1 | namespace Eigen {
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2 |
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3 | namespace internal {
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4 |
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5 | template <typename Scalar>
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6 | void lmpar(
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7 | Matrix< Scalar, Dynamic, Dynamic > &r,
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8 | const VectorXi &ipvt,
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9 | const Matrix< Scalar, Dynamic, 1 > &diag,
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10 | const Matrix< Scalar, Dynamic, 1 > &qtb,
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11 | Scalar delta,
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12 | Scalar &par,
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13 | Matrix< Scalar, Dynamic, 1 > &x)
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14 | {
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15 | using std::abs;
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16 | using std::sqrt;
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17 | typedef DenseIndex Index;
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18 |
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19 | /* Local variables */
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20 | Index i, j, l;
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21 | Scalar fp;
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22 | Scalar parc, parl;
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23 | Index iter;
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24 | Scalar temp, paru;
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25 | Scalar gnorm;
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26 | Scalar dxnorm;
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27 |
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28 |
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29 | /* Function Body */
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30 | const Scalar dwarf = (std::numeric_limits<Scalar>::min)();
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31 | const Index n = r.cols();
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32 | eigen_assert(n==diag.size());
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33 | eigen_assert(n==qtb.size());
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34 | eigen_assert(n==x.size());
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35 |
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36 | Matrix< Scalar, Dynamic, 1 > wa1, wa2;
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37 |
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38 | /* compute and store in x the gauss-newton direction. if the */
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39 | /* jacobian is rank-deficient, obtain a least squares solution. */
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40 | Index nsing = n-1;
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41 | wa1 = qtb;
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42 | for (j = 0; j < n; ++j) {
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43 | if (r(j,j) == 0. && nsing == n-1)
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44 | nsing = j - 1;
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45 | if (nsing < n-1)
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46 | wa1[j] = 0.;
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47 | }
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48 | for (j = nsing; j>=0; --j) {
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49 | wa1[j] /= r(j,j);
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50 | temp = wa1[j];
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51 | for (i = 0; i < j ; ++i)
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52 | wa1[i] -= r(i,j) * temp;
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53 | }
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54 |
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55 | for (j = 0; j < n; ++j)
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56 | x[ipvt[j]] = wa1[j];
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57 |
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58 | /* initialize the iteration counter. */
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59 | /* evaluate the function at the origin, and test */
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60 | /* for acceptance of the gauss-newton direction. */
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61 | iter = 0;
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62 | wa2 = diag.cwiseProduct(x);
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63 | dxnorm = wa2.blueNorm();
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64 | fp = dxnorm - delta;
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65 | if (fp <= Scalar(0.1) * delta) {
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66 | par = 0;
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67 | return;
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68 | }
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69 |
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70 | /* if the jacobian is not rank deficient, the newton */
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71 | /* step provides a lower bound, parl, for the zero of */
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72 | /* the function. otherwise set this bound to zero. */
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73 | parl = 0.;
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74 | if (nsing >= n-1) {
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75 | for (j = 0; j < n; ++j) {
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76 | l = ipvt[j];
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77 | wa1[j] = diag[l] * (wa2[l] / dxnorm);
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78 | }
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79 | // it's actually a triangularView.solveInplace(), though in a weird
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80 | // way:
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81 | for (j = 0; j < n; ++j) {
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82 | Scalar sum = 0.;
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83 | for (i = 0; i < j; ++i)
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84 | sum += r(i,j) * wa1[i];
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85 | wa1[j] = (wa1[j] - sum) / r(j,j);
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86 | }
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87 | temp = wa1.blueNorm();
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88 | parl = fp / delta / temp / temp;
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89 | }
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90 |
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91 | /* calculate an upper bound, paru, for the zero of the function. */
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92 | for (j = 0; j < n; ++j)
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93 | wa1[j] = r.col(j).head(j+1).dot(qtb.head(j+1)) / diag[ipvt[j]];
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94 |
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95 | gnorm = wa1.stableNorm();
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96 | paru = gnorm / delta;
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97 | if (paru == 0.)
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98 | paru = dwarf / (std::min)(delta,Scalar(0.1));
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99 |
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100 | /* if the input par lies outside of the interval (parl,paru), */
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101 | /* set par to the closer endpoint. */
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102 | par = (std::max)(par,parl);
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103 | par = (std::min)(par,paru);
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104 | if (par == 0.)
