unsupported/test/NonLinearOptimization.cpp

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Thomas Capricelli <orzel@freehackers.org>

#include <stdio.h>

#include "main.h"
#include <unsupported/Eigen/NonLinearOptimization>

// This disables some useless Warnings on MSVC.
// It is intended to be done for this test only.
#include <Eigen/src/Core/util/DisableStupidWarnings.h>

// tolerance for checking number of iterations
#define LM_EVAL_COUNT_TOL 2

#define LM_CHECK_N_ITERS(SOLVER, NFEV, NJEV)         \
  {                                                  \
    VERIFY(SOLVER.nfev <= NFEV * LM_EVAL_COUNT_TOL); \
    VERIFY(SOLVER.njev <= NJEV * LM_EVAL_COUNT_TOL); \
  }

int fcn_chkder(const VectorXd &x, VectorXd &fvec, MatrixXd &fjac, int iflag) {
  /*      subroutine fcn for chkder example. */

  int i;
  assert(15 == fvec.size());
  assert(3 == x.size());
  double tmp1, tmp2, tmp3, tmp4;
  static const double y[15] = {1.4e-1, 1.8e-1, 2.2e-1, 2.5e-1, 2.9e-1, 3.2e-1, 3.5e-1, 3.9e-1,
                               3.7e-1, 5.8e-1, 7.3e-1, 9.6e-1, 1.34,   2.1,    4.39};

  if (iflag == 0) return 0;

  if (iflag != 2)
    for (i = 0; i < 15; i++) {
      tmp1 = i + 1;
      tmp2 = 16 - i - 1;
      tmp3 = tmp1;
      if (i >= 8) tmp3 = tmp2;
      fvec[i] = y[i] - (x[0] + tmp1 / (x[1] * tmp2 + x[2] * tmp3));
    }
  else {
    for (i = 0; i < 15; i++) {
      tmp1 = i + 1;
      tmp2 = 16 - i - 1;

      /* error introduced into next statement for illustration. */
      /* corrected statement should read    tmp3 = tmp1 . */

      tmp3 = tmp2;
      if (i >= 8) tmp3 = tmp2;
      tmp4 = (x[1] * tmp2 + x[2] * tmp3);
      tmp4 = tmp4 * tmp4;
      fjac(i, 0) = -1.;
      fjac(i, 1) = tmp1 * tmp2 / tmp4;
      fjac(i, 2) = tmp1 * tmp3 / tmp4;
    }
  }
  return 0;
}

void testChkder() {
  const int m = 15, n = 3;
  VectorXd x(n), fvec(m), xp, fvecp(m), err;
  MatrixXd fjac(m, n);
  VectorXi ipvt;

  /*      the following values should be suitable for */
  /*      checking the jacobian matrix. */
  x << 9.2e-1, 1.3e-1, 5.4e-1;

  internal::chkder(x, fvec, fjac, xp, fvecp, 1, err);
  fcn_chkder(x, fvec, fjac, 1);
  fcn_chkder(x, fvec, fjac, 2);
  fcn_chkder(xp, fvecp, fjac, 1);
  internal::chkder(x, fvec, fjac, xp, fvecp, 2, err);

  fvecp -= fvec;

  // check those
  VectorXd fvec_ref(m), fvecp_ref(m), err_ref(m);
  fvec_ref << -1.181606, -1.429655, -1.606344, -1.745269, -1.840654, -1.921586, -1.984141, -2.022537, -2.468977,
      -2.827562, -3.473582, -4.437612, -6.047662, -9.267761, -18.91806;
  fvecp_ref << -7.724666e-09, -3.432406e-09, -2.034843e-10, 2.313685e-09, 4.331078e-09, 5.984096e-09, 7.363281e-09,
      8.53147e-09, 1.488591e-08, 2.33585e-08, 3.522012e-08, 5.301255e-08, 8.26666e-08, 1.419747e-07, 3.19899e-07;
  err_ref << 0.1141397, 0.09943516, 0.09674474, 0.09980447, 0.1073116, 0.1220445, 0.1526814, 1, 1, 1, 1, 1, 1, 1, 1;

  VERIFY_IS_APPROX(fvec, fvec_ref);
  VERIFY_IS_APPROX(fvecp, fvecp_ref);
  VERIFY_IS_APPROX(err, err_ref);
}

// Generic functor
template <typename Scalar_, int NX = Dynamic, int NY = Dynamic>
struct Functor {
  typedef Scalar_ Scalar;
  enum { InputsAtCompileTime = NX, ValuesAtCompileTime = NY };
  typedef Matrix<Scalar, InputsAtCompileTime, 1> InputType;
  typedef Matrix<Scalar, ValuesAtCompileTime, 1> ValueType;
  typedef Matrix<Scalar, ValuesAtCompileTime, InputsAtCompileTime> JacobianType;

  const int m_inputs, m_values;

  Functor() : m_inputs(InputsAtCompileTime), m_values(ValuesAtCompileTime) {}
  Functor(int inputs, int values) : m_inputs(inputs), m_values(values) {}

  int inputs() const { return m_inputs; }
  int values() const { return m_values; }

