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annotate liboctave/oct-norm.cc @ 10050:dc88a0b6472c

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support old style jacobian for fsolve
author Jaroslav Hajek <highegg@gmail.com>
date Sat, 02 Jan 2010 21:55:17 +0100
parents 16f53d29049f
children 829e69ec3110
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1 /*
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2
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3 Copyright (C) 2008, 2009 VZLU Prague, a.s.
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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 // author: Jaroslav Hajek <highegg@gmail.com>
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24
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25 #ifdef HAVE_CONFIG_H
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26 #include <config.h>
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27 #endif
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28
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29 #include <cassert>
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30 #include <cfloat>
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31 #include <cmath>
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32
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33 #include <iostream>
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34 #include <vector>
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35
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36 #include "oct-cmplx.h"
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37 #include "lo-error.h"
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38 #include "lo-ieee.h"
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39 #include "Array.h"
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40 #include "Array-util.h"
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41 #include "CMatrix.h"
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42 #include "dMatrix.h"
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43 #include "fCMatrix.h"
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44 #include "fMatrix.h"
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45 #include "CColVector.h"
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46 #include "dColVector.h"
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47 #include "CRowVector.h"
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48 #include "dRowVector.h"
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49 #include "fCColVector.h"
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50 #include "fColVector.h"
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51 #include "fCRowVector.h"
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52 #include "fRowVector.h"
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53 #include "CSparse.h"
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54 #include "dSparse.h"
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55 #include "dbleSVD.h"
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56 #include "CmplxSVD.h"
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57 #include "floatSVD.h"
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58 #include "fCmplxSVD.h"
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59
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60 // Theory: norm accumulator is an object that has an accum method able to handle
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61 // both real and complex element, and a cast operator returning the intermediate
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62 // norm.
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63
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64 // norm accumulator for the p-norm
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65 template <class R>
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66 class norm_accumulator_p
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67 {
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68 R p,scl,sum;
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69 public:
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70 norm_accumulator_p () {} // we need this one for Array
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71 norm_accumulator_p (R pp) : p(pp), scl(0), sum(1) {}
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72
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73 template<class U>
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74 void accum (U val)
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75 {
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76 R t = std::abs (val);
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77 if (scl == t) // we need this to handle Infs properly
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78 sum += 1;
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79 else if (scl < t)
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80 {
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81 sum *= std::pow (scl/t, p);
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82 sum += 1;
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83 scl = t;
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84 }
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85 else if (t != 0)
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86 sum += std::pow (t/scl, p);
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87 }
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88 operator R () { return scl * std::pow (sum, 1/p); }
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89 };
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90
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91 // norm accumulator for the minus p-pseudonorm
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92 template <class R>
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93 class norm_accumulator_mp
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94 {
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95 R p,scl,sum;
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96 public:
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97 norm_accumulator_mp () {} // we need this one for Array
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98 norm_accumulator_mp (R pp) : p(pp), scl(0), sum(1) {}
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99
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100 template<class U>
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101 void accum (U val)
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102 {
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103 R t = 1 / std::abs (val);
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104 if (scl == t)
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105 sum += 1;
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106 else if (scl < t)
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107 {
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108 sum *= std::pow (scl/t, p);
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109 sum += 1;
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110 scl = t;
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111 }
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112 else if (t != 0)
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113 sum += std::pow (t/scl, p);
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114 }
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115 operator R () { return scl * std::pow (sum, -1/p); }
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116 };
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117
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118 // norm accumulator for the 2-norm (euclidean)
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119 template <class R>
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120 class norm_accumulator_2
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121 {
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122 R scl,sum;
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123 static R pow2 (R x) { return x*x; }
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124 public:
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125 norm_accumulator_2 () : scl(0), sum(1) {}
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126
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127 void accum (R val)