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105 | par = gnorm / dxnorm;
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106 |
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107 | /* beginning of an iteration. */
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108 | while (true) {
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109 | ++iter;
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110 |
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111 | /* evaluate the function at the current value of par. */
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112 | if (par == 0.)
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113 | par = (std::max)(dwarf,Scalar(.001) * paru); /* Computing MAX */
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114 | wa1 = sqrt(par)* diag;
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115 |
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116 | Matrix< Scalar, Dynamic, 1 > sdiag(n);
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117 | qrsolv<Scalar>(r, ipvt, wa1, qtb, x, sdiag);
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118 |
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119 | wa2 = diag.cwiseProduct(x);
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120 | dxnorm = wa2.blueNorm();
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121 | temp = fp;
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122 | fp = dxnorm - delta;
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123 |
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124 | /* if the function is small enough, accept the current value */
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125 | /* of par. also test for the exceptional cases where parl */
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126 | /* is zero or the number of iterations has reached 10. */
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127 | if (abs(fp) <= Scalar(0.1) * delta || (parl == 0. && fp <= temp && temp < 0.) || iter == 10)
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128 | break;
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129 |
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130 | /* compute the newton correction. */
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131 | for (j = 0; j < n; ++j) {
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132 | l = ipvt[j];
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133 | wa1[j] = diag[l] * (wa2[l] / dxnorm);
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134 | }
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135 | for (j = 0; j < n; ++j) {
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136 | wa1[j] /= sdiag[j];
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137 | temp = wa1[j];
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138 | for (i = j+1; i < n; ++i)
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139 | wa1[i] -= r(i,j) * temp;
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140 | }
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141 | temp = wa1.blueNorm();
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142 | parc = fp / delta / temp / temp;
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143 |
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144 | /* depending on the sign of the function, update parl or paru. */
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145 | if (fp > 0.)
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146 | parl = (std::max)(parl,par);
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147 | if (fp < 0.)
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148 | paru = (std::min)(paru,par);
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149 |
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150 | /* compute an improved estimate for par. */
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151 | /* Computing MAX */
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152 | par = (std::max)(parl,par+parc);
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153 |
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154 | /* end of an iteration. */
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155 | }
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156 |
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157 | /* termination. */
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158 | if (iter == 0)
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159 | par = 0.;
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160 | return;
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161 | }
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162 |
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163 | template <typename Scalar>
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164 | void lmpar2(
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165 | const ColPivHouseholderQR<Matrix< Scalar, Dynamic, Dynamic> > &qr,
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166 | const Matrix< Scalar, Dynamic, 1 > &diag,
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167 | const Matrix< Scalar, Dynamic, 1 > &qtb,
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168 | Scalar delta,
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169 | Scalar &par,
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170 | Matrix< Scalar, Dynamic, 1 > &x)
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171 |
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172 | {
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173 | using std::sqrt;
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174 | using std::abs;
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175 | typedef DenseIndex Index;
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176 |
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177 | /* Local variables */
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178 | Index j;
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179 | Scalar fp;
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180 | Scalar parc, parl;
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181 | Index iter;
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182 | Scalar temp, paru;
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183 | Scalar gnorm;
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184 | Scalar dxnorm;
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185 |
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186 |
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187 | /* Function Body */
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188 | const Scalar dwarf = (std::numeric_limits<Scalar>::min)();
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189 | const Index n = qr.matrixQR().cols();
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190 | eigen_assert(n==diag.size());
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191 | eigen_assert(n==qtb.size());
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192 |
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193 | Matrix< Scalar, Dynamic, 1 > wa1, wa2;
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194 |
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195 | /* compute and store in x the gauss-newton direction. if the */
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196 | /* jacobian is rank-deficient, obtain a least squares solution. */
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197 |
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198 | // const Index rank = qr.nonzeroPivots(); // exactly double(0.)