  // you should define that in the subclass :
  //  void operator() (const InputType& x, ValueType* v, JacobianType* _j=0) const;
};

struct lmder_functor : Functor<double> {
  lmder_functor(void) : Functor<double>(3, 15) {}
  int operator()(const VectorXd &x, VectorXd &fvec) const {
    double tmp1, tmp2, tmp3;
    static const double y[15] = {1.4e-1, 1.8e-1, 2.2e-1, 2.5e-1, 2.9e-1, 3.2e-1, 3.5e-1, 3.9e-1,
                                 3.7e-1, 5.8e-1, 7.3e-1, 9.6e-1, 1.34,   2.1,    4.39};

    for (int i = 0; i < values(); i++) {
      tmp1 = i + 1;
      tmp2 = 16 - i - 1;
      tmp3 = (i >= 8) ? tmp2 : tmp1;
      fvec[i] = y[i] - (x[0] + tmp1 / (x[1] * tmp2 + x[2] * tmp3));
    }
    return 0;
  }

  int df(const VectorXd &x, MatrixXd &fjac) const {
    double tmp1, tmp2, tmp3, tmp4;
    for (int i = 0; i < values(); i++) {
      tmp1 = i + 1;
      tmp2 = 16 - i - 1;
      tmp3 = (i >= 8) ? tmp2 : tmp1;
      tmp4 = (x[1] * tmp2 + x[2] * tmp3);
      tmp4 = tmp4 * tmp4;
      fjac(i, 0) = -1;
      fjac(i, 1) = tmp1 * tmp2 / tmp4;
      fjac(i, 2) = tmp1 * tmp3 / tmp4;
    }
    return 0;
  }
};

void testLmder1() {
  int n = 3, info;

  VectorXd x;

  /* the following starting values provide a rough fit. */
  x.setConstant(n, 1.);

  // do the computation
  lmder_functor functor;
  LevenbergMarquardt<lmder_functor> lm(functor);
  info = lm.lmder1(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 6, 5);

  // check norm
  VERIFY_IS_APPROX(lm.fvec.blueNorm(), 0.09063596);

  // check x
  VectorXd x_ref(n);
  x_ref << 0.08241058, 1.133037, 2.343695;
  VERIFY_IS_APPROX(x, x_ref);
}

void testLmder() {
  const int m = 15, n = 3;
  int info;
  double fnorm, covfac;
  VectorXd x;

  /* the following starting values provide a rough fit. */
  x.setConstant(n, 1.);

  // do the computation
  lmder_functor functor;
  LevenbergMarquardt<lmder_functor> lm(functor);
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return values
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 6, 5);

  // check norm
  fnorm = lm.fvec.blueNorm();
  VERIFY_IS_APPROX(fnorm, 0.09063596);

  // check x
  VectorXd x_ref(n);
  x_ref << 0.08241058, 1.133037, 2.343695;
  VERIFY_IS_APPROX(x, x_ref);

  // check covariance
  covfac = fnorm * fnorm / (m - n);
  internal::covar(lm.fjac, lm.permutation.indices());  // TODO : move this as a function of lm

  MatrixXd cov_ref(n, n);
  cov_ref << 0.0001531202, 0.002869941, -0.002656662, 0.002869941, 0.09480935, -0.09098995, -0.002656662, -0.09098995,
      0.08778727;

  //  std::cout << fjac*covfac << std::endl;

  MatrixXd cov;
  cov = covfac * lm.fjac.topLeftCorner<n, n>();
  VERIFY_IS_APPROX(cov, cov_ref);
  // TODO: why isn't this allowed ? :
  // VERIFY_IS_APPROX( covfac*fjac.topLeftCorner<n,n>() , cov_ref);
}

struct hybrj_functor : Functor<double> {
  hybrj_functor(void) : Functor<double>(9, 9) {}

  int operator()(const VectorXd &x, VectorXd &fvec) {
    double temp, temp1, temp2;
    const VectorXd::Index n = x.size();
    assert(fvec.size() == n);
    for (VectorXd::Index k = 0; k < n; k++) {
      temp = (3. - 2. * x[k]) * x[k];
      temp1 = 0.;
      if (k) temp1 = x[k - 1];
      temp2 = 0.;
      if (k != n - 1) temp2 = x[k + 1];
      fvec[k] = temp - temp1 - 2. * temp2 + 1.;
    }
    return 0;
  }
  int df(const VectorXd &x, MatrixXd &fjac) {
    const VectorXd::Index n = x.size();
    assert(fjac.rows() == n);
    assert(fjac.cols() == n);
    for (VectorXd::Index k = 0; k < n; k++) {
      for (VectorXd::Index j = 0; j < n; j++) fjac(k, j) = 0.;
      fjac(k, k) = 3. - 4. * x[k];
      if (k) fjac(k, k - 1) = -1.;
      if (k != n - 1) fjac(k, k + 1) = -2.;
    }
    return 0;
  }
};

void testHybrj1() {
  const int n = 9;
  int info;
  VectorXd x(n);

  /* the following starting values provide a rough fit. */
  x.setConstant(n, -1.);

  // do the computation
  hybrj_functor functor;
  HybridNonLinearSolver<hybrj_functor> solver(functor);
  info = solver.hybrj1(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(solver, 11, 1);

  // check norm
  VERIFY_IS_APPROX(solver.fvec.blueNorm(), 1.192636e-08);

  // check x
  VectorXd x_ref(n);
  x_ref << -0.5706545, -0.6816283, -0.7017325, -0.7042129, -0.701369, -0.6918656, -0.665792, -0.5960342, -0.4164121;
  VERIFY_IS_APPROX(x, x_ref);
}

void testHybrj() {
  const int n = 9;
  int info;
  VectorXd x(n);

  /* the following starting values provide a rough fit. */
  x.setConstant(n, -1.);

  // do the computation
  hybrj_functor functor;
  HybridNonLinearSolver<hybrj_functor> solver(functor);
  solver.diag.setConstant(n, 1.);
  solver.useExternalScaling = true;
  info = solver.solve(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(solver, 11, 1);

  // check norm
  VERIFY_IS_APPROX(solver.fvec.blueNorm(), 1.192636e-08);

