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template <typename Scalar>
void ei_lmpar(
Matrix< Scalar, Dynamic, Dynamic > &r,
VectorXi &ipvt, // TODO : const once ipvt mess fixed
const Matrix< Scalar, Dynamic, 1 > &diag,
const Matrix< Scalar, Dynamic, 1 > &qtb,
Scalar delta,
Scalar &par,
Matrix< Scalar, Dynamic, 1 > &x)
{
/* Local variables */
int i, j, l;
Scalar fp;
Scalar parc, parl;
int iter;
Scalar temp, paru;
Scalar gnorm;
Scalar dxnorm;
/* Function Body */
const Scalar dwarf = std::numeric_limits<Scalar>::min();
const int n = r.cols();
assert(n==diag.size());
assert(n==qtb.size());
assert(n==x.size());
Matrix< Scalar, Dynamic, 1 > wa1, wa2;
/* compute and store in x the gauss-newton direction. if the */
/* jacobian is rank-deficient, obtain a least squares solution. */
int nsing = n-1;
wa1 = qtb;
for (j = 0; j < n; ++j) {
if (r(j,j) == 0. && nsing == n-1)
nsing = j - 1;
if (nsing < n-1)
wa1[j] = 0.;
}
for (j = nsing; j>=0; --j) {
wa1[j] /= r(j,j);
temp = wa1[j];
for (i = 0; i < j ; ++i)
wa1[i] -= r(i,j) * temp;
}
for (j = 0; j < n; ++j)
x[ipvt[j]] = wa1[j];
/* initialize the iteration counter. */
/* evaluate the function at the origin, and test */
/* for acceptance of the gauss-newton direction. */
iter = 0;
wa2 = diag.cwise() * x;
dxnorm = wa2.blueNorm();
fp = dxnorm - delta;
if (fp <= Scalar(0.1) * delta) {
par = 0;
return;
}
/* if the jacobian is not rank deficient, the newton */
/* step provides a lower bound, parl, for the zero of */
/* the function. otherwise set this bound to zero. */
parl = 0.;
if (nsing >= n-1) {
for (j = 0; j < n; ++j) {
l = ipvt[j];
wa1[j] = diag[l] * (wa2[l] / dxnorm);
}
// it's actually a triangularView.solveInplace(), though in a weird
// way:
for (j = 0; j < n; ++j) {
Scalar sum = 0.;
for (i = 0; i < j; ++i)
sum += r(i,j) * wa1[i];
wa1[j] = (wa1[j] - sum) / r(j,j);
}
temp = wa1.blueNorm();
parl = fp / delta / temp / temp;
}
/* calculate an upper bound, paru, for the zero of the function. */
for (j = 0; j < n; ++j)
wa1[j] = r.col(j).head(j+1).dot(qtb.head(j+1)) / diag[ipvt[j]];
gnorm = wa1.stableNorm();
paru = gnorm / delta;
if (paru == 0.)
paru = dwarf / std::min(delta,Scalar(0.1));
/* if the input par lies outside of the interval (parl,paru), */
/* set par to the closer endpoint. */
par = std::max(par,parl);
par = std::min(par,paru);
if (par == 0.)
par = gnorm / dxnorm;
/* beginning of an iteration. */
while (true) {
++iter;
/* evaluate the function at the current value of par. */
if (par == 0.)
par = std::max(dwarf,Scalar(.001) * paru); /* Computing MAX */
wa1 = ei_sqrt(par)* diag;
Matrix< Scalar, Dynamic, 1 > sdiag(n);
ei_qrsolv<Scalar>(r, ipvt, wa1, qtb, x, sdiag);
wa2 = diag.cwise() * x;
dxnorm = wa2.blueNorm();
temp = fp;
fp = dxnorm - delta;
/* if the function is small enough, accept the current value */
/* of par. also test for the exceptional cases where parl */
/* is zero or the number of iterations has reached 10. */
if (ei_abs(fp) <= Scalar(0.1) * delta || (parl == 0. && fp <= temp && temp < 0.) || iter == 10)
break;
/* compute the newton correction. */
for (j = 0; j < n; ++j) {
l = ipvt[j];
wa1[j] = diag[l] * (wa2[l] / dxnorm);
}
for (j = 0; j < n; ++j) {
wa1[j] /= sdiag[j];
temp = wa1[j];
for (i = j+1; i < n; ++i)
wa1[i] -= r(i,j) * temp;
}
temp = wa1.blueNorm();
parc = fp / delta / temp / temp;
/* depending on the sign of the function, update parl or paru. */
if (fp > 0.)
parl = std::max(parl,par);
if (fp < 0.)
paru = std::min(paru,par);
/* compute an improved estimate for par. */
/* Computing MAX */
par = std::max(parl,par+parc);
/* end of an iteration. */
}
/* termination. */
if (iter == 0)
par = 0.;
return;
}
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