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128 {
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129 R t = std::abs (val);
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130 if (scl == t)
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131 sum += 1;
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132 else if (scl < t)
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133 {
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134 sum *= pow2 (scl/t);
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135 sum += 1;
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136 scl = t;
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137 }
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138 else if (t != 0)
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139 sum += pow2 (t/scl);
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140 }
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141
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142 void accum (std::complex<R> val)
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143 {
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144 accum (val.real ());
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145 accum (val.imag ());
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146 }
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147
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148 operator R () { return scl * std::sqrt (sum); }
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149 };
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150
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151 // norm accumulator for the 1-norm (city metric)
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152 template <class R>
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153 class norm_accumulator_1
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154 {
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155 R sum;
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156 public:
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157 norm_accumulator_1 () : sum (0) {}
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158 template<class U>
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159 void accum (U val)
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160 {
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161 sum += std::abs (val);
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162 }
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163 operator R () { return sum; }
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164 };
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165
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166 // norm accumulator for the inf-norm (max metric)
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167 template <class R>
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168 class norm_accumulator_inf
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169 {
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170 R max;
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171 public:
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172 norm_accumulator_inf () : max (0) {}
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173 template<class U>
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174 void accum (U val)
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175 {
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176 max = std::max (max, std::abs (val));
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177 }
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178 operator R () { return max; }
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179 };
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180
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181 // norm accumulator for the -inf pseudonorm (min abs value)
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182 template <class R>
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183 class norm_accumulator_minf
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184 {
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185 R min;
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186 public:
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187 norm_accumulator_minf () : min (octave_Inf) {}
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188 template<class U>
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189 void accum (U val)
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190 {
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191 min = std::min (min, std::abs (val));
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192 }
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193 operator R () { return min; }
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194 };
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195
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196 // norm accumulator for the 0-pseudonorm (hamming distance)
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197 template <class R>
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198 class norm_accumulator_0
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199 {
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200 unsigned int num;
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201 public:
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202 norm_accumulator_0 () : num (0) {}
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203 template<class U>
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204 void accum (U val)
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205 {
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206 if (val != static_cast<U> (0)) ++num;
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207 }
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208 operator R () { return num; }
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209 };
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210
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211
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212 // OK, we're armed :) Now let's go for the fun
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213
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214 template <class T, class R, class ACC>
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215 inline void vector_norm (const Array<T>& v, R& res, ACC acc)
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216 {
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217 for (octave_idx_type i = 0; i < v.numel (); i++)
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218 acc.accum (v(i));
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219
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220 res = acc;
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221 }
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222
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223 // dense versions
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224 template <class T, class R, class ACC>
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225 void column_norms (const MArray2<T>& m, MArray<R>& res, ACC acc)
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226 {
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227 res = MArray2<R> (1, m.columns ());
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228 for (octave_idx_type j = 0; j < m.columns (); j++)
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229 {
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230 ACC accj = acc;
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231 for (octave_idx_type i = 0; i < m.rows (); i++)
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232 accj.accum (m(i, j));
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233
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234 res.xelem (j) = accj;
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235 }
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236 }
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237
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238 template <class T, class R, class ACC>
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239 void row_norms (const MArray2<T>& m, MArray<R>& res, ACC acc)
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240 {
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241 res = MArray2<R> (m.rows (), 1);
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242 std::vector<ACC> acci (m.rows (), acc);