  // check x
  VectorXd x_ref(n);
  x_ref << -0.5706545, -0.6816283, -0.7017325, -0.7042129, -0.701369, -0.6918656, -0.665792, -0.5960342, -0.4164121;
  VERIFY_IS_APPROX(x, x_ref);
}

struct hybrd_functor : Functor<double> {
  hybrd_functor(void) : Functor<double>(9, 9) {}
  int operator()(const VectorXd &x, VectorXd &fvec) const {
    double temp, temp1, temp2;
    const VectorXd::Index n = x.size();

    assert(fvec.size() == n);
    for (VectorXd::Index k = 0; k < n; k++) {
      temp = (3. - 2. * x[k]) * x[k];
      temp1 = 0.;
      if (k) temp1 = x[k - 1];
      temp2 = 0.;
      if (k != n - 1) temp2 = x[k + 1];
      fvec[k] = temp - temp1 - 2. * temp2 + 1.;
    }
    return 0;
  }
};

void testHybrd1() {
  int n = 9, info;
  VectorXd x(n);

  /* the following starting values provide a rough solution. */
  x.setConstant(n, -1.);

  // do the computation
  hybrd_functor functor;
  HybridNonLinearSolver<hybrd_functor> solver(functor);
  info = solver.hybrd1(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  VERIFY(solver.nfev <= 20 * LM_EVAL_COUNT_TOL);

  // check norm
  VERIFY_IS_APPROX(solver.fvec.blueNorm(), 1.192636e-08);

  // check x
  VectorXd x_ref(n);
  x_ref << -0.5706545, -0.6816283, -0.7017325, -0.7042129, -0.701369, -0.6918656, -0.665792, -0.5960342, -0.4164121;
  VERIFY_IS_APPROX(x, x_ref);
}

void testHybrd() {
  const int n = 9;
  int info;
  VectorXd x;

  /* the following starting values provide a rough fit. */
  x.setConstant(n, -1.);

  // do the computation
  hybrd_functor functor;
  HybridNonLinearSolver<hybrd_functor> solver(functor);
  solver.parameters.nb_of_subdiagonals = 1;
  solver.parameters.nb_of_superdiagonals = 1;
  solver.diag.setConstant(n, 1.);
  solver.useExternalScaling = true;
  info = solver.solveNumericalDiff(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  VERIFY(solver.nfev <= 14 * LM_EVAL_COUNT_TOL);

  // check norm
  VERIFY_IS_APPROX(solver.fvec.blueNorm(), 1.192636e-08);

  // check x
  VectorXd x_ref(n);
  x_ref << -0.5706545, -0.6816283, -0.7017325, -0.7042129, -0.701369, -0.6918656, -0.665792, -0.5960342, -0.4164121;
  VERIFY_IS_APPROX(x, x_ref);
}

struct lmstr_functor : Functor<double> {
  lmstr_functor(void) : Functor<double>(3, 15) {}
  int operator()(const VectorXd &x, VectorXd &fvec) {
    /*  subroutine fcn for lmstr1 example. */
    double tmp1, tmp2, tmp3;
    static const double y[15] = {1.4e-1, 1.8e-1, 2.2e-1, 2.5e-1, 2.9e-1, 3.2e-1, 3.5e-1, 3.9e-1,
                                 3.7e-1, 5.8e-1, 7.3e-1, 9.6e-1, 1.34,   2.1,    4.39};

    assert(15 == fvec.size());
    assert(3 == x.size());

    for (int i = 0; i < 15; i++) {
      tmp1 = i + 1;
      tmp2 = 16 - i - 1;
      tmp3 = (i >= 8) ? tmp2 : tmp1;
      fvec[i] = y[i] - (x[0] + tmp1 / (x[1] * tmp2 + x[2] * tmp3));
    }
    return 0;
  }
  int df(const VectorXd &x, VectorXd &jac_row, VectorXd::Index rownb) {
    assert(x.size() == 3);
    assert(jac_row.size() == x.size());
    double tmp1, tmp2, tmp3, tmp4;

    VectorXd::Index i = rownb - 2;
    tmp1 = i + 1;
    tmp2 = 16 - i - 1;
    tmp3 = (i >= 8) ? tmp2 : tmp1;
    tmp4 = (x[1] * tmp2 + x[2] * tmp3);
    tmp4 = tmp4 * tmp4;
    jac_row[0] = -1;
    jac_row[1] = tmp1 * tmp2 / tmp4;
    jac_row[2] = tmp1 * tmp3 / tmp4;
    return 0;
  }
};

void testLmstr1() {
  const int n = 3;
  int info;

  VectorXd x(n);

  /* the following starting values provide a rough fit. */
  x.setConstant(n, 1.);

  // do the computation
  lmstr_functor functor;
  LevenbergMarquardt<lmstr_functor> lm(functor);
  info = lm.lmstr1(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 6, 5);

  // check norm
  VERIFY_IS_APPROX(lm.fvec.blueNorm(), 0.09063596);

  // check x
  VectorXd x_ref(n);
  x_ref << 0.08241058, 1.133037, 2.343695;
  VERIFY_IS_APPROX(x, x_ref);
}

void testLmstr() {
  const int n = 3;
  int info;
  double fnorm;
  VectorXd x(n);

  /* the following starting values provide a rough fit. */
  x.setConstant(n, 1.);

  // do the computation
  lmstr_functor functor;
  LevenbergMarquardt<lmstr_functor> lm(functor);
  info = lm.minimizeOptimumStorage(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return values
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 6, 5);

  // check norm
  fnorm = lm.fvec.blueNorm();
  VERIFY_IS_APPROX(fnorm, 0.09063596);

