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annotate liboctave/numeric/oct-norm.cc @ 20830:b65888ec820e draft default tip gccjit

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dmalcom gcc jit import
author Stefan Mahr <dac922@gmx.de>
date Fri, 27 Feb 2015 16:59:36 +0100
parents 277b12eed117
children
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1 /*
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2
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3 Copyright (C) 2008-2015 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 "mx-cm-s.h"
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40 #include "mx-s-cm.h"
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41 #include "mx-fcm-fs.h"
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42 #include "mx-fs-fcm.h"
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43 #include "Array.h"
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44 #include "Array-util.h"
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45 #include "CMatrix.h"
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46 #include "dMatrix.h"
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47 #include "fCMatrix.h"
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48 #include "fMatrix.h"
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49 #include "CColVector.h"
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50 #include "dColVector.h"
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51 #include "CRowVector.h"
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52 #include "dRowVector.h"
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53 #include "fCColVector.h"
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54 #include "fColVector.h"
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55 #include "fCRowVector.h"
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56 #include "fRowVector.h"
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57 #include "CSparse.h"
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58 #include "dSparse.h"
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59 #include "dbleSVD.h"
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60 #include "CmplxSVD.h"
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61 #include "floatSVD.h"
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62 #include "fCmplxSVD.h"
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63
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64 // Theory: norm accumulator is an object that has an accum method able
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65 // to handle both real and complex element, and a cast operator
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66 // returning the intermediate norm. Reference: Higham, N. "Estimating
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67 // the Matrix p-Norm." Numer. Math. 62, 539-555, 1992.
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68
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69 // norm accumulator for the p-norm
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70 template <class R>
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71 class norm_accumulator_p
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72 {
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73 R p,scl,sum;
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74 public:
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75 norm_accumulator_p () {} // we need this one for Array
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76 norm_accumulator_p (R pp) : p(pp), scl(0), sum(1) {}
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77
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78 template<class U>
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79 void accum (U val)
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80 {
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81 octave_quit ();
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82 R t = std::abs (val);
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83 if (scl == t) // we need this to handle Infs properly
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84 sum += 1;
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85 else if (scl < t)
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86 {
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87 sum *= std::pow (scl/t, p);
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88 sum += 1;
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89 scl = t;
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90 }
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91 else if (t != 0)
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92 sum += std::pow (t/scl, p);
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93 }
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94 operator R () { return scl * std::pow (sum, 1/p); }
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95 };
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96
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97 // norm accumulator for the minus p-pseudonorm
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98 template <class R>
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99 class norm_accumulator_mp
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100 {
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101 R p,scl,sum;
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102 public:
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103 norm_accumulator_mp () {} // we need this one for Array
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104 norm_accumulator_mp (R pp) : p(pp), scl(0), sum(1) {}
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105
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106 template<class U>
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107 void accum (U val)
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108 {
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109 octave_quit ();
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110 R t = 1 / std::abs (val);
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111 if (scl == t)
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112 sum += 1;
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113 else if (scl < t)
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114 {
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115 sum *= std::pow (scl/t, p);
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116 sum += 1;
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117 scl = t;
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118 }
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119 else if (t != 0)
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120 sum += std::pow (t/scl, p);
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121 }
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122 operator R () { return scl * std::pow (sum, -1/p); }
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123 };
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124
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125 // norm accumulator for the 2-norm (euclidean)
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126 template <class R>
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127 class norm_accumulator_2
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128 {
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129 R scl,sum;
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130 static R pow2 (R x) { return x*x; }
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131 public:
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132 norm_accumulator_2 () : scl(0), sum(1) {}
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133
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134 void accum (R val)
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135 {
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136 R t = std::abs (val);
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137 if (scl == t)
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138 sum += 1;
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139 else if (scl < t)
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140 {
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141 sum *= pow2 (scl/t);