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243 for (octave_idx_type j = 0; j < m.columns (); j++)
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244 {
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245 for (octave_idx_type i = 0; i < m.rows (); i++)
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246 acci[i].accum (m(i, j));
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247 }
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248
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249 for (octave_idx_type i = 0; i < m.rows (); i++)
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250 res.xelem (i) = acci[i];
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251 }
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252
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253 // sparse versions
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254 template <class T, class R, class ACC>
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255 void column_norms (const MSparse<T>& m, MArray<R>& res, ACC acc)
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256 {
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257 res = MArray2<R> (1, m.columns ());
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258 for (octave_idx_type j = 0; j < m.columns (); j++)
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259 {
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260 ACC accj = acc;
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261 for (octave_idx_type k = m.cidx (j); k < m.cidx (j+1); k++)
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262 accj.accum (m.data (k));
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263
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264 res.xelem (j) = accj;
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265 }
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266 }
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267
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268 template <class T, class R, class ACC>
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269 void row_norms (const MSparse<T>& m, MArray<R>& res, ACC acc)
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270 {
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271 res = MArray2<R> (m.rows (), 1);
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272 std::vector<ACC> acci (m.rows (), acc);
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273 for (octave_idx_type j = 0; j < m.columns (); j++)
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274 {
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275 for (octave_idx_type k = m.cidx (j); k < m.cidx (j+1); k++)
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276 acci[m.ridx (k)].accum (m.data (k));
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277 }
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278
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279 for (octave_idx_type i = 0; i < m.rows (); i++)
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280 res.xelem (i) = acci[i];
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281 }
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282
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283 // now the dispatchers
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284 #define DEFINE_DISPATCHER(FUNC_NAME, ARG_TYPE, RES_TYPE) \
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285 template <class T, class R> \
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286 RES_TYPE FUNC_NAME (const ARG_TYPE& v, R p) \
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287 { \
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288 RES_TYPE res; \
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289 if (p == 2) \
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290 FUNC_NAME (v, res, norm_accumulator_2<R> ()); \
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291 else if (p == 1) \
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292 FUNC_NAME (v, res, norm_accumulator_1<R> ()); \
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293 else if (lo_ieee_isinf (p)) \
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294 { \
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295 if (p > 0) \
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296 FUNC_NAME (v, res, norm_accumulator_inf<R> ()); \
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297 else \
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298 FUNC_NAME (v, res, norm_accumulator_minf<R> ()); \
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299 } \
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300 else if (p == 0) \
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301 FUNC_NAME (v, res, norm_accumulator_0<R> ()); \
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302 else if (p > 0) \
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303 FUNC_NAME (v, res, norm_accumulator_p<R> (p)); \
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304 else \
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305 FUNC_NAME (v, res, norm_accumulator_mp<R> (p)); \
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306 return res; \
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307 }
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308
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309 DEFINE_DISPATCHER (vector_norm, MArray<T>, R)
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310 DEFINE_DISPATCHER (vector_norm, MArray2<T>, R)
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311 DEFINE_DISPATCHER (column_norms, MArray2<T>, MArray<R>)
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312 DEFINE_DISPATCHER (row_norms, MArray2<T>, MArray<R>)
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313 DEFINE_DISPATCHER (column_norms, MSparse<T>, MArray<R>)
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314 DEFINE_DISPATCHER (row_norms, MSparse<T>, MArray<R>)
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315
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316 // The approximate subproblem in Higham's method. Find lambda and mu such that
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317 // norm ([lambda, mu], p) == 1 and norm (y*lambda + col*mu, p) is maximized.
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318 // Real version. As in Higham's paper.
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319 template <class ColVectorT, class R>
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320 static void
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321 higham_subp (const ColVectorT& y, const ColVectorT& col,
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322 octave_idx_type nsamp, R p, R& lambda, R& mu)
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323 {
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324 R nrm = 0;
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325 for (octave_idx_type i = 0; i < nsamp; i++)
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326 {
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327 R fi = i*M_PI/nsamp, lambda1 = cos (fi), mu1 = sin (fi);
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328 R lmnr = std::pow (std::pow (std::abs (lambda1), p) +
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329 std::pow (std::abs (mu1), p), 1/p);
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330 lambda1 /= lmnr; mu1 /= lmnr;
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331 R nrm1 = vector_norm (lambda1 * y + mu1 * col, p);
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332 if (nrm1 > nrm)
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333 {
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334 lambda = lambda1;
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335 mu = mu1;
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336 nrm = nrm1;
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337 }
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338 }
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339 }
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340
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341 // Complex version. Higham's paper does not deal with complex case, so we use a simple
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342 // extension. First, guess the magnitudes as in real version, then try to rotate lambda
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343 // to improve further.