  // check x
  VectorXd x_ref(n);
  x_ref << 0.08241058, 1.133037, 2.343695;
  VERIFY_IS_APPROX(x, x_ref);
}

struct lmdif_functor : Functor<double> {
  lmdif_functor(void) : Functor<double>(3, 15) {}
  int operator()(const VectorXd &x, VectorXd &fvec) const {
    int i;
    double tmp1, tmp2, tmp3;
    static const double y[15] = {1.4e-1, 1.8e-1, 2.2e-1, 2.5e-1, 2.9e-1, 3.2e-1, 3.5e-1, 3.9e-1,
                                 3.7e-1, 5.8e-1, 7.3e-1, 9.6e-1, 1.34e0, 2.1e0,  4.39e0};

    assert(x.size() == 3);
    assert(fvec.size() == 15);
    for (i = 0; i < 15; i++) {
      tmp1 = i + 1;
      tmp2 = 15 - i;
      tmp3 = tmp1;

      if (i >= 8) tmp3 = tmp2;
      fvec[i] = y[i] - (x[0] + tmp1 / (x[1] * tmp2 + x[2] * tmp3));
    }
    return 0;
  }
};

void testLmdif1() {
  const int n = 3;
  int info;

  VectorXd x(n), fvec(15);

  /* the following starting values provide a rough fit. */
  x.setConstant(n, 1.);

  // do the computation
  lmdif_functor functor;
  DenseIndex nfev = -1;  // initialize to avoid maybe-uninitialized warning
  info = LevenbergMarquardt<lmdif_functor>::lmdif1(functor, x, &nfev);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  VERIFY(nfev <= 26 * LM_EVAL_COUNT_TOL);

  // check norm
  functor(x, fvec);
  VERIFY_IS_APPROX(fvec.blueNorm(), 0.09063596);

  // check x
  VectorXd x_ref(n);
  x_ref << 0.0824106, 1.1330366, 2.3436947;
  VERIFY_IS_APPROX(x, x_ref);
}

void testLmdif() {
  const int m = 15, n = 3;
  int info;
  double fnorm, covfac;
  VectorXd x(n);

  /* the following starting values provide a rough fit. */
  x.setConstant(n, 1.);

  // do the computation
  lmdif_functor functor;
  NumericalDiff<lmdif_functor> numDiff(functor);
  LevenbergMarquardt<NumericalDiff<lmdif_functor> > lm(numDiff);
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return values
  // VERIFY_IS_EQUAL(info, 1);
  VERIFY(lm.nfev <= 26 * LM_EVAL_COUNT_TOL);

  // check norm
  fnorm = lm.fvec.blueNorm();
  VERIFY_IS_APPROX(fnorm, 0.09063596);

  // check x
  VectorXd x_ref(n);
  x_ref << 0.08241058, 1.133037, 2.343695;
  VERIFY_IS_APPROX(x, x_ref);

  // check covariance
  covfac = fnorm * fnorm / (m - n);
  internal::covar(lm.fjac, lm.permutation.indices());  // TODO : move this as a function of lm

  MatrixXd cov_ref(n, n);
  cov_ref << 0.0001531202, 0.002869942, -0.002656662, 0.002869942, 0.09480937, -0.09098997, -0.002656662, -0.09098997,
      0.08778729;

  //  std::cout << fjac*covfac << std::endl;

  MatrixXd cov;
  cov = covfac * lm.fjac.topLeftCorner<n, n>();
  VERIFY_IS_APPROX(cov, cov_ref);
  // TODO: why isn't this allowed ? :
  // VERIFY_IS_APPROX( covfac*fjac.topLeftCorner<n,n>() , cov_ref);
}

struct chwirut2_functor : Functor<double> {
  chwirut2_functor(void) : Functor<double>(3, 54) {}
  static const double m_x[54];
  static const double m_y[54];
  int operator()(const VectorXd &b, VectorXd &fvec) {
    int i;

    assert(b.size() == 3);
    assert(fvec.size() == 54);
    for (i = 0; i < 54; i++) {
      double x = m_x[i];
      fvec[i] = exp(-b[0] * x) / (b[1] + b[2] * x) - m_y[i];
    }
    return 0;
  }
  int df(const VectorXd &b, MatrixXd &fjac) {
    assert(b.size() == 3);
    assert(fjac.rows() == 54);
    assert(fjac.cols() == 3);
    for (int i = 0; i < 54; i++) {
      double x = m_x[i];
      double factor = 1. / (b[1] + b[2] * x);
      double e = exp(-b[0] * x);
      fjac(i, 0) = -x * e * factor;
      fjac(i, 1) = -e * factor * factor;
      fjac(i, 2) = -x * e * factor * factor;
    }
    return 0;
  }
};
const double chwirut2_functor::m_x[54] = {
    0.500E0, 1.000E0, 1.750E0, 3.750E0, 5.750E0, 0.875E0, 2.250E0, 3.250E0, 5.250E0, 0.750E0, 1.750E0,
    2.750E0, 4.750E0, 0.625E0, 1.250E0, 2.250E0, 4.250E0, .500E0,  3.000E0, .750E0,  3.000E0, 1.500E0,
    6.000E0, 3.000E0, 6.000E0, 1.500E0, 3.000E0, .500E0,  2.000E0, 4.000E0, .750E0,  2.000E0, 5.000E0,
    .750E0,  2.250E0, 3.750E0, 5.750E0, 3.000E0, .750E0,  2.500E0, 4.000E0, .750E0,  2.500E0, 4.000E0,
    .750E0,  2.500E0, 4.000E0, .500E0,  6.000E0, 3.000E0, .500E0,  2.750E0, .500E0,  1.750E0};
const double chwirut2_functor::m_y[54] = {
    92.9000E0, 57.1000E0, 31.0500E0, 11.5875E0, 8.0250E0,  63.6000E0, 21.4000E0, 14.2500E0, 8.4750E0,
    63.8000E0, 26.8000E0, 16.4625E0, 7.1250E0,  67.3000E0, 41.0000E0, 21.1500E0, 8.1750E0,  81.5000E0,
    13.1200E0, 59.9000E0, 14.6200E0, 32.9000E0, 5.4400E0,  12.5600E0, 5.4400E0,  32.0000E0, 13.9500E0,
    75.8000E0, 20.0000E0, 10.4200E0, 59.5000E0, 21.6700E0, 8.5500E0,  62.0000E0, 20.2000E0, 7.7600E0,
    3.7500E0,  11.8100E0, 54.7000E0, 23.7000E0, 11.5500E0, 61.3000E0, 17.7000E0, 8.7400E0,  59.2000E0,
    16.3000E0, 8.6200E0,  81.0000E0, 4.8700E0,  14.6200E0, 81.7000E0, 17.1700E0, 81.3000E0, 28.9000E0};