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199 | const Index rank = qr.rank(); // use a threshold
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200 | wa1 = qtb;
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201 | wa1.tail(n-rank).setZero();
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202 | qr.matrixQR().topLeftCorner(rank, rank).template triangularView<Upper>().solveInPlace(wa1.head(rank));
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203 |
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204 | x = qr.colsPermutation()*wa1;
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205 |
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206 | /* initialize the iteration counter. */
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207 | /* evaluate the function at the origin, and test */
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208 | /* for acceptance of the gauss-newton direction. */
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209 | iter = 0;
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210 | wa2 = diag.cwiseProduct(x);
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211 | dxnorm = wa2.blueNorm();
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212 | fp = dxnorm - delta;
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213 | if (fp <= Scalar(0.1) * delta) {
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214 | par = 0;
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215 | return;
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216 | }
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217 |
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218 | /* if the jacobian is not rank deficient, the newton */
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219 | /* step provides a lower bound, parl, for the zero of */
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220 | /* the function. otherwise set this bound to zero. */
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221 | parl = 0.;
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222 | if (rank==n) {
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223 | wa1 = qr.colsPermutation().inverse() * diag.cwiseProduct(wa2)/dxnorm;
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224 | qr.matrixQR().topLeftCorner(n, n).transpose().template triangularView<Lower>().solveInPlace(wa1);
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225 | temp = wa1.blueNorm();
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226 | parl = fp / delta / temp / temp;
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227 | }
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228 |
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229 | /* calculate an upper bound, paru, for the zero of the function. */
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230 | for (j = 0; j < n; ++j)
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231 | wa1[j] = qr.matrixQR().col(j).head(j+1).dot(qtb.head(j+1)) / diag[qr.colsPermutation().indices()(j)];
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232 |
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233 | gnorm = wa1.stableNorm();
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234 | paru = gnorm / delta;
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235 | if (paru == 0.)
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236 | paru = dwarf / (std::min)(delta,Scalar(0.1));
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237 |
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238 | /* if the input par lies outside of the interval (parl,paru), */
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239 | /* set par to the closer endpoint. */
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240 | par = (std::max)(par,parl);
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241 | par = (std::min)(par,paru);
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242 | if (par == 0.)
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243 | par = gnorm / dxnorm;
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244 |
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245 | /* beginning of an iteration. */
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246 | Matrix< Scalar, Dynamic, Dynamic > s = qr.matrixQR();
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247 | while (true) {
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248 | ++iter;
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249 |
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250 | /* evaluate the function at the current value of par. */
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251 | if (par == 0.)
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252 | par = (std::max)(dwarf,Scalar(.001) * paru); /* Computing MAX */
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253 | wa1 = sqrt(par)* diag;
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254 |
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255 | Matrix< Scalar, Dynamic, 1 > sdiag(n);
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256 | qrsolv<Scalar>(s, qr.colsPermutation().indices(), wa1, qtb, x, sdiag);
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257 |
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258 | wa2 = diag.cwiseProduct(x);
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259 | dxnorm = wa2.blueNorm();
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260 | temp = fp;
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261 | fp = dxnorm - delta;
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262 |
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263 | /* if the function is small enough, accept the current value */
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264 | /* of par. also test for the exceptional cases where parl */
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265 | /* is zero or the number of iterations has reached 10. */
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266 | if (abs(fp) <= Scalar(0.1) * delta || (parl == 0. && fp <= temp && temp < 0.) || iter == 10)
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267 | break;
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268 |
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269 | /* compute the newton correction. */
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270 | wa1 = qr.colsPermutation().inverse() * diag.cwiseProduct(wa2/dxnorm);
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271 | // we could almost use this here, but the diagonal is outside qr, in sdiag[]
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272 | // qr.matrixQR().topLeftCorner(n, n).transpose().template triangularView<Lower>().solveInPlace(wa1);
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273 | for (j = 0; j < n; ++j) {
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274 | wa1[j] /= sdiag[j];
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275 | temp = wa1[j];
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276 | for (Index i = j+1; i < n; ++i)
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277 | wa1[i] -= s(i,j) * temp;
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278 | }
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279 | temp = wa1.blueNorm();
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280 | parc = fp / delta / temp / temp;
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281 |
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282 | /* depending on the sign of the function, update parl or paru. */
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283 | if (fp > 0.)
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284 | parl = (std::max)(parl,par);
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285 | if (fp < 0.)
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286 | paru = (std::min)(paru,par);
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287 |
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288 | /* compute an improved estimate for par. */
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289 | par = (std::max)(parl,par+parc);
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290 | }
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291 | if (iter == 0)
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292 | par = 0.;
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293 | return;
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294 | }
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295 |
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296 | } // end namespace internal
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297 |
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298 | } // end namespace Eigen
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