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344 template <class ColVectorT, class R>
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345 static void
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346 higham_subp (const ColVectorT& y, const ColVectorT& col,
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347 octave_idx_type nsamp, R p,
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348 std::complex<R>& lambda, std::complex<R>& mu)
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349 {
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350 typedef std::complex<R> CR;
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351 R nrm = 0;
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352 lambda = 1.0;
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353 CR lamcu = lambda / std::abs (lambda);
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354 // Probe magnitudes
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355 for (octave_idx_type i = 0; i < nsamp; i++)
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356 {
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357 R fi = i*M_PI/nsamp, lambda1 = cos (fi), mu1 = sin (fi);
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358 R lmnr = std::pow (std::pow (std::abs (lambda1), p) +
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359 std::pow (std::abs (mu1), p), 1/p);
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360 lambda1 /= lmnr; mu1 /= lmnr;
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361 R nrm1 = vector_norm (lambda1 * lamcu * y + mu1 * col, p);
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362 if (nrm1 > nrm)
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363 {
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364 lambda = lambda1 * lamcu;
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365 mu = mu1;
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366 nrm = nrm1;
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367 }
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368 }
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369 R lama = std::abs (lambda);
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370 // Probe orientation
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371 for (octave_idx_type i = 0; i < nsamp; i++)
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372 {
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373 R fi = i*M_PI/nsamp;
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374 lamcu = CR (cos (fi), sin (fi));
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375 R nrm1 = vector_norm (lama * lamcu * y + mu * col, p);
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376 if (nrm1 > nrm)
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377 {
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378 lambda = lama * lamcu;
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379 nrm = nrm1;
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380 }
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381 }
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382 }
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383
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384 // the p-dual element (should work for both real and complex)
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385 template <class T, class R>
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386 inline T elem_dual_p (T x, R p)
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387 {
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388 return signum (x) * std::pow (std::abs (x), p-1);
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389 }
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390
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391 // the VectorT is used for vectors, but actually it has to be
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392 // a Matrix type to allow all the operations. For instance SparseMatrix
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393 // does not support multiplication with column/row vectors.
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394 // the dual vector
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395 template <class VectorT, class R>
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396 VectorT dual_p (const VectorT& x, R p, R q)
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397 {
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398 VectorT res (x.dims ());
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399 for (octave_idx_type i = 0; i < x.numel (); i++)
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400 res.xelem (i) = elem_dual_p (x(i), p);
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401 return res / vector_norm (res, q);
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402 }
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403
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404 // Higham's hybrid method
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405 template <class MatrixT, class VectorT, class R>
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406 R higham (const MatrixT& m, R p, R tol, int maxiter,
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407 VectorT& x)
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408 {
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409 x.resize (m.columns (), 1);
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410 // the OSE part
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411 VectorT y(m.rows (), 1, 0), z(m.rows (), 1);
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412 typedef typename VectorT::element_type RR;
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413 RR lambda = 0, mu = 0;
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414 for (octave_idx_type k = 0; k < m.columns (); k++)
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415 {
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416 OCTAVE_QUIT;
8303
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417 VectorT col (m.column (k));
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418 if (k > 0)
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419 higham_subp (y, col, 4*k, p, lambda, mu);
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420 for (octave_idx_type i = 0; i < k; i++)
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421 x(i) *= lambda;
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422 x(k) = mu;
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423 y = lambda * y + mu * col;
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424 }
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425
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426 // the PM part
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427 x = x / vector_norm (x, p);
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428 R q = p/(p-1);
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429
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430 R gamma = 0, gamma1;
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431 int iter = 0;
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432 while (iter < maxiter)
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433 {
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434 OCTAVE_QUIT;
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435 y = m*x;
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436 gamma1 = gamma;
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437 gamma = vector_norm (y, p);
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438 z = dual_p (y, p, q);
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439 z = z.hermitian ();
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440 z = z * m;
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441
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442 if (iter > 0 && (vector_norm (z, q) <= gamma
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443 || (gamma - gamma1) <= tol*gamma))
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444 break;
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445
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446 z = z.hermitian ();
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447 x = dual_p (z, q, p);
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448 iter ++;
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449 }
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450
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451 return gamma;
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452 }
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453
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454 // derive column vector and SVD types
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455
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456 static const char *p_less1_gripe = "xnorm: p must be at least 1";
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457
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458 // Static constant to control the maximum number of iterations. 100 seems to be a good value.
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459 // Eventually, we can provide a means to change this constant from Octave.