// http://www.itl.nist.gov/div898/strd/nls/data/chwirut2.shtml
void testNistChwirut2(void) {
  const int n = 3;
  int info;

  VectorXd x(n);

  /*
   * First try
   */
  x << 0.1, 0.01, 0.02;
  // do the computation
  chwirut2_functor functor;
  LevenbergMarquardt<chwirut2_functor> lm(functor);
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 10, 8);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 5.1304802941E+02);
  // check x
  VERIFY_IS_APPROX(x[0], 1.6657666537E-01);
  VERIFY_IS_APPROX(x[1], 5.1653291286E-03);
  VERIFY_IS_APPROX(x[2], 1.2150007096E-02);

  /*
   * Second try
   */
  x << 0.15, 0.008, 0.010;
  // do the computation
  lm.resetParameters();
  lm.parameters.ftol = 1.E6 * NumTraits<double>::epsilon();
  lm.parameters.xtol = 1.E6 * NumTraits<double>::epsilon();
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 7, 6);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 5.1304802941E+02);
  // check x
  VERIFY_IS_APPROX(x[0], 1.6657666537E-01);
  VERIFY_IS_APPROX(x[1], 5.1653291286E-03);
  VERIFY_IS_APPROX(x[2], 1.2150007096E-02);
}

struct misra1a_functor : Functor<double> {
  misra1a_functor(void) : Functor<double>(2, 14) {}
  static const double m_x[14];
  static const double m_y[14];
  int operator()(const VectorXd &b, VectorXd &fvec) {
    assert(b.size() == 2);
    assert(fvec.size() == 14);
    for (int i = 0; i < 14; i++) {
      fvec[i] = b[0] * (1. - exp(-b[1] * m_x[i])) - m_y[i];
    }
    return 0;
  }
  int df(const VectorXd &b, MatrixXd &fjac) {
    assert(b.size() == 2);
    assert(fjac.rows() == 14);
    assert(fjac.cols() == 2);
    for (int i = 0; i < 14; i++) {
      fjac(i, 0) = (1. - exp(-b[1] * m_x[i]));
      fjac(i, 1) = (b[0] * m_x[i] * exp(-b[1] * m_x[i]));
    }
    return 0;
  }
};
const double misra1a_functor::m_x[14] = {77.6E0,  114.9E0, 141.1E0, 190.8E0, 239.9E0, 289.0E0, 332.8E0,
                                         378.4E0, 434.8E0, 477.3E0, 536.8E0, 593.1E0, 689.1E0, 760.0E0};
const double misra1a_functor::m_y[14] = {10.07E0, 14.73E0, 17.94E0, 23.93E0, 29.61E0, 35.18E0, 40.02E0,
                                         44.82E0, 50.76E0, 55.05E0, 61.01E0, 66.40E0, 75.47E0, 81.78E0};

// http://www.itl.nist.gov/div898/strd/nls/data/misra1a.shtml
void testNistMisra1a(void) {
  const int n = 2;
  int info;

  VectorXd x(n);

  /*
   * First try
   */
  x << 500., 0.0001;
  // do the computation
  misra1a_functor functor;
  LevenbergMarquardt<misra1a_functor> lm(functor);
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 19, 15);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 1.2455138894E-01);
  // check x
  VERIFY_IS_APPROX(x[0], 2.3894212918E+02);
  VERIFY_IS_APPROX(x[1], 5.5015643181E-04);

  /*
   * Second try
   */
  x << 250., 0.0005;
  // do the computation
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 5, 4);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 1.2455138894E-01);
  // check x
  VERIFY_IS_APPROX(x[0], 2.3894212918E+02);
  VERIFY_IS_APPROX(x[1], 5.5015643181E-04);
}

