1 | // This file is part of Eigen, a lightweight C++ template library
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2 | // for linear algebra.
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3 | //
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4 | // Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
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5 | //
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6 | // This Source Code Form is subject to the terms of the Mozilla
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7 | // Public License v. 2.0. If a copy of the MPL was not distributed
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8 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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9 |
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10 | #ifndef EIGEN_STABLENORM_H
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11 | #define EIGEN_STABLENORM_H
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12 |
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13 | namespace Eigen {
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14 |
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15 | namespace internal {
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16 |
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17 | template<typename ExpressionType, typename Scalar>
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18 | inline void stable_norm_kernel(const ExpressionType& bl, Scalar& ssq, Scalar& scale, Scalar& invScale)
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19 | {
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20 | using std::max;
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21 | Scalar maxCoeff = bl.cwiseAbs().maxCoeff();
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22 |
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23 | if (maxCoeff>scale)
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24 | {
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25 | ssq = ssq * numext::abs2(scale/maxCoeff);
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26 | Scalar tmp = Scalar(1)/maxCoeff;
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27 | if(tmp > NumTraits<Scalar>::highest())
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28 | {
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29 | invScale = NumTraits<Scalar>::highest();
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30 | scale = Scalar(1)/invScale;
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31 | }
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32 | else
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33 | {
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34 | scale = maxCoeff;
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35 | invScale = tmp;
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36 | }
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37 | }
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38 |
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39 | // TODO if the maxCoeff is much much smaller than the current scale,
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40 | // then we can neglect this sub vector
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41 | if(scale>Scalar(0)) // if scale==0, then bl is 0
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42 | ssq += (bl*invScale).squaredNorm();
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43 | }
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44 |
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45 | template<typename Derived>
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46 | inline typename NumTraits<typename traits<Derived>::Scalar>::Real
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47 | blueNorm_impl(const EigenBase<Derived>& _vec)
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48 | {
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49 | typedef typename Derived::RealScalar RealScalar;
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50 | typedef typename Derived::Index Index;
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51 | using std::pow;
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52 | using std::min;
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53 | using std::max;
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54 | using std::sqrt;
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55 | using std::abs;
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56 | const Derived& vec(_vec.derived());
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57 | static bool initialized = false;
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58 | static RealScalar b1, b2, s1m, s2m, overfl, rbig, relerr;
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59 | if(!initialized)
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60 | {
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61 | int ibeta, it, iemin, iemax, iexp;
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62 | RealScalar eps;
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63 | // This program calculates the machine-dependent constants
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64 | // bl, b2, slm, s2m, relerr overfl
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65 | // from the "basic" machine-dependent numbers
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66 | // nbig, ibeta, it, iemin, iemax, rbig.
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67 | // The following define the basic machine-dependent constants.
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68 | // For portability, the PORT subprograms "ilmaeh" and "rlmach"
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69 | // are used. For any specific computer, each of the assignment
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70 | // statements can be replaced
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71 | ibeta = std::numeric_limits<RealScalar>::radix; // base for floating-point numbers
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72 | it = std::numeric_limits<RealScalar>::digits; // number of base-beta digits in mantissa
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73 | iemin = std::numeric_limits<RealScalar>::min_exponent; // minimum exponent
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74 | iemax = std::numeric_limits<RealScalar>::max_exponent; // maximum exponent
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75 | rbig = (std::numeric_limits<RealScalar>::max)(); // largest floating-point number
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76 |
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77 | iexp = -((1-iemin)/2);
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78 | b1 = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // lower boundary of midrange
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79 | iexp = (iemax + 1 - it)/2;
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80 | b2 = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // upper boundary of midrange
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81 |
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82 | iexp = (2-iemin)/2;
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83 | s1m = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // scaling factor for lower range
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84 | iexp = - ((iemax+it)/2);
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85 | s2m = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // scaling factor for upper range
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86 |
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87 | overfl = rbig*s2m; // overflow boundary for abig
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88 | eps = RealScalar(pow(double(ibeta), 1-it));
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89 | relerr = sqrt(eps); // tolerance for neglecting asml
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90 | initialized = true;
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91 | }
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92 | Index n = vec.size();
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93 | RealScalar ab2 = b2 / RealScalar(n);
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94 | RealScalar asml = RealScalar(0);
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95 | RealScalar amed = RealScalar(0);
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96 | RealScalar abig = RealScalar(0);
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97 | for(typename Derived::InnerIterator it(vec, 0); it; ++it)
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98 | {
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99 | RealScalar ax = abs(it.value());
