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1 /* |
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2 |
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3 Copyright (C) 2001 Ross Lippert and Paul Kienzle |
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4 |
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5 This program is free software; you can redistribute it and/or modify |
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6 it under the terms of the GNU General Public License as published by |
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7 the Free Software Foundation; either version 2 of the License, or |
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8 (at your option) any later version. |
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9 |
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10 This program is distributed in the hope that it will be useful, |
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11 but WITHOUT ANY WARRANTY; without even the implied warranty of |
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12 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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13 GNU General Public License for more details. |
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14 |
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15 You should have received a copy of the GNU General Public License |
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16 along with this program; if not, write to the Free Software |
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17 Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA |
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18 |
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19 */ |
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20 |
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21 #ifdef HAVE_CONFIG_H |
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22 #include <config.h> |
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23 #endif |
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24 |
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25 #include <float.h> |
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26 |
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27 #include "CmplxSCHUR.h" |
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28 #include "lo-ieee.h" |
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29 #include "lo-mappers.h" |
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30 |
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31 #include "defun-dld.h" |
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32 #include "error.h" |
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33 #include "gripes.h" |
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34 #include "utils.h" |
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35 |
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36 static inline double |
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37 getmin (double x, double y) |
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38 { |
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39 return x < y ? x : y; |
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40 } |
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41 |
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42 static inline double |
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43 getmax (double x, double y) |
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44 { |
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45 return x > y ? x : y; |
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46 } |
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47 |
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48 static double |
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49 frobnorm (const ComplexMatrix& A) |
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50 { |
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51 double sum = 0; |
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52 |
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53 for (int i = 0; i < A.rows (); i++) |
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54 for (int j = 0; j < A.columns (); j++) |
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55 sum += real (A(i,j) * conj (A(i,j))); |
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56 |
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57 return sqrt (sum); |
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58 } |
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59 |
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60 static double |
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61 frobnorm (const Matrix& A) |
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62 { |
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63 double sum = 0; |
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64 for (int i = 0; i < A.rows (); i++) |
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65 for (int j = 0; j < A.columns (); j++) |
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66 sum += A(i,j) * A(i,j); |
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67 |
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68 return sqrt (sum); |
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69 } |
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70 |
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71 |
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72 static ComplexMatrix |
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73 sqrtm_from_schur (const ComplexMatrix& U, const ComplexMatrix& T) |
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74 { |
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75 const int n = U.rows (); |
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76 |
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77 ComplexMatrix R (n, n, 0.0); |
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78 |
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79 for (int j = 0; j < n; j++) |
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80 R(j,j) = sqrt (T(j,j)); |
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81 |
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82 const double fudge = sqrt (DBL_MIN); |
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83 |
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84 for (int p = 0; p < n-1; p++) |
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85 { |
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86 for (int i = 0; i < n-(p+1); i++) |
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87 { |
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88 const int j = i + p + 1; |
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89 |
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90 Complex s = T(i,j); |
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91 |
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92 for (int k = i+1; k < j; k++) |
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93 s -= R(i,k) * R(k,j); |
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94 |
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95 // dividing |
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96 // R(i,j) = s/(R(i,i)+R(j,j)); |
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97 // screwing around to not / 0 |
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98 |
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99 const Complex d = R(i,i) + R(j,j) + fudge; |
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100 const Complex conjd = conj (d); |
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101 |
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102 R(i,j) = (s*conjd)/(d*conjd); |
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103 } |
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104 } |
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105 |
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106 return U * R * U.hermitian (); |
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107 } |
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108 |
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109 DEFUN_DLD (sqrtm, args, nargout, |
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110 "-*- texinfo -*-\n\ |
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111 @deftypefn {Loadable Function} {[@var{result}, @var{error_estimate}] =} sqrtm (@var{a})\n\ |
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112 Compute the matrix square root of the square matrix @var{a}.\n\ |
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113 \n\ |
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114 Ref: Nicholas J. Higham. A new sqrtm for MATLAB. Numerical Analysis\n\ |
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115 Report No. 336, Manchester Centre for Computational Mathematics,\n\ |
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116 Manchester, England, January 1999.\n\ |
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117 @end deftypefn\n\ |
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118 @seealso{expm, logm, and funm}") |
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119 { |
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120 octave_value_list retval; |
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121 |
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122 int nargin = args.length (); |
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123 |
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124 if (nargin != 1) |
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125 { |
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126 print_usage ("sqrtm"); |
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127 return retval; |
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128 } |
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129 |
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130 octave_value arg = args(0); |
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131 |
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132 int n = arg.rows (); |
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133 int nc = arg.columns (); |
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134 |
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135 int arg_is_empty = empty_arg ("sqrtm", n, nc); |
