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1 /* |
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2 |
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3 Copyright (C) 1996, 1997, 1999, 2000, 2005, 2006, 2007 John W. Eaton |
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4 |
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5 This file is part of Octave. |
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6 |
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7 Octave is free software; you can redistribute it and/or modify it |
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8 under the terms of the GNU General Public License as published by the |
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9 Free Software Foundation; either version 3 of the License, or (at your |
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10 option) any later version. |
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11 |
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12 Octave is distributed in the hope that it will be useful, but WITHOUT |
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13 ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
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14 FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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15 for more details. |
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16 |
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17 You should have received a copy of the GNU General Public License |
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18 along with Octave; see the file COPYING. If not, see |
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19 <http://www.gnu.org/licenses/>. |
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20 |
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21 */ |
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22 |
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23 #ifdef HAVE_CONFIG_H |
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24 #include <config.h> |
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25 #endif |
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26 |
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27 #include "CmplxQR.h" |
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28 #include "CmplxQRP.h" |
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29 #include "dbleQR.h" |
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30 #include "dbleQRP.h" |
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31 |
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32 #include "defun-dld.h" |
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33 #include "error.h" |
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34 #include "gripes.h" |
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35 #include "oct-obj.h" |
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36 #include "utils.h" |
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37 |
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38 // [Q, R] = qr (X): form Q unitary and R upper triangular such |
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39 // that Q * R = X |
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40 // |
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41 // [Q, R] = qr (X, 0): form the economy decomposition such that if X is |
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42 // m by n then only the first n columns of Q are |
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43 // computed. |
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44 // |
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45 // [Q, R, P] = qr (X): form QRP factorization of X where |
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46 // P is a permutation matrix such that |
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47 // A * P = Q * R |
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48 // |
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49 // [Q, R, P] = qr (X, 0): form the economy decomposition with |
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50 // permutation vector P such that Q * R = X (:, P) |
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51 // |
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52 // qr (X) alone returns the output of the LAPACK routine dgeqrf, such |
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53 // that R = triu (qr (X)) |
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54 |
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55 DEFUN_DLD (qr, args, nargout, |
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56 "-*- texinfo -*-\n\ |
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57 @deftypefn {Loadable Function} {[@var{q}, @var{r}, @var{p}] =} qr (@var{a})\n\ |
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58 @cindex QR factorization\n\ |
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59 Compute the QR factorization of @var{a}, using standard @sc{Lapack}\n\ |
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60 subroutines. For example, given the matrix @code{a = [1, 2; 3, 4]},\n\ |
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61 \n\ |
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62 @example\n\ |
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63 [q, r] = qr (a)\n\ |
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64 @end example\n\ |
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65 \n\ |
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66 @noindent\n\ |
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67 returns\n\ |
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68 \n\ |
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69 @example\n\ |
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70 q =\n\ |
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71 \n\ |
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72 -0.31623 -0.94868\n\ |
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73 -0.94868 0.31623\n\ |
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74 \n\ |
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75 r =\n\ |
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76 \n\ |
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77 -3.16228 -4.42719\n\ |
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78 0.00000 -0.63246\n\ |
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79 @end example\n\ |
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80 \n\ |
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81 The @code{qr} factorization has applications in the solution of least\n\ |
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82 squares problems\n\ |
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83 @iftex\n\ |
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84 @tex\n\ |
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85 $$\n\ |
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86 \\min_x \\left\\Vert A x - b \\right\\Vert_2\n\ |
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87 $$\n\ |
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88 @end tex\n\ |
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89 @end iftex\n\ |
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90 @ifinfo\n\ |
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91 \n\ |
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92 @example\n\ |
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93 @code{min norm(A x - b)}\n\ |
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94 @end example\n\ |
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95 \n\ |
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96 @end ifinfo\n\ |
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97 for overdetermined systems of equations (i.e.,\n\ |
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98 @iftex\n\ |
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99 @tex\n\ |
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100 $A$\n\ |
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101 @end tex\n\ |
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102 @end iftex\n\ |
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103 @ifinfo\n\ |
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104 @code{a}\n\ |
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105 @end ifinfo\n\ |
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106 is a tall, thin matrix). The QR factorization is\n\ |
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107 @iftex\n\ |
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108 @tex\n\ |
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109 $QR = A$ where $Q$ is an orthogonal matrix and $R$ is upper triangular.\n\ |
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110 @end tex\n\ |
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111 @end iftex\n\ |
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112 @ifinfo\n\ |
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113 @code{q * r = a} where @code{q} is an orthogonal matrix and @code{r} is\n\ |
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114 upper triangular.\n\ |
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115 @end ifinfo\n\ |
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116 \n\ |
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117 The permuted QR factorization @code{[@var{q}, @var{r}, @var{p}] =\n\ |
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118 qr (@var{a})} forms the QR factorization such that the diagonal\n\ |
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119 entries of @code{r} are decreasing in magnitude order. For example,\n\ |
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120 given the matrix @code{a = [1, 2; 3, 4]},\n\ |
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121 \n\ |
