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