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100 | if(ax > ab2) abig += numext::abs2(ax*s2m);
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101 | else if(ax < b1) asml += numext::abs2(ax*s1m);
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102 | else amed += numext::abs2(ax);
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103 | }
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104 | if(abig > RealScalar(0))
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105 | {
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106 | abig = sqrt(abig);
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107 | if(abig > overfl)
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108 | {
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109 | return rbig;
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110 | }
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111 | if(amed > RealScalar(0))
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112 | {
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113 | abig = abig/s2m;
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114 | amed = sqrt(amed);
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115 | }
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116 | else
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117 | return abig/s2m;
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118 | }
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119 | else if(asml > RealScalar(0))
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120 | {
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121 | if (amed > RealScalar(0))
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122 | {
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123 | abig = sqrt(amed);
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124 | amed = sqrt(asml) / s1m;
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125 | }
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126 | else
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127 | return sqrt(asml)/s1m;
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128 | }
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129 | else
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130 | return sqrt(amed);
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131 | asml = (min)(abig, amed);
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132 | abig = (max)(abig, amed);
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133 | if(asml <= abig*relerr)
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134 | return abig;
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135 | else
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136 | return abig * sqrt(RealScalar(1) + numext::abs2(asml/abig));
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137 | }
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138 |
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139 | } // end namespace internal
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140 |
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141 | /** \returns the \em l2 norm of \c *this avoiding underflow and overflow.
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142 | * This version use a blockwise two passes algorithm:
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143 | * 1 - find the absolute largest coefficient \c s
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144 | * 2 - compute \f$ s \Vert \frac{*this}{s} \Vert \f$ in a standard way
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145 | *
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146 | * For architecture/scalar types supporting vectorization, this version
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147 | * is faster than blueNorm(). Otherwise the blueNorm() is much faster.
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148 | *
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149 | * \sa norm(), blueNorm(), hypotNorm()
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150 | */
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151 | template<typename Derived>
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152 | inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real
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153 | MatrixBase<Derived>::stableNorm() const
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154 | {
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155 | using std::min;
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156 | using std::sqrt;
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157 | const Index blockSize = 4096;
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158 | RealScalar scale(0);
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159 | RealScalar invScale(1);
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160 | RealScalar ssq(0); // sum of square
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161 | enum {
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162 | Alignment = (int(Flags)&DirectAccessBit) || (int(Flags)&AlignedBit) ? 1 : 0
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163 | };
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164 | Index n = size();
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165 | Index bi = internal::first_aligned(derived());
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166 | if (bi>0)
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167 | internal::stable_norm_kernel(this->head(bi), ssq, scale, invScale);
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168 | for (; bi<n; bi+=blockSize)
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169 | internal::stable_norm_kernel(this->segment(bi,(min)(blockSize, n - bi)).template forceAlignedAccessIf<Alignment>(), ssq, scale, invScale);
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170 | return scale * sqrt(ssq);
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171 | }
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172 |
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173 | /** \returns the \em l2 norm of \c *this using the Blue's algorithm.
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174 | * A Portable Fortran Program to Find the Euclidean Norm of a Vector,
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175 | * ACM TOMS, Vol 4, Issue 1, 1978.
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176 | *
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177 | * For architecture/scalar types without vectorization, this version
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178 | * is much faster than stableNorm(). Otherwise the stableNorm() is faster.
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179 | *
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180 | * \sa norm(), stableNorm(), hypotNorm()
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181 | */
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182 | template<typename Derived>
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183 | inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real
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184 | MatrixBase<Derived>::blueNorm() const
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185 | {
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186 | return internal::blueNorm_impl(*this);
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187 | }
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188 |
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189 | /** \returns the \em l2 norm of \c *this avoiding undeflow and overflow.
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190 | * This version use a concatenation of hypot() calls, and it is very slow.
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191 | *
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192 | * \sa norm(), stableNorm()
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193 | */
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194 | template<typename Derived>
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195 | inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real
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196 | MatrixBase<Derived>::hypotNorm() const
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197 | {
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198 | return this->cwiseAbs().redux(internal::scalar_hypot_op<RealScalar>());
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199 | }
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200 |
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201 | } // end namespace Eigen
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202 |
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203 | #endif // EIGEN_STABLENORM_H
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