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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, 1995 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, 59 Temple Place - Suite 330, Boston, MA 02111-1307, 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 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 <cmath> |
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31 |
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32 #include "CColVector.h" |
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33 #include "CMatrix.h" |
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34 #include "CmplxAEPBAL.h" |
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35 #include "dMatrix.h" |
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36 #include "dbleAEPBAL.h" |
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37 #include "f77-uscore.h" |
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38 |
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39 #include "defun-dld.h" |
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40 #include "error.h" |
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41 #include "gripes.h" |
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42 #include "help.h" |
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43 #include "tree-const.h" |
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44 #include "user-prefs.h" |
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45 #include "utils.h" |
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46 |
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47 extern "C" |
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48 { |
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49 double F77_FCN (dlange, DLANGE) (const char*, const int&, |
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50 const int&, const double*, |
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51 const int&, double*); |
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52 |
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53 double F77_FCN (zlange, ZLANGE) (const char*, const int&, |
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54 const int&, const Complex*, |
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55 const int&, double*); |
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56 } |
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57 |
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58 DEFUN_DLD_BUILTIN ("expm", Fexpm, Sexpm, 2, 1, |
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59 "expm (X): matrix exponential, e^A") |
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60 { |
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61 Octave_object retval; |
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62 |
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63 int nargin = args.length (); |
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64 |
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65 if (nargin != 1) |
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66 { |
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67 print_usage ("expm"); |
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68 return retval; |
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69 } |
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70 |
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71 tree_constant arg = args(0); |
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72 |
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73 // Constants for matrix exponential calculation. |
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74 |
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75 static double padec [] = |
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76 { |
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77 5.0000000000000000e-1, |
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78 1.1666666666666667e-1, |
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79 1.6666666666666667e-2, |
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80 1.6025641025641026e-3, |
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81 1.0683760683760684e-4, |
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82 4.8562548562548563e-6, |
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83 1.3875013875013875e-7, |
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84 1.9270852604185938e-9, |
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85 }; |
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86 |
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87 int nr = arg.rows (); |
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88 int nc = arg.columns (); |
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89 |
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90 int arg_is_empty = empty_arg ("expm", nr, nc); |
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91 |
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92 if (arg_is_empty < 0) |
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93 return retval; |
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94 if (arg_is_empty > 0) |
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95 return Matrix (); |
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96 |
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97 if (nr != nc) |
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98 { |
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99 gripe_square_matrix_required ("expm"); |
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100 return retval; |
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101 } |
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102 |
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103 int i, j; |
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104 |
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105 char* balance_job = "B"; // variables for balancing |
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106 |
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107 int sqpow; // power for scaling and squaring |
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108 double inf_norm; // norm of preconditioned matrix |
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109 int minus_one_j; // used in computing pade approx |
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110 |
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111 if (arg.is_real_type ()) |
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112 { |
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113 |
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114 // Compute the exponential. |
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115 |
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116 Matrix m = arg.matrix_value (); |
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117 |
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118 if (error_state) |
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119 return retval; |
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120 |
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121 double trshift = 0; // trace shift value |
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122 |
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123 // Preconditioning step 1: trace normalization. |
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124 |
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125 for (i = 0; i < nc; i++) |
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126 trshift += m.elem (i, i); |
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127 trshift /= nc; |
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128 for (i = 0; i < nc; i++) |
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129 m.elem (i, i) -= trshift; |
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130 |
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131 // Preconditioning step 2: balancing. |
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132 |
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133 AEPBALANCE mbal (m, balance_job); |
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134 m = mbal.balanced_matrix (); |
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135 Matrix d = mbal.balancing_matrix (); |
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136 |
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137 // Preconditioning step 3: scaling. |
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138 |
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139 ColumnVector work(nc); |
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140 inf_norm = F77_FCN (dlange, DLANGE) ("I", nc, nc, |
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141 m.fortran_vec (), nc, |
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142 work.fortran_vec ()); |
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143 |
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144 sqpow = (int) (1.0 + log (inf_norm) / log (2.0)); |
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145 |
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146 // Check whether we need to square at all. |
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147 |
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148 if (sqpow < 0) |
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149 sqpow = 0; |
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150 else |
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151 { |
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152 for (inf_norm = 1.0, i = 0; i < sqpow; i++) |
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153 inf_norm *= 2.0; |
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154 |
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155 m = m / inf_norm; |
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156 } |
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157 |
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158 // npp, dpp: pade' approx polynomial matrices. |
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159 |
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160 Matrix npp (nc, nc, 0.0); |
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161 Matrix dpp = npp; |
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162 |
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163 // now powers a^8 ... a^1. |
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164 |
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165 minus_one_j = -1; |
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166 for (j = 7; j >= 0; j--) |
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167 { |
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168 npp = m * npp + m * padec[j]; |
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169 dpp = m * dpp + m * (minus_one_j * padec[j]); |
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170 minus_one_j *= -1; |
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171 } |