struct hahn1_functor : Functor<double> {
  hahn1_functor(void) : Functor<double>(7, 236) {}
  static const double m_x[236];
  int operator()(const VectorXd &b, VectorXd &fvec) {
    static const double m_y[236] = {
        .591E0,   1.547E0,  2.902E0,  2.894E0,  4.703E0,  6.307E0,  7.03E0,   7.898E0,  9.470E0,  9.484E0,  10.072E0,
        10.163E0, 11.615E0, 12.005E0, 12.478E0, 12.982E0, 12.970E0, 13.926E0, 14.452E0, 14.404E0, 15.190E0, 15.550E0,
        15.528E0, 15.499E0, 16.131E0, 16.438E0, 16.387E0, 16.549E0, 16.872E0, 16.830E0, 16.926E0, 16.907E0, 16.966E0,
        17.060E0, 17.122E0, 17.311E0, 17.355E0, 17.668E0, 17.767E0, 17.803E0, 17.765E0, 17.768E0, 17.736E0, 17.858E0,
        17.877E0, 17.912E0, 18.046E0, 18.085E0, 18.291E0, 18.357E0, 18.426E0, 18.584E0, 18.610E0, 18.870E0, 18.795E0,
        19.111E0, .367E0,   .796E0,   0.892E0,  1.903E0,  2.150E0,  3.697E0,  5.870E0,  6.421E0,  7.422E0,  9.944E0,
        11.023E0, 11.87E0,  12.786E0, 14.067E0, 13.974E0, 14.462E0, 14.464E0, 15.381E0, 15.483E0, 15.59E0,  16.075E0,
        16.347E0, 16.181E0, 16.915E0, 17.003E0, 16.978E0, 17.756E0, 17.808E0, 17.868E0, 18.481E0, 18.486E0, 19.090E0,
        16.062E0, 16.337E0, 16.345E0, 16.388E0, 17.159E0, 17.116E0, 17.164E0, 17.123E0, 17.979E0, 17.974E0, 18.007E0,
        17.993E0, 18.523E0, 18.669E0, 18.617E0, 19.371E0, 19.330E0, 0.080E0,  0.248E0,  1.089E0,  1.418E0,  2.278E0,
        3.624E0,  4.574E0,  5.556E0,  7.267E0,  7.695E0,  9.136E0,  9.959E0,  9.957E0,  11.600E0, 13.138E0, 13.564E0,
        13.871E0, 13.994E0, 14.947E0, 15.473E0, 15.379E0, 15.455E0, 15.908E0, 16.114E0, 17.071E0, 17.135E0, 17.282E0,
        17.368E0, 17.483E0, 17.764E0, 18.185E0, 18.271E0, 18.236E0, 18.237E0, 18.523E0, 18.627E0, 18.665E0, 19.086E0,
        0.214E0,  0.943E0,  1.429E0,  2.241E0,  2.951E0,  3.782E0,  4.757E0,  5.602E0,  7.169E0,  8.920E0,  10.055E0,
        12.035E0, 12.861E0, 13.436E0, 14.167E0, 14.755E0, 15.168E0, 15.651E0, 15.746E0, 16.216E0, 16.445E0, 16.965E0,
        17.121E0, 17.206E0, 17.250E0, 17.339E0, 17.793E0, 18.123E0, 18.49E0,  18.566E0, 18.645E0, 18.706E0, 18.924E0,
        19.1E0,   0.375E0,  0.471E0,  1.504E0,  2.204E0,  2.813E0,  4.765E0,  9.835E0,  10.040E0, 11.946E0, 12.596E0,
        13.303E0, 13.922E0, 14.440E0, 14.951E0, 15.627E0, 15.639E0, 15.814E0, 16.315E0, 16.334E0, 16.430E0, 16.423E0,
        17.024E0, 17.009E0, 17.165E0, 17.134E0, 17.349E0, 17.576E0, 17.848E0, 18.090E0, 18.276E0, 18.404E0, 18.519E0,
        19.133E0, 19.074E0, 19.239E0, 19.280E0, 19.101E0, 19.398E0, 19.252E0, 19.89E0,  20.007E0, 19.929E0, 19.268E0,
        19.324E0, 20.049E0, 20.107E0, 20.062E0, 20.065E0, 19.286E0, 19.972E0, 20.088E0, 20.743E0, 20.83E0,  20.935E0,
        21.035E0, 20.93E0,  21.074E0, 21.085E0, 20.935E0};

    //        int called=0; printf("call hahn1_functor with  iflag=%d, called=%d\n", iflag, called); if (iflag==1)
    //        called++;

    assert(b.size() == 7);
    assert(fvec.size() == 236);
    for (int i = 0; i < 236; i++) {
      double x = m_x[i], xx = x * x, xxx = xx * x;
      fvec[i] = (b[0] + b[1] * x + b[2] * xx + b[3] * xxx) / (1. + b[4] * x + b[5] * xx + b[6] * xxx) - m_y[i];
    }
    return 0;
  }