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142 sum += 1;
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143 scl = t;
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144 }
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145 else if (t != 0)
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146 sum += pow2 (t/scl);
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147 }
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148
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149 void accum (std::complex<R> val)
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150 {
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151 accum (val.real ());
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152 accum (val.imag ());
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153 }
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154
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155 operator R () { return scl * std::sqrt (sum); }
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156 };
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157
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158 // norm accumulator for the 1-norm (city metric)
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159 template <class R>
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160 class norm_accumulator_1
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161 {
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162 R sum;
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163 public:
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164 norm_accumulator_1 () : sum (0) {}
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165 template<class U>
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166 void accum (U val)
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167 {
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168 sum += std::abs (val);
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169 }
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170 operator R () { return sum; }
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171 };
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172
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173 // norm accumulator for the inf-norm (max metric)
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174 template <class R>
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175 class norm_accumulator_inf
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176 {
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177 R max;
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178 public:
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179 norm_accumulator_inf () : max (0) {}
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180 template<class U>
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181 void accum (U val)
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182 {
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183 max = std::max (max, std::abs (val));
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184 }
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185 operator R () { return max; }
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186 };
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187
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188 // norm accumulator for the -inf pseudonorm (min abs value)
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189 template <class R>
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190 class norm_accumulator_minf
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191 {
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192 R min;
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193 public:
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194 norm_accumulator_minf () : min (octave_Inf) {}
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195 template<class U>
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196 void accum (U val)
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197 {
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198 min = std::min (min, std::abs (val));
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199 }
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200 operator R () { return min; }
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201 };
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202
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203 // norm accumulator for the 0-pseudonorm (hamming distance)
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204 template <class R>
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205 class norm_accumulator_0
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206 {
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207 unsigned int num;
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208 public:
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209 norm_accumulator_0 () : num (0) {}
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210 template<class U>
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211 void accum (U val)
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212 {
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213 if (val != static_cast<U> (0)) ++num;
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214 }
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215 operator R () { return num; }
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216 };
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217
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218
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219 // OK, we're armed :) Now let's go for the fun
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220
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221 template <class T, class R, class ACC>
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222 inline void vector_norm (const Array<T>& v, R& res, ACC acc)
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223 {
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224 for (octave_idx_type i = 0; i < v.numel (); i++)
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225 acc.accum (v(i));
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226
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227 res = acc;
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228 }
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229
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230 // dense versions
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231 template <class T, class R, class ACC>
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232 void column_norms (const MArray<T>& m, MArray<R>& res, ACC acc)
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233 {
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234 res = MArray<R> (dim_vector (1, m.columns ()));
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235 for (octave_idx_type j = 0; j < m.columns (); j++)
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236 {
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237 ACC accj = acc;
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238 for (octave_idx_type i = 0; i < m.rows (); i++)
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239 accj.accum (m(i, j));
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240
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241 res.xelem (j) = accj;
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242 }
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243 }
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244
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245 template <class T, class R, class ACC>
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246 void row_norms (const MArray<T>& m, MArray<R>& res, ACC acc)
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247 {
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248 res = MArray<R> (dim_vector (m.rows (), 1));
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249 std::vector<ACC> acci (m.rows (), acc);
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250 for (octave_idx_type j = 0; j < m.columns (); j++)
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251 {
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252 for (octave_idx_type i = 0; i < m.rows (); i++)