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172 // Zero power. |
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173 |
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174 dpp = -dpp; |
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175 for(j = 0; j < nc; j++) |
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176 { |
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177 npp.elem (j, j) += 1.0; |
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178 dpp.elem (j, j) += 1.0; |
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179 } |
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180 |
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181 // Compute pade approximation = inverse (dpp) * npp. |
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182 |
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183 Matrix result = dpp.solve (npp); |
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184 |
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185 // Reverse preconditioning step 3: repeated squaring. |
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186 |
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187 while (sqpow) |
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188 { |
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189 result = result * result; |
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190 sqpow--; |
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191 } |
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192 |
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193 // Reverse preconditioning step 2: inverse balancing. |
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194 |
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195 result = result.transpose(); |
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196 d = d.transpose (); |
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197 result = result * d; |
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198 result = d.solve (result); |
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199 result = result.transpose (); |
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200 |
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201 // Reverse preconditioning step 1: fix trace normalization. |
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202 |
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203 result = result * exp (trshift); |
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204 |
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205 retval = result; |
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206 } |
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207 else if (arg.is_complex_type ()) |
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208 { |
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209 ComplexMatrix m = arg.complex_matrix_value (); |
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210 |
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211 if (error_state) |
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212 return retval; |
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213 |
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214 Complex trshift = 0.0; // trace shift value |
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215 |
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216 // Preconditioning step 1: trace normalization. |
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217 |
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218 for (i = 0; i < nc; i++) |
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219 trshift += m.elem (i, i); |
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220 trshift /= nc; |
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221 for (i = 0; i < nc; i++) |
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222 m.elem (i, i) -= trshift; |
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223 |
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224 // Preconditioning step 2: eigenvalue balancing. |
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225 |
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226 ComplexAEPBALANCE mbal (m, balance_job); |
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227 m = mbal.balanced_matrix (); |
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228 ComplexMatrix d = mbal.balancing_matrix (); |
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229 |
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230 // Preconditioning step 3: scaling. |
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231 |
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232 ColumnVector work (nc); |
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233 inf_norm = F77_FCN (zlange, ZLANGE) ("I", nc, nc, |
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234 m.fortran_vec (), nc, |
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235 work.fortran_vec ()); |
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236 |
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237 sqpow = (int) (1.0 + log (inf_norm) / log (2.0)); |
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238 |
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239 // Check whether we need to square at all. |
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240 |
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241 if (sqpow < 0) |
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242 sqpow = 0; |
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243 else |
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244 { |
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245 for (inf_norm = 1.0, i = 0; i < sqpow; i++) |
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246 inf_norm *= 2.0; |
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247 |
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248 m = m / inf_norm; |
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249 } |
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250 |
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251 // npp, dpp: pade' approx polynomial matrices. |
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252 |
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253 ComplexMatrix npp (nc, nc, 0.0); |
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254 ComplexMatrix dpp = npp; |
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255 |
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256 // Now powers a^8 ... a^1. |
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257 |
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258 minus_one_j = -1; |
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259 for (j = 7; j >= 0; j--) |
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260 { |
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261 npp = m * npp + m * padec[j]; |
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262 dpp = m * dpp + m * (minus_one_j * padec[j]); |
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263 minus_one_j *= -1; |
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264 } |
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265 |
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266 // Zero power. |
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267 |
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268 dpp = -dpp; |
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269 for (j = 0; j < nc; j++) |
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270 { |
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271 npp.elem (j, j) += 1.0; |
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272 dpp.elem (j, j) += 1.0; |
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273 } |
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274 |
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275 // Compute pade approximation = inverse (dpp) * npp. |
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276 |
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277 ComplexMatrix result = dpp.solve (npp); |
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278 |
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279 // Reverse preconditioning step 3: repeated squaring. |
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280 |
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281 while (sqpow) |
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282 { |
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283 result = result * result; |
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284 sqpow--; |
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285 } |
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286 |
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287 // reverse preconditioning step 2: inverse balancing XXX FIXME XXX: |
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288 // should probably do this with lapack calls instead of a complete |
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289 // matrix inversion. |
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290 |
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291 result = result.transpose (); |
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292 d = d.transpose (); |
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293 result = result * d; |
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294 result = d.solve (result); |
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295 result = result.transpose (); |
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296 |
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297 // Reverse preconditioning step 1: fix trace normalization. |
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298 |
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299 result = result * exp (trshift); |
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300 |
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301 retval = result; |
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302 } |
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303 else |
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304 { |
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305 gripe_wrong_type_arg ("expm", arg); |
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306 } |
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307 |
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308 return retval; |
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309 } |
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310 |
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311 /* |
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312 ;;; Local Variables: *** |
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313 ;;; mode: C++ *** |
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314 ;;; page-delimiter: "^/\\*" *** |
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315 ;;; End: *** |
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316 */ |
