i < n_cols; i++) |
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118 trshift += m.elem (i, i); |
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119 trshift /= n_cols; |
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120 for (i = 0; i < n_cols; i++) |
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121 m.elem (i, i) -= trshift; |
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122 |
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123 // Preconditioning step 2: eigenvalue balancing. |
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124 |
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125 ComplexAEPBALANCE mbal (m, balance_job); |
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126 m = mbal.balanced_matrix (); |
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127 ComplexMatrix d = mbal.balancing_matrix (); |
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128 |
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129 // Preconditioning step 3: scaling. |
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130 |
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131 ColumnVector work (n_cols); |
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132 inf_norm = F77_FCN (zlange) ("I", &n_cols, &n_cols, m. |
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133 fortran_vec (), &n_cols, |
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134 work.fortran_vec ()); |
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135 |
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136 sqpow = (int) (1.0 + log (inf_norm) / log (2.0)); |
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137 |
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138 // Check whether we need to square at all. |
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139 |
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140 if (sqpow < 0) |
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141 sqpow = 0; |
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142 else |
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143 { |
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144 for (inf_norm = 1.0, i = 0; i < sqpow; i++) |
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145 inf_norm *= 2.0; |
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146 |
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147 m = m / inf_norm; |
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148 } |
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149 |
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150 // npp, dpp: pade' approx polynomial matrices. |
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151 |
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152 ComplexMatrix npp (n_cols, n_cols, 0.0); |
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153 ComplexMatrix dpp = npp; |
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154 |
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155 // Now powers a^8 ... a^1. |
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156 |
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157 minus_one_j = -1; |
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158 for (j = 7; j >= 0; j--) |
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159 { |
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160 npp = m * npp + m * padec[j]; |
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161 dpp = m * dpp + m * (minus_one_j * padec[j]); |
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162 minus_one_j *= -1; |
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163 } |
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164 |
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165 // Zero power. |
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166 |
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167 dpp = -dpp; |
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168 for (j = 0; j < n_cols; j++) |
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169 { |
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170 npp.elem (j, j) += 1.0; |
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171 dpp.elem (j, j) += 1.0; |
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172 } |
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173 |
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174 // Compute pade approximation = inverse (dpp) * npp. |
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175 |
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176 ComplexMatrix result = dpp.solve (npp); |
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177 |
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178 // Reverse preconditioning step 3: repeated squaring. |
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179 |
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180 while (sqpow) |
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181 { |
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182 result = result * result; |
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183 sqpow--; |
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184 } |
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185 |
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186 // reverse preconditioning step 2: inverse balancing XXX FIXME XXX: |
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187 // should probably do this with lapack calls instead of a complete |
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188 // matrix inversion. |
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189 |
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190 result = result.transpose (); |
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191 d = d.transpose (); |
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192 result = result * d; |
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193 result = d.solve (result); |
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194 result = result.transpose (); |
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195 |
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196 // Reverse preconditioning step 1: fix trace normalization. |
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197 |
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198 result = result * exp (trshift); |
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199 |
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200 retval = result; |
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201 } |
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202 break; |
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203 case tree_constant_rep::complex_scalar_constant: |
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204 { |
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205 Complex c = tmp.complex_value (); |
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206 retval = exp (c); |
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207 } |
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208 break; |
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209 case tree_constant_rep::matrix_constant: |
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210 { |
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211 |
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212 // Compute the exponential. |
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213 |
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214 Matrix m = tmp.matrix_value (); |
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215 |
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216 double trshift = 0; // trace shift value |
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217 |
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218 // Preconditioning step 1: trace normalization. |
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219 |
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220 for (i = 0; i < n_cols; i++) |
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221 trshift += m.elem (i, i); |
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222 trshift /= n_cols; |
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223 for (i = 0; i < n_cols; i++) |
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224 m.elem (i, i) -= trshift; |
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225 |
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226 // Preconditioning step 2: balancing. |
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227 |
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228 AEPBALANCE mbal (m, balance_job); |
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229 m = mbal.balanced_matrix (); |
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230 Matrix d = mbal.balancing_matrix (); |
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231 |
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232 // Preconditioning step 3: scaling. |
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233 |
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234 ColumnVector work(n_cols); |
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235 inf_norm = F77_FCN (dlange) ("I", &n_cols, &n_cols, |
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236 m.fortran_vec (), &n_cols, |
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237 work.fortran_vec ()); |
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238 |
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239 sqpow = (int) (1.0 + log (inf_norm) / log (2.0)); |
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240 |
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241 // Check whether we need to square at all. |
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242 |
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243 if (sqpow < 0) |
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244 sqpow = 0; |
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245 else |
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246 { |
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247 for (inf_norm = 1.0, i = 0; i < sqpow; i++) |
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248 inf_norm *= 2.0; |
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249 |
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250 m = m / inf_norm; |
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251 } |
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252 |
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253 // npp, dpp: pade' approx polynomial matrices. |
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254 |
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255 Matrix npp (n_cols, n_cols, 0.0); |
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256 Matrix dpp = npp; |
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257 |
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258 // now powers a^8 ... a^1. |
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259 |
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260 minus_one_j = -1; |
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261 for (j = 7; j >= 0; j--) |
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262 { |
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263 npp = m * npp + m * padec[j]; |
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264 dpp = m * dpp + m * (minus_one_j * padec[j]); |
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265 minus_one_j *= -1; |
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266 } |
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267 // Zero power. |
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268 |
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269 dpp = -dpp; |
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270 for(j = 0; j < n_cols; j++) |
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271 { |
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272 npp.elem (j, j) += 1.0; |
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273 dpp.elem (j, j) += 1.0; |
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274 } |
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275 |
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276 // Compute pade approximation = inverse (dpp) * npp. |
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277 |
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278 Matrix result = dpp.solve (npp); |
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279 |
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280 // Reverse preconditioning step 3: repeated squaring. |
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281 |
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282 while(sqpow) |
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283 { |
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284 result = result * result; |
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285 sqpow--; |
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286 } |
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287 |
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288 // Reverse preconditioning step 2: inverse balancing. |
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289 |
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290 result = result.transpose(); |
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291 d = d.transpose (); |
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292 result = result * d; |
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293 result = d.solve (result); |
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294 result = result.transpose (); |
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295 |
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296 // Reverse preconditioning step 1: fix trace normalization. |
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297 |
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298 result = result * exp (trshift); |
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299 |
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300 retval = result; |
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301 } |
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302 break; |
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303 case tree_constant_rep::scalar_constant: |
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304 { |
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305 double d = tmp.double_value (); |
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306 retval = exp (d); |
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307 } |
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308 break; |
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309 default: |
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310 panic_impossible(); |
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311 break; |
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312 } |
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313 } |
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314 return retval; |
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315 } |
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316 |
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317 /* |
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318 ;;; Local Variables: *** |
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319 ;;; mode: C++ *** |
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320 ;;; page-delimiter: "^/\\*" *** |
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321 ;;; End: *** |
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322 */ |