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253 acci[i].accum (m(i, j));
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254 }
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255
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256 for (octave_idx_type i = 0; i < m.rows (); i++)
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257 res.xelem (i) = acci[i];
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258 }
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259
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260 // sparse versions
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261 template <class T, class R, class ACC>
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262 void column_norms (const MSparse<T>& m, MArray<R>& res, ACC acc)
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263 {
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264 res = MArray<R> (dim_vector (1, m.columns ()));
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265 for (octave_idx_type j = 0; j < m.columns (); j++)
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266 {
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267 ACC accj = acc;
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268 for (octave_idx_type k = m.cidx (j); k < m.cidx (j+1); k++)
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269 accj.accum (m.data (k));
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270
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271 res.xelem (j) = accj;
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272 }
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273 }
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274
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275 template <class T, class R, class ACC>
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276 void row_norms (const MSparse<T>& m, MArray<R>& res, ACC acc)
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277 {
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278 res = MArray<R> (dim_vector (m.rows (), 1));
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279 std::vector<ACC> acci (m.rows (), acc);
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280 for (octave_idx_type j = 0; j < m.columns (); j++)
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281 {
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282 for (octave_idx_type k = m.cidx (j); k < m.cidx (j+1); k++)
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283 acci[m.ridx (k)].accum (m.data (k));
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284 }
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285
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286 for (octave_idx_type i = 0; i < m.rows (); i++)
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287 res.xelem (i) = acci[i];
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288 }
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289
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290 // now the dispatchers
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291 #define DEFINE_DISPATCHER(FUNC_NAME, ARG_TYPE, RES_TYPE) \
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292 template <class T, class R> \
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293 RES_TYPE FUNC_NAME (const ARG_TYPE& v, R p) \
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294 { \
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295 RES_TYPE res; \
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296 if (p == 2) \
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297 FUNC_NAME (v, res, norm_accumulator_2<R> ()); \
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298 else if (p == 1) \
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299 FUNC_NAME (v, res, norm_accumulator_1<R> ()); \
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300 else if (lo_ieee_isinf (p)) \
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301 { \
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302 if (p > 0) \
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303 FUNC_NAME (v, res, norm_accumulator_inf<R> ()); \
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304 else \
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305 FUNC_NAME (v, res, norm_accumulator_minf<R> ()); \
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306 } \
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307 else if (p == 0) \
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308 FUNC_NAME (v, res, norm_accumulator_0<R> ()); \
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309 else if (p > 0) \
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310 FUNC_NAME (v, res, norm_accumulator_p<R> (p)); \
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311 else \
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312 FUNC_NAME (v, res, norm_accumulator_mp<R> (p)); \
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313 return res; \
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314 }
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315
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316 DEFINE_DISPATCHER (vector_norm, MArray<T>, R)
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317 DEFINE_DISPATCHER (column_norms, MArray<T>, MArray<R>)
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318 DEFINE_DISPATCHER (row_norms, MArray<T>, MArray<R>)
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319 DEFINE_DISPATCHER (column_norms, MSparse<T>, MArray<R>)
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320 DEFINE_DISPATCHER (row_norms, MSparse<T>, MArray<R>)
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321
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322 // The approximate subproblem in Higham's method. Find lambda and mu such that
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323 // norm ([lambda, mu], p) == 1 and norm (y*lambda + col*mu, p) is maximized.
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324 // Real version. As in Higham's paper.
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325 template <class ColVectorT, class R>
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326 static void
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327 higham_subp (const ColVectorT& y, const ColVectorT& col,
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328 octave_idx_type nsamp, R p, R& lambda, R& mu)
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329 {
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330 R nrm = 0;
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331 for (octave_idx_type i = 0; i < nsamp; i++)
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332 {
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333 octave_quit ();
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334 R fi = i * static_cast<R> (M_PI) / nsamp;
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335 R lambda1 = cos (fi);
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336 R mu1 = sin (fi);
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337 R lmnr = std::pow (std::pow (std::abs (lambda1), p) +
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338 std::pow (std::abs (mu1), p), 1/p);
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339 lambda1 /= lmnr; mu1 /= lmnr;
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340 R nrm1 = vector_norm (lambda1 * y + mu1 * col, p);
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341 if (nrm1 > nrm)
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342 {
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343 lambda = lambda1;
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344 mu = mu1;
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345 nrm = nrm1;
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346 }
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347 }
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348 }
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349
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350 // Complex version. Higham's paper does not deal with complex case, so we use a
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351 // simple extension. First, guess the magnitudes as in real version, then try
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352 // to rotate lambda to improve further.