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460 static int max_norm_iter = 100;
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461
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462 // version with SVD for dense matrices
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463 template <class MatrixT, class VectorT, class SVDT, class R>
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464 R matrix_norm (const MatrixT& m, R p, VectorT, SVDT)
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465 {
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466 R res = 0;
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467 if (p == 2)
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468 {
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469 SVDT svd (m, SVD::sigma_only);
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470 res = svd.singular_values () (0,0);
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471 }
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472 else if (p == 1)
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473 res = xcolnorms (m, 1).max ();
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474 else if (lo_ieee_isinf (p))
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475 res = xrownorms (m, 1).max ();
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476 else if (p > 1)
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477 {
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478 VectorT x;
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479 const R sqrteps = std::sqrt (std::numeric_limits<R>::epsilon ());
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480 res = higham (m, p, sqrteps, max_norm_iter, x);
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481 }
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482 else
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483 (*current_liboctave_error_handler) (p_less1_gripe);
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484
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485 return res;
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486 }
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487
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488 // SVD-free version for sparse matrices
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489 template <class MatrixT, class VectorT, class R>
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490 R matrix_norm (const MatrixT& m, R p, VectorT)
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491 {
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492 R res = 0;
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493 if (p == 1)
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494 res = xcolnorms (m, 1).max ();
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495 else if (lo_ieee_isinf (p))
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496 res = xrownorms (m, 1).max ();
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497 else if (p > 1)
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498 {
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499 VectorT x;
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500 const R sqrteps = std::sqrt (std::numeric_limits<R>::epsilon ());
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501 res = higham (m, p, sqrteps, max_norm_iter, x);
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502 }
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503 else
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504 (*current_liboctave_error_handler) (p_less1_gripe);
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505
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506 return res;
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507 }
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508
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509 // and finally, here's what we've promised in the header file
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510
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511 #define DEFINE_XNORM_FUNCS(PREFIX, RTYPE) \
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512 OCTAVE_API RTYPE xnorm (const PREFIX##ColumnVector& x, RTYPE p) \
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513 { return vector_norm (x, p); } \
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514 OCTAVE_API RTYPE xnorm (const PREFIX##RowVector& x, RTYPE p) \
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515 { return vector_norm (x, p); } \
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516 OCTAVE_API RTYPE xnorm (const PREFIX##Matrix& x, RTYPE p) \
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517 { return matrix_norm (x, p, PREFIX##Matrix (), PREFIX##SVD ()); } \
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518 OCTAVE_API RTYPE xfrobnorm (const PREFIX##Matrix& x) \
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519 { return vector_norm (x, static_cast<RTYPE> (2)); }
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520
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521 DEFINE_XNORM_FUNCS(, double)
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522 DEFINE_XNORM_FUNCS(Complex, double)
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523 DEFINE_XNORM_FUNCS(Float, float)
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524 DEFINE_XNORM_FUNCS(FloatComplex, float)
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525
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526 // this is needed to avoid copying the sparse matrix for xfrobnorm
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527 template <class T, class R>
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528 inline void array_norm_2 (const T* v, octave_idx_type n, R& res)
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529 {
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530 norm_accumulator_2<R> acc;
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531 for (octave_idx_type i = 0; i < n; i++)
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532 acc.accum (v[i]);
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533
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534 res = acc;
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535 }
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536
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537 #define DEFINE_XNORM_SPARSE_FUNCS(PREFIX, RTYPE) \
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538 OCTAVE_API RTYPE xnorm (const Sparse##PREFIX##Matrix& x, RTYPE p) \
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539 { return matrix_norm (x, p, PREFIX##Matrix ()); } \
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540 OCTAVE_API RTYPE xfrobnorm (const Sparse##PREFIX##Matrix& x) \
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541 { \
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542 RTYPE res; \
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543 array_norm_2 (x.data (), x.nnz (), res); \
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544 return res; \
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545 }
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546
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547 DEFINE_XNORM_SPARSE_FUNCS(, double)
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548 DEFINE_XNORM_SPARSE_FUNCS(Complex, double)
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549
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550 #define DEFINE_COLROW_NORM_FUNCS(PREFIX, RPREFIX, RTYPE) \
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551 extern OCTAVE_API RPREFIX##RowVector xcolnorms (const PREFIX##Matrix& m, RTYPE p) \
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552 { return column_norms (m, p); } \
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553 extern OCTAVE_API RPREFIX##ColumnVector xrownorms (const PREFIX##Matrix& m, RTYPE p) \
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554 { return row_norms (m, p); } \
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555
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556 DEFINE_COLROW_NORM_FUNCS(, , double)
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557 DEFINE_COLROW_NORM_FUNCS(Complex, , double)
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558 DEFINE_COLROW_NORM_FUNCS(Float, Float, float)
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559 DEFINE_COLROW_NORM_FUNCS(FloatComplex, Float, float)
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560
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561 DEFINE_COLROW_NORM_FUNCS(Sparse, , double)
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562 DEFINE_COLROW_NORM_FUNCS(SparseComplex, , double)
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563