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136 |
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137 if (arg_is_empty < 0) |
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138 return retval; |
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139 else if (arg_is_empty > 0) |
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140 return octave_value (Matrix ()); |
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141 |
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142 if (n != nc) |
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143 { |
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144 gripe_square_matrix_required ("sqrtm"); |
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145 return retval; |
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146 } |
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147 |
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148 retval(1) = lo_ieee_inf_value (); |
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149 retval(0) = lo_ieee_nan_value (); |
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150 |
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151 if (arg.is_real_scalar ()) |
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152 { |
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153 double d = arg.double_value (); |
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154 if (d > 0.0) |
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155 { |
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156 retval(0) = sqrt (d); |
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157 retval(1) = 0.0; |
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158 } |
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159 else |
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160 { |
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161 retval(0) = Complex (0.0, sqrt (d)); |
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162 retval(1) = 0.0; |
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163 } |
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164 } |
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165 else if (arg.is_complex_scalar ()) |
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166 { |
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167 Complex c = arg.complex_value (); |
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168 retval(0) = sqrt (c); |
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169 retval(1) = 0.0; |
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170 } |
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171 else if (arg.is_matrix_type ()) |
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172 { |
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173 double err, minT; |
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174 |
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175 if (arg.is_real_matrix ()) |
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176 { |
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177 Matrix A = arg.matrix_value(); |
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178 |
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179 if (error_state) |
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180 return retval; |
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181 |
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182 // XXX FIXME XXX -- eventually, ComplexSCHUR will accept a |
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183 // real matrix arg. |
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184 |
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185 ComplexMatrix Ac (A); |
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186 |
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187 const ComplexSCHUR schur (Ac, std::string ()); |
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188 |
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189 if (error_state) |
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190 return retval; |
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191 |
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192 const ComplexMatrix U (schur.unitary_matrix ()); |
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193 const ComplexMatrix T (schur.schur_matrix ()); |
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194 const ComplexMatrix X (sqrtm_from_schur (U, T)); |
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195 |
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196 // Check for minimal imaginary part |
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197 double normX = 0.0; |
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198 double imagX = 0.0; |
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199 for (int i = 0; i < n; i++) |
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200 for (int j = 0; j < n; j++) |
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201 { |
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202 imagX = getmax (imagX, imag (X(i,j))); |
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203 normX = getmax (normX, abs (X(i,j))); |
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204 } |
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205 |
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206 if (imagX < normX * 100 * DBL_EPSILON) |
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207 retval(0) = real (X); |
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208 else |
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209 retval(0) = X; |
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210 |
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211 // Compute error |
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212 // XXX FIXME XXX can we estimate the error without doing the |
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213 // matrix multiply? |
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214 |
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215 err = frobnorm (X*X - ComplexMatrix (A)) / frobnorm (A); |
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216 |
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217 if (xisnan (err)) |
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218 err = lo_ieee_inf_value (); |
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219 |
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220 // Find min diagonal |
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221 minT = lo_ieee_inf_value (); |
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222 for (int i=0; i < n; i++) |
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223 minT = getmin(minT, abs(T(i,i))); |
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224 } |
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225 else |
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226 { |
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227 ComplexMatrix A = arg.complex_matrix_value (); |
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228 |
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229 if (error_state) |
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230 return retval; |
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231 |
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232 const ComplexSCHUR schur (A, std::string ()); |
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233 |
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234 if (error_state) |
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235 return retval; |
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236 |
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237 const ComplexMatrix U (schur.unitary_matrix ()); |
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238 const ComplexMatrix T (schur.schur_matrix ()); |
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239 const ComplexMatrix X (sqrtm_from_schur (U, T)); |
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240 |
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241 retval(0) = X; |
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242 |
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243 err = frobnorm (X*X - A) / frobnorm (A); |
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244 |
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245 if (xisnan (err)) |
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246 err = lo_ieee_inf_value (); |
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247 |
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248 minT = lo_ieee_inf_value (); |
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249 for (int i = 0; i < n; i++) |
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250 minT = getmin (minT, abs (T(i,i))); |
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251 } |
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252 |
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253 retval(1) = err; |
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254 |
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255 if (nargout < 2) |
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256 { |
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257 if (err > 100*(minT+DBL_EPSILON)*n) |
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258 { |
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259 if (minT == 0.0) |
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260 error ("sqrtm: A is singular, sqrt may not exist"); |
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261 else if (minT <= sqrt (DBL_MIN)) |
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262 error ("sqrtm: A is nearly singular, failed to find sqrt"); |
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263 else |
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264 error ("sqrtm: failed to find sqrt"); |
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265 } |
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266 } |
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267 } |
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268 else |
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269 gripe_wrong_type_arg ("sqrtm", arg); |
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270 |
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271 return retval; |
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272 } |