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353 template <class ColVectorT, class R>
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354 static void
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355 higham_subp (const ColVectorT& y, const ColVectorT& col,
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356 octave_idx_type nsamp, R p,
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357 std::complex<R>& lambda, std::complex<R>& mu)
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358 {
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359 typedef std::complex<R> CR;
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360 R nrm = 0;
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361 lambda = 1.0;
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362 CR lamcu = lambda / std::abs (lambda);
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363 // Probe magnitudes
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364 for (octave_idx_type i = 0; i < nsamp; i++)
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365 {
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366 octave_quit ();
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367 R fi = i * static_cast<R> (M_PI) / nsamp;
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368 R lambda1 = cos (fi);
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369 R mu1 = sin (fi);
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370 R lmnr = std::pow (std::pow (std::abs (lambda1), p) +
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371 std::pow (std::abs (mu1), p), 1/p);
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372 lambda1 /= lmnr; mu1 /= lmnr;
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373 R nrm1 = vector_norm (lambda1 * lamcu * y + mu1 * col, p);
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374 if (nrm1 > nrm)
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375 {
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376 lambda = lambda1 * lamcu;
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377 mu = mu1;
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378 nrm = nrm1;
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379 }
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380 }
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381 R lama = std::abs (lambda);
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382 // Probe orientation
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383 for (octave_idx_type i = 0; i < nsamp; i++)
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384 {
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385 octave_quit ();
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386 R fi = i * static_cast<R> (M_PI) / nsamp;
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387 lamcu = CR (cos (fi), sin (fi));
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388 R nrm1 = vector_norm (lama * lamcu * y + mu * col, p);
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389 if (nrm1 > nrm)
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390 {
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391 lambda = lama * lamcu;
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392 nrm = nrm1;
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393 }
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394 }
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395 }
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396
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397 // the p-dual element (should work for both real and complex)
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398 template <class T, class R>
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399 inline T elem_dual_p (T x, R p)
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400 {
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401 return signum (x) * std::pow (std::abs (x), p-1);
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402 }
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403
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404 // the VectorT is used for vectors, but actually it has to be
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405 // a Matrix type to allow all the operations. For instance SparseMatrix
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406 // does not support multiplication with column/row vectors.
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407 // the dual vector
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diff changeset
408 template <class VectorT, class R>
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409 VectorT dual_p (const VectorT& x, R p, R q)
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410 {
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411 VectorT res (x.dims ());
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412 for (octave_idx_type i = 0; i < x.numel (); i++)
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413 res.xelem (i) = elem_dual_p (x(i), p);
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414 return res / vector_norm (res, q);
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415 }
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416
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417 // Higham's hybrid method
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418 template <class MatrixT, class VectorT, class R>
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419 R higham (const MatrixT& m, R p, R tol, int maxiter,
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420 VectorT& x)
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421 {
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422 x.resize (m.columns (), 1);
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423 // the OSE part
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424 VectorT y(m.rows (), 1, 0), z(m.rows (), 1);
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425 typedef typename VectorT::element_type RR;
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426 RR lambda = 0;
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diff changeset
427 RR mu = 1;
8303
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428 for (octave_idx_type k = 0; k < m.columns (); k++)
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429 {
10142
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430 octave_quit ();
8303
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diff changeset
431 VectorT col (m.column (k));
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diff changeset
432 if (k > 0)
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433 higham_subp (y, col, 4*k, p, lambda, mu);
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diff changeset
434 for (octave_idx_type i = 0; i < k; i++)
