LMpar.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// This code initially comes from MINPACK whose original authors are:
// Copyright Jorge More - Argonne National Laboratory
// Copyright Burt Garbow - Argonne National Laboratory
// Copyright Ken Hillstrom - Argonne National Laboratory
//
// This Source Code Form is subject to the terms of the Minpack license
// (a BSD-like license) described in the campaigned CopyrightMINPACK.txt file.

#ifndef EIGEN_LMPAR_H
#define EIGEN_LMPAR_H

// IWYU pragma: private
#include "./InternalHeaderCheck.h"

namespace Eigen {

namespace internal {

template <typename QRSolver, typename VectorType>
void lmpar2(const QRSolver &qr, const VectorType &diag, const VectorType &qtb, typename VectorType::Scalar m_delta,
            typename VectorType::Scalar &par, VectorType &x)

{
  using std::abs;
  using std::sqrt;
  typedef typename QRSolver::MatrixType MatrixType;
  typedef typename QRSolver::Scalar Scalar;
  //    typedef typename QRSolver::StorageIndex StorageIndex;

  /* Local variables */
  Index j;
  Scalar fp;
  Scalar parc, parl;
  Index iter;
  Scalar temp, paru;
  Scalar gnorm;
  Scalar dxnorm;

  // Make a copy of the triangular factor.
  // This copy is modified during call the qrsolv
  MatrixType s;
  s = qr.matrixR();

  /* Function Body */
  const Scalar dwarf = (std::numeric_limits<Scalar>::min)();
  const Index n = qr.matrixR().cols();
  eigen_assert(n == diag.size());
  eigen_assert(n == qtb.size());

  VectorType wa1, wa2;

  /* compute and store in x the gauss-newton direction. if the */
  /* jacobian is rank-deficient, obtain a least squares solution. */

  //    const Index rank = qr.nonzeroPivots(); // exactly double(0.)
  const Index rank = qr.rank();  // use a threshold
  wa1 = qtb;
  wa1.tail(n - rank).setZero();
  // FIXME There is no solve in place for sparse triangularView
  wa1.head(rank) = s.topLeftCorner(rank, rank).template triangularView<Upper>().solve(qtb.head(rank));

  x = qr.colsPermutation() * wa1;

  /* initialize the iteration counter. */
  /* evaluate the function at the origin, and test */
  /* for acceptance of the gauss-newton direction. */
  iter = 0;
  wa2 = diag.cwiseProduct(x);
  dxnorm = wa2.blueNorm();
  fp = dxnorm - m_delta;
  if (fp <= Scalar(0.1) * m_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 (rank == n) {
    wa1 = qr.colsPermutation().inverse() * diag.cwiseProduct(wa2) / dxnorm;
    s.topLeftCorner(n, n).transpose().template triangularView<Lower>().solveInPlace(wa1);
    temp = wa1.blueNorm();
    parl = fp / m_delta / temp / temp;
  }

  /* calculate an upper bound, paru, for the zero of the function. */
  for (j = 0; j < n; ++j) wa1[j] = s.col(j).head(j + 1).dot(qtb.head(j + 1)) / diag[qr.colsPermutation().indices()(j)];

  gnorm = wa1.stableNorm();
  paru = gnorm / m_delta;
  if (paru == 0.) paru = dwarf / (std::min)(m_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 = sqrt(par) * diag;

    VectorType sdiag(n);
    lmqrsolv(s, qr.colsPermutation(), wa1, qtb, x, sdiag);

    wa2 = diag.cwiseProduct(x);
    dxnorm = wa2.blueNorm();
    temp = fp;
    fp = dxnorm - m_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 (abs(fp) <= Scalar(0.1) * m_delta || (parl == 0. && fp <= temp && temp < 0.) || iter == 10) break;

    /* compute the newton correction. */
    wa1 = qr.colsPermutation().inverse() * diag.cwiseProduct(wa2 / dxnorm);
    // we could almost use this here, but the diagonal is outside qr, in sdiag[]
    for (j = 0; j < n; ++j) {
      wa1[j] /= sdiag[j];
      temp = wa1[j];
      for (Index i = j + 1; i < n; ++i) wa1[i] -= s.coeff(i, j) * temp;
    }
    temp = wa1.blueNorm();
    parc = fp / m_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. */
    par = (std::max)(parl, par + parc);
  }
  if (iter == 0) par = 0.;
  return;
}
}  // end namespace internal

}  // end namespace Eigen

#endif  // EIGEN_LMPAR_H