| 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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