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diff changeset
435 x(i) *= lambda;
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diff changeset
436 x(k) = mu;
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diff changeset
437 y = lambda * y + mu * col;
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diff changeset
438 }
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439
8303
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440 // the PM part
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441 x = x / vector_norm (x, p);
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442 R q = p/(p-1);
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443
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444 R gamma = 0, gamma1;
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445 int iter = 0;
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446 while (iter < maxiter)
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447 {
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448 octave_quit ();
8303
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449 y = m*x;
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450 gamma1 = gamma;
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diff changeset
451 gamma = vector_norm (y, p);
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452 z = dual_p (y, p, q);
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diff changeset
453 z = z.hermitian ();
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454 z = z * m;
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455
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456 if (iter > 0 && (vector_norm (z, q) <= gamma
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457 || (gamma - gamma1) <= tol*gamma))
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458 break;
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459
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diff changeset
460 z = z.hermitian ();
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461 x = dual_p (z, q, p);
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462 iter ++;
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463 }
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464
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465 return gamma;
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466 }
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467
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468 // derive column vector and SVD types
8303
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469
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470 static const char *p_less1_gripe = "xnorm: p must be at least 1";
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471
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49a5a4be04a1 maint: Use GNU style coding conventions for code in liboctave/
Rik <rik@octave.org>
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472 // Static constant to control the maximum number of iterations. 100 seems to
49a5a4be04a1 maint: Use GNU style coding conventions for code in liboctave/
Rik <rik@octave.org>
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473 // be a good value. Eventually, we can provide a means to change this
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474 // constant from Octave.
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475 static int max_norm_iter = 100;
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476
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477 // version with SVD for dense matrices
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478 template <class MatrixT, class VectorT, class SVDT, class R>
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479 R matrix_norm (const MatrixT& m, R p, VectorT, SVDT)
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480 {
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481 R res = 0;
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482 if (p == 2)
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483 {
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484 SVDT svd (m, SVD::sigma_only);
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485 res = svd.singular_values () (0,0);
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486 }
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487 else if (p == 1)
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488 res = xcolnorms (m, 1).max ();
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489 else if (lo_ieee_isinf (p))
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490 res = xrownorms (m, 1).max ();
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491 else if (p > 1)
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492 {
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493 VectorT x;
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494 const R sqrteps = std::sqrt (std::numeric_limits<R>::epsilon ());
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495 res = higham (m, p, sqrteps, max_norm_iter, x);
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496 }
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497 else
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498 (*current_liboctave_error_handler) (p_less1_gripe);
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499
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500 return res;
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501 }
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502
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503 // SVD-free version for sparse matrices
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504 template <class MatrixT, class VectorT, class R>
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505 R matrix_norm (const MatrixT& m, R p, VectorT)
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506 {
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507 R res = 0;
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508 if (p == 1)
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509 res = xcolnorms (m, 1).max ();
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510 else if (lo_ieee_isinf (p))
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511 res = xrownorms (m, 1).max ();
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512 else if (p > 1)
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513 {
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514 VectorT x;
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515 const R sqrteps = std::sqrt (std::numeric_limits<R>::epsilon ());
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516 res = higham (m, p, sqrteps, max_norm_iter, x);
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517 }