  int df(const VectorXd &b, MatrixXd &fjac) {
    assert(b.size() == 7);
    assert(fjac.rows() == 236);
    assert(fjac.cols() == 7);
    for (int i = 0; i < 236; i++) {
      double x = m_x[i], xx = x * x, xxx = xx * x;
      double fact = 1. / (1. + b[4] * x + b[5] * xx + b[6] * xxx);
      fjac(i, 0) = 1. * fact;
      fjac(i, 1) = x * fact;
      fjac(i, 2) = xx * fact;
      fjac(i, 3) = xxx * fact;
      fact = -(b[0] + b[1] * x + b[2] * xx + b[3] * xxx) * fact * fact;
      fjac(i, 4) = x * fact;
      fjac(i, 5) = xx * fact;
      fjac(i, 6) = xxx * fact;
    }
    return 0;
  }
};
const double hahn1_functor::m_x[236] = {
    24.41E0,  34.82E0,  44.09E0,  45.07E0,  54.98E0,  65.51E0,  70.53E0,  75.70E0,  89.57E0,  91.14E0,  96.40E0,
    97.19E0,  114.26E0, 120.25E0, 127.08E0, 133.55E0, 133.61E0, 158.67E0, 172.74E0, 171.31E0, 202.14E0, 220.55E0,
    221.05E0, 221.39E0, 250.99E0, 268.99E0, 271.80E0, 271.97E0, 321.31E0, 321.69E0, 330.14E0, 333.03E0, 333.47E0,
    340.77E0, 345.65E0, 373.11E0, 373.79E0, 411.82E0, 419.51E0, 421.59E0, 422.02E0, 422.47E0, 422.61E0, 441.75E0,
    447.41E0, 448.7E0,  472.89E0, 476.69E0, 522.47E0, 522.62E0, 524.43E0, 546.75E0, 549.53E0, 575.29E0, 576.00E0,
    625.55E0, 20.15E0,  28.78E0,  29.57E0,  37.41E0,  39.12E0,  50.24E0,  61.38E0,  66.25E0,  73.42E0,  95.52E0,
    107.32E0, 122.04E0, 134.03E0, 163.19E0, 163.48E0, 175.70E0, 179.86E0, 211.27E0, 217.78E0, 219.14E0, 262.52E0,
    268.01E0, 268.62E0, 336.25E0, 337.23E0, 339.33E0, 427.38E0, 428.58E0, 432.68E0, 528.99E0, 531.08E0, 628.34E0,
    253.24E0, 273.13E0, 273.66E0, 282.10E0, 346.62E0, 347.19E0, 348.78E0, 351.18E0, 450.10E0, 450.35E0, 451.92E0,
    455.56E0, 552.22E0, 553.56E0, 555.74E0, 652.59E0, 656.20E0, 14.13E0,  20.41E0,  31.30E0,  33.84E0,  39.70E0,
    48.83E0,  54.50E0,  60.41E0,  72.77E0,  75.25E0,  86.84E0,  94.88E0,  96.40E0,  117.37E0, 139.08E0, 147.73E0,
    158.63E0, 161.84E0, 192.11E0, 206.76E0, 209.07E0, 213.32E0, 226.44E0, 237.12E0, 330.90E0, 358.72E0, 370.77E0,
    372.72E0, 396.24E0, 416.59E0, 484.02E0, 495.47E0, 514.78E0, 515.65E0, 519.47E0, 544.47E0, 560.11E0, 620.77E0,
    18.97E0,  28.93E0,  33.91E0,  40.03E0,  44.66E0,  49.87E0,  55.16E0,  60.90E0,  72.08E0,  85.15E0,  97.06E0,
    119.63E0, 133.27E0, 143.84E0, 161.91E0, 180.67E0, 198.44E0, 226.86E0, 229.65E0, 258.27E0, 273.77E0, 339.15E0,
    350.13E0, 362.75E0, 371.03E0, 393.32E0, 448.53E0, 473.78E0, 511.12E0, 524.70E0, 548.75E0, 551.64E0, 574.02E0,
    623.86E0, 21.46E0,  24.33E0,  33.43E0,  39.22E0,  44.18E0,  55.02E0,  94.33E0,  96.44E0,  118.82E0, 128.48E0,
    141.94E0, 156.92E0, 171.65E0, 190.00E0, 223.26E0, 223.88E0, 231.50E0, 265.05E0, 269.44E0, 271.78E0, 273.46E0,
    334.61E0, 339.79E0, 349.52E0, 358.18E0, 377.98E0, 394.77E0, 429.66E0, 468.22E0, 487.27E0, 519.54E0, 523.03E0,
    612.99E0, 638.59E0, 641.36E0, 622.05E0, 631.50E0, 663.97E0, 646.9E0,  748.29E0, 749.21E0, 750.14E0, 647.04E0,
    646.89E0, 746.9E0,  748.43E0, 747.35E0, 749.27E0, 647.61E0, 747.78E0, 750.51E0, 851.37E0, 845.97E0, 847.54E0,
    849.93E0, 851.61E0, 849.75E0, 850.98E0, 848.23E0};

// http://www.itl.nist.gov/div898/strd/nls/data/hahn1.shtml
void testNistHahn1(void) {
  const int n = 7;
  int info;

  VectorXd x(n);

  /*
   * First try
   */
  x << 10., -1., .05, -.00001, -.05, .001, -.000001;
  // do the computation
  hahn1_functor functor;
  LevenbergMarquardt<hahn1_functor> lm(functor);
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 11, 10);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 1.5324382854E+00);
  // check x
  VERIFY_IS_APPROX(x[0], 1.0776351733E+00);
  VERIFY_IS_APPROX(x[1], -1.2269296921E-01);
  VERIFY_IS_APPROX(x[2], 4.0863750610E-03);
  VERIFY_IS_APPROX(x[3], -1.426264e-06);  // shoulde be : -1.4262662514E-06
  VERIFY_IS_APPROX(x[4], -5.7609940901E-03);
  VERIFY_IS_APPROX(x[5], 2.4053735503E-04);
  VERIFY_IS_APPROX(x[6], -1.2314450199E-07);

  /*
   * Second try
   */
  x << .1, -.1, .005, -.000001, -.005, .0001, -.0000001;
  // do the computation
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 11, 10);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 1.5324382854E+00);
  // check x
  VERIFY_IS_APPROX(x[0], 1.077640);       // should be :  1.0776351733E+00
  VERIFY_IS_APPROX(x[1], -0.1226933);     // should be : -1.2269296921E-01
  VERIFY_IS_APPROX(x[2], 0.004086383);    // should be : 4.0863750610E-03
  VERIFY_IS_APPROX(x[3], -1.426277e-06);  // shoulde be : -1.4262662514E-06
  VERIFY_IS_APPROX(x[4], -5.7609940901E-03);
  VERIFY_IS_APPROX(x[5], 0.00024053772);  // should be : 2.4053735503E-04
  VERIFY_IS_APPROX(x[6], -1.231450e-07);  // should be : -1.2314450199E-07
}

