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 | Scalar max = bl.cwiseAbs().maxCoeff(); |
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21 | if (max>scale) |
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22 | { |
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23 | ssq = ssq * abs2(scale/max); |
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24 | scale = max; |
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25 | invScale = Scalar(1)/scale; |
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26 | } |
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27 | // TODO if the max is much much smaller than the current scale, |
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28 | // then we can neglect this sub vector |
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29 | ssq += (bl*invScale).squaredNorm(); |
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30 | } |
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31 | } |
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32 | |
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33 | /** \returns the \em l2 norm of \c *this avoiding underflow and overflow. |
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34 | * This version use a blockwise two passes algorithm: |
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35 | * 1 - find the absolute largest coefficient \c s |
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36 | * 2 - compute \f$ s \Vert \frac{*this}{s} \Vert \f$ in a standard way |
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37 | * |
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38 | * For architecture/scalar types supporting vectorization, this version |
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39 | * is faster than blueNorm(). Otherwise the blueNorm() is much faster. |
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40 | * |
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41 | * \sa norm(), blueNorm(), hypotNorm() |
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42 | */ |
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43 | template<typename Derived> |
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44 | inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real |
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45 | MatrixBase<Derived>::stableNorm() const |
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46 | { |
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47 | using std::min; |
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48 | const Index blockSize = 4096; |
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49 | RealScalar scale(0); |
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50 | RealScalar invScale(1); |
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51 | RealScalar ssq(0); // sum of square |
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52 | enum { |
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53 | Alignment = (int(Flags)&DirectAccessBit) || (int(Flags)&AlignedBit) ? 1 : 0 |
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54 | }; |
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55 | Index n = size(); |
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56 | Index bi = internal::first_aligned(derived()); |
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57 | if (bi>0) |
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58 | internal::stable_norm_kernel(this->head(bi), ssq, scale, invScale); |
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59 | for (; bi<n; bi+=blockSize) |
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60 | internal::stable_norm_kernel(this->segment(bi,(min)(blockSize, n - bi)).template forceAlignedAccessIf<Alignment>(), ssq, scale, invScale); |
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61 | return scale * internal::sqrt(ssq); |
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62 | } |
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63 | |
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64 | /** \returns the \em l2 norm of \c *this using the Blue's algorithm. |
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65 | * A Portable Fortran Program to Find the Euclidean Norm of a Vector, |
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66 | * ACM TOMS, Vol 4, Issue 1, 1978. |
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67 | * |
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68 | * For architecture/scalar types without vectorization, this version |
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69 | * is much faster than stableNorm(). Otherwise the stableNorm() is faster. |
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70 | * |
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71 | * \sa norm(), stableNorm(), hypotNorm() |
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72 | */ |
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73 | template<typename Derived> |
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74 | inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real |
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75 | MatrixBase<Derived>::blueNorm() const |
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76 | { |
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77 | using std::pow; |
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78 | using std::min; |
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79 | using std::max; |
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80 | static bool initialized = false; |
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81 | static RealScalar b1, b2, s1m, s2m, overfl, rbig, relerr; |
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82 | if(!initialized) |
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83 | { |
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84 | int ibeta, it, iemin, iemax, iexp; |
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85 | RealScalar abig, eps; |
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86 | // This program calculates the machine-dependent constants |
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87 | // bl, b2, slm, s2m, relerr overfl |
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88 | // from the "basic" machine-dependent numbers |
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89 | // ibeta, it, iemin, iemax, rbig. |
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90 | // The following define the basic machine-dependent constants. |
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91 | // For portability, the PORT subprograms "ilmaeh" and "rlmach" |
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92 | // are used. For any specific computer, each of the assignment |
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93 | // statements can be replaced |
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94 | ibeta = std::numeric_limits<RealScalar>::radix; // base for floating-point numbers |
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95 | it = std::numeric_limits<RealScalar>::digits; // number of base-beta digits in mantissa |
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96 | iemin = std::numeric_limits<RealScalar>::min_exponent; // minimum exponent |
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97 | iemax = std::numeric_limits<RealScalar>::max_exponent; // maximum exponent |
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98 | rbig = (std::numeric_limits<RealScalar>::max)(); // largest floating-point number |
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99 | |
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100 | iexp = -((1-iemin)/2); |
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101 | b1 = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // lower boundary of midrange |
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102 | iexp = (iemax + 1 - it)/2; |
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103 | b2 = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // upper boundary of midrange |
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104 | |
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105 | iexp = (2-iemin)/2; |
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106 | s1m = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // scaling factor for lower range |
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107 | iexp = - ((iemax+it)/2); |
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108 | s2m = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // scaling factor for upper range |
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109 | |
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110 | overfl = rbig*s2m; // overflow boundary for abig |
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111 | eps = RealScalar(pow(double(ibeta), 1-it)); |
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112 | relerr = internal::sqrt(eps); // tolerance for neglecting asml |
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113 | abig = RealScalar(1.0/eps - 1.0); |
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114 | initialized = true; |
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115 | } |
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116 | Index n = size(); |
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117 | RealScalar ab2 = b2 / RealScalar(n); |
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118 | RealScalar asml = RealScalar(0); |
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119 | RealScalar amed = RealScalar(0); |
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120 | RealScalar abig = RealScalar(0); |
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121 | for(Index j=0; j<n; ++j) |
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122 | { |
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123 | RealScalar ax = internal::abs(coeff(j)); |
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124 | if(ax > ab2) abig += internal::abs2(ax*s2m); |
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125 | else if(ax < b1) asml += internal::abs2(ax*s1m); |
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126 | else amed += internal::abs2(ax); |
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127 | } |
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128 | if(abig > RealScalar(0)) |
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129 | { |
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130 | abig = internal::sqrt(abig); |
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131 | if(abig > overfl) |
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132 | { |
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133 | return rbig; |
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134 | } |
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135 | if(amed > RealScalar(0)) |
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136 | { |
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137 | abig = abig/s2m; |
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138 | amed = internal::sqrt(amed); |
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139 | } |
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140 | else |
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141 | return abig/s2m; |
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142 | } |
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143 | else if(asml > RealScalar(0)) |
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144 | { |
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145 | if (amed > RealScalar(0)) |
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146 | { |
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147 | abig = internal::sqrt(amed); |
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148 | amed = internal::sqrt(asml) / s1m; |
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149 | } |
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150 | else |
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151 | return internal::sqrt(asml)/s1m; |
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152 | } |
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153 | else |
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154 | return internal::sqrt(amed); |
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155 | asml = (min)(abig, amed); |
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156 | abig = (max)(abig, amed); |
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157 | if(asml <= abig*relerr) |
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158 | return abig; |
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159 | else |
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160 | return abig * internal::sqrt(RealScalar(1) + internal::abs2(asml/abig)); |
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161 | } |
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162 | |
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163 | /** \returns the \em l2 norm of \c *this avoiding undeflow and overflow. |
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164 | * This version use a concatenation of hypot() calls, and it is very slow. |
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165 | * |
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166 | * \sa norm(), stableNorm() |
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167 | */ |
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168 | template<typename Derived> |
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169 | inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real |
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170 | MatrixBase<Derived>::hypotNorm() const |
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171 | { |
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172 | return this->cwiseAbs().redux(internal::scalar_hypot_op<RealScalar>()); |
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173 | } |
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174 | |
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175 | } // end namespace Eigen |
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176 | |
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177 | #endif // EIGEN_STABLENORM_H |
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