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518 else
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519 (*current_liboctave_error_handler) (p_less1_gripe);
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520
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521 return res;
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522 }
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523
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524 // and finally, here's what we've promised in the header file
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525
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526 #define DEFINE_XNORM_FUNCS(PREFIX, RTYPE) \
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527 OCTAVE_API RTYPE xnorm (const PREFIX##ColumnVector& x, RTYPE p) \
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528 { return vector_norm (x, p); } \
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529 OCTAVE_API RTYPE xnorm (const PREFIX##RowVector& x, RTYPE p) \
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530 { return vector_norm (x, p); } \
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Michael Goffioul <michael.goffioul@gmail.com>
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531 OCTAVE_API RTYPE xnorm (const PREFIX##Matrix& x, RTYPE p) \
8303
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532 { return matrix_norm (x, p, PREFIX##Matrix (), PREFIX##SVD ()); } \
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533 OCTAVE_API RTYPE xfrobnorm (const PREFIX##Matrix& x) \
8303
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534 { return vector_norm (x, static_cast<RTYPE> (2)); }
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535
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536 DEFINE_XNORM_FUNCS(, double)
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537 DEFINE_XNORM_FUNCS(Complex, double)
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parents:
diff changeset
538 DEFINE_XNORM_FUNCS(Float, float)
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parents:
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539 DEFINE_XNORM_FUNCS(FloatComplex, float)
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parents:
diff changeset
540
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parents:
diff changeset
541 // this is needed to avoid copying the sparse matrix for xfrobnorm
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542 template <class T, class R>
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diff changeset
543 inline void array_norm_2 (const T* v, octave_idx_type n, R& res)
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544 {
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545 norm_accumulator_2<R> acc;
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Jaroslav Hajek <highegg@gmail.com>
parents:
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546 for (octave_idx_type i = 0; i < n; i++)
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diff changeset
547 acc.accum (v[i]);
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parents:
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548
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549 res = acc;
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550 }
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551
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parents:
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552 #define DEFINE_XNORM_SPARSE_FUNCS(PREFIX, RTYPE) \
8319
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Michael Goffioul <michael.goffioul@gmail.com>
parents: 8309
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553 OCTAVE_API RTYPE xnorm (const Sparse##PREFIX##Matrix& x, RTYPE p) \
8303
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554 { return matrix_norm (x, p, PREFIX##Matrix ()); } \
8319
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parents: 8309
diff changeset
555 OCTAVE_API RTYPE xfrobnorm (const Sparse##PREFIX##Matrix& x) \
8303
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parents:
diff changeset
556 { \
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parents:
diff changeset
557 RTYPE res; \
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parents:
diff changeset
558 array_norm_2 (x.data (), x.nnz (), res); \
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parents:
diff changeset
559 return res; \
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diff changeset
560 }
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parents:
diff changeset
561
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562 DEFINE_XNORM_SPARSE_FUNCS(, double)
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parents:
diff changeset
563 DEFINE_XNORM_SPARSE_FUNCS(Complex, double)
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parents:
diff changeset
564
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Jaroslav Hajek <highegg@gmail.com>
parents:
diff changeset
565 #define DEFINE_COLROW_NORM_FUNCS(PREFIX, RPREFIX, RTYPE) \
8319
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Michael Goffioul <michael.goffioul@gmail.com>
parents: 8309
diff changeset
566 extern OCTAVE_API RPREFIX##RowVector xcolnorms (const PREFIX##Matrix& m, RTYPE p) \
8303
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parents:
diff changeset
567 { return column_norms (m, p); } \
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Michael Goffioul <michael.goffioul@gmail.com>
parents: 8309
diff changeset
568 extern OCTAVE_API RPREFIX##ColumnVector xrownorms (const PREFIX##Matrix& m, RTYPE p) \
8303
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parents:
diff changeset
569 { return row_norms (m, p); } \
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parents:
diff changeset
570
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parents:
diff changeset
571 DEFINE_COLROW_NORM_FUNCS(, , double)
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parents:
diff changeset
572 DEFINE_COLROW_NORM_FUNCS(Complex, , double)
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parents:
diff changeset
573 DEFINE_COLROW_NORM_FUNCS(Float, Float, float)
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parents:
diff changeset
574 DEFINE_COLROW_NORM_FUNCS(FloatComplex, Float, float)
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parents:
diff changeset
575
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Jaroslav Hajek <highegg@gmail.com>
parents:
diff changeset
576 DEFINE_COLROW_NORM_FUNCS(Sparse, , double)
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parents:
diff changeset
577 DEFINE_COLROW_NORM_FUNCS(SparseComplex, , double)
b11c31849b44 improve norm computation capabilities
Jaroslav Hajek <highegg@gmail.com>
parents:
diff changeset
578