struct misra1d_functor : Functor<double> {
  misra1d_functor(void) : Functor<double>(2, 14) {}
  static const double x[14];
  static const double y[14];
  int operator()(const VectorXd &b, VectorXd &fvec) {
    assert(b.size() == 2);
    assert(fvec.size() == 14);
    for (int i = 0; i < 14; i++) {
      fvec[i] = b[0] * b[1] * x[i] / (1. + b[1] * x[i]) - y[i];
    }
    return 0;
  }
  int df(const VectorXd &b, MatrixXd &fjac) {
    assert(b.size() == 2);
    assert(fjac.rows() == 14);
    assert(fjac.cols() == 2);
    for (int i = 0; i < 14; i++) {
      double den = 1. + b[1] * x[i];
      fjac(i, 0) = b[1] * x[i] / den;
      fjac(i, 1) = b[0] * x[i] * (den - b[1] * x[i]) / den / den;
    }
    return 0;
  }
};
const double misra1d_functor::x[14] = {77.6E0,  114.9E0, 141.1E0, 190.8E0, 239.9E0, 289.0E0, 332.8E0,
                                       378.4E0, 434.8E0, 477.3E0, 536.8E0, 593.1E0, 689.1E0, 760.0E0};
const double misra1d_functor::y[14] = {10.07E0, 14.73E0, 17.94E0, 23.93E0, 29.61E0, 35.18E0, 40.02E0,
                                       44.82E0, 50.76E0, 55.05E0, 61.01E0, 66.40E0, 75.47E0, 81.78E0};

// http://www.itl.nist.gov/div898/strd/nls/data/misra1d.shtml
void testNistMisra1d(void) {
  const int n = 2;
  int info;

  VectorXd x(n);

  /*
   * First try
   */
  x << 500., 0.0001;
  // do the computation
  misra1d_functor functor;
  LevenbergMarquardt<misra1d_functor> lm(functor);
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 3);
  LM_CHECK_N_ITERS(lm, 9, 7);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 5.6419295283E-02);
  // check x
  VERIFY_IS_APPROX(x[0], 4.3736970754E+02);
  VERIFY_IS_APPROX(x[1], 3.0227324449E-04);

  /*
   * Second try
   */
  x << 450., 0.0003;
  // do the computation
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 4, 3);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 5.6419295283E-02);
  // check x
  VERIFY_IS_APPROX(x[0], 4.3736970754E+02);
  VERIFY_IS_APPROX(x[1], 3.0227324449E-04);
}

struct lanczos1_functor : Functor<double> {
  lanczos1_functor(void) : Functor<double>(6, 24) {}
  static const double x[24];
  static const double y[24];
  int operator()(const VectorXd &b, VectorXd &fvec) {
    assert(b.size() == 6);
    assert(fvec.size() == 24);
    for (int i = 0; i < 24; i++)
      fvec[i] = b[0] * exp(-b[1] * x[i]) + b[2] * exp(-b[3] * x[i]) + b[4] * exp(-b[5] * x[i]) - y[i];
    return 0;
  }
  int df(const VectorXd &b, MatrixXd &fjac) {
    assert(b.size() == 6);
    assert(fjac.rows() == 24);
    assert(fjac.cols() == 6);
    for (int i = 0; i < 24; i++) {
      fjac(i, 0) = exp(-b[1] * x[i]);
      fjac(i, 1) = -b[0] * x[i] * exp(-b[1] * x[i]);
      fjac(i, 2) = exp(-b[3] * x[i]);
      fjac(i, 3) = -b[2] * x[i] * exp(-b[3] * x[i]);
      fjac(i, 4) = exp(-b[5] * x[i]);
      fjac(i, 5) = -b[4] * x[i] * exp(-b[5] * x[i]);
    }
    return 0;
  }
};
const double lanczos1_functor::x[24] = {0.000000000000E+00, 5.000000000000E-02, 1.000000000000E-01, 1.500000000000E-01,
                                        2.000000000000E-01, 2.500000000000E-01, 3.000000000000E-01, 3.500000000000E-01,
                                        4.000000000000E-01, 4.500000000000E-01, 5.000000000000E-01, 5.500000000000E-01,
                                        6.000000000000E-01, 6.500000000000E-01, 7.000000000000E-01, 7.500000000000E-01,
                                        8.000000000000E-01, 8.500000000000E-01, 9.000000000000E-01, 9.500000000000E-01,
                                        1.000000000000E+00, 1.050000000000E+00, 1.100000000000E+00, 1.150000000000E+00};
const double lanczos1_functor::y[24] = {2.513400000000E+00, 2.044333373291E+00, 1.668404436564E+00, 1.366418021208E+00,
                                        1.123232487372E+00, 9.268897180037E-01, 7.679338563728E-01, 6.388775523106E-01,
                                        5.337835317402E-01, 4.479363617347E-01, 3.775847884350E-01, 3.197393199326E-01,
                                        2.720130773746E-01, 2.324965529032E-01, 1.996589546065E-01, 1.722704126914E-01,
                                        1.493405660168E-01, 1.300700206922E-01, 1.138119324644E-01, 1.000415587559E-01,
                                        8.833209084540E-02, 7.833544019350E-02, 6.976693743449E-02, 6.239312536719E-02};

// http://www.itl.nist.gov/div898/strd/nls/data/lanczos1.shtml
void testNistLanczos1(void) {
  const int n = 6;
  int info;

  VectorXd x(n);

  /*
   * First try
   */
  x << 1.2, 0.3, 5.6, 5.5, 6.5, 7.6;
  // do the computation
  lanczos1_functor functor;
  LevenbergMarquardt<lanczos1_functor> lm(functor);
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 2);
  LM_CHECK_N_ITERS(lm, 79, 72);
  // check norm^2
  // std::cout.precision(30);
  // std::cout << lm.fvec.squaredNorm() << "\n";
  VERIFY(lm.fvec.squaredNorm() <= 1.44E-25);
  // check x
  VERIFY_IS_APPROX(x[0], 9.5100000027E-02);
  VERIFY_IS_APPROX(x[1], 1.0000000001E+00);
  VERIFY_IS_APPROX(x[2], 8.6070000013E-01);
  VERIFY_IS_APPROX(x[3], 3.0000000002E+00);
  VERIFY_IS_APPROX(x[4], 1.5575999998E+00);
  VERIFY_IS_