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122 @example\n\ |
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123 [q, r, p] = qr(a)\n\ |
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124 @end example\n\ |
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125 \n\ |
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126 @noindent\n\ |
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127 returns\n\ |
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128 \n\ |
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129 @example\n\ |
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130 q = \n\ |
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131 \n\ |
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132 -0.44721 -0.89443\n\ |
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133 -0.89443 0.44721\n\ |
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134 \n\ |
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135 r =\n\ |
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136 \n\ |
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137 -4.47214 -3.13050\n\ |
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138 0.00000 0.44721\n\ |
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139 \n\ |
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140 p =\n\ |
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141 \n\ |
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142 0 1\n\ |
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143 1 0\n\ |
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144 @end example\n\ |
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145 \n\ |
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146 The permuted @code{qr} factorization @code{[q, r, p] = qr (a)}\n\ |
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147 factorization allows the construction of an orthogonal basis of\n\ |
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148 @code{span (a)}.\n\ |
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149 \n\ |
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150 If given a second argument, @code{qr} returns an economy-sized\n\ |
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151 QR factorization, omitting zero rows of @var{R} and\n\ |
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152 the corresponding columns of @var{Q}.\n\ |
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153 @end deftypefn") |
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154 { |
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155 octave_value_list retval; |
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156 |
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157 int nargin = args.length (); |
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158 |
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159 if (nargin < 1 || nargin > 2 || nargout > 3) |
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160 { |
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161 print_usage (); |
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162 return retval; |
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163 } |
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164 |
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165 octave_value arg = args(0); |
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166 |
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167 int arg_is_empty = empty_arg ("qr", arg.rows (), arg.columns ()); |
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168 |
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169 if (arg_is_empty < 0) |
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170 return retval; |
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171 else if (arg_is_empty > 0) |
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172 return octave_value_list (3, Matrix ()); |
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173 |
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174 QR::type type = (nargout == 0 || nargout == 1) ? QR::raw |
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175 : (nargin == 2 ? QR::economy : QR::std); |
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176 |
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177 if (arg.is_real_type ()) |
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178 { |
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179 Matrix m = arg.matrix_value (); |
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180 |
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181 if (! error_state) |
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182 { |
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183 switch (nargout) |
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184 { |
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185 case 0: |
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186 case 1: |
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187 { |
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188 QR fact (m, type); |
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189 retval(0) = fact.R (); |
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190 } |
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191 break; |
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192 |
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193 case 2: |
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194 { |
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195 QR fact (m, type); |
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196 retval(1) = fact.R (); |
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197 retval(0) = fact.Q (); |
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198 } |
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199 break; |
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200 |
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201 default: |
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202 { |
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203 QRP fact (m, type); |
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204 retval(2) = fact.P (); |
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205 retval(1) = fact.R (); |
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206 retval(0) = fact.Q (); |
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207 } |
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208 break; |
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209 } |
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210 } |
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211 } |
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212 else if (arg.is_complex_type ()) |
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213 { |
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214 ComplexMatrix m = arg.complex_matrix_value (); |
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215 |
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216 if (! error_state) |
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217 { |
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218 switch (nargout) |
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219 { |
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220 case 0: |
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221 case 1: |
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222 { |
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223 ComplexQR fact (m, type); |
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224 retval(0) = fact.R (); |
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225 } |
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226 break; |
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227 |
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228 case 2: |
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229 { |
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230 ComplexQR fact (m, type); |
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231 retval(1) = fact.R (); |
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232 retval(0) = fact.Q (); |
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233 } |
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234 break; |
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235 |
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236 default: |
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237 { |
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238 ComplexQRP fact (m, type); |
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239 retval(2) = fact.P (); |
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240 retval(1) = fact.R (); |
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241 retval(0) = fact.Q (); |
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242 } |
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243 break; |
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244 } |
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245 } |
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246 } |
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247 else |
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248 { |
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249 gripe_wrong_type_arg ("qr", arg); |
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250 } |
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251 |
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252 return retval; |
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253 } |
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254 |
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255 /* |
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256 ;;; Local Variables: *** |
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257 ;;; mode: C++ *** |
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258 ;;; End: *** |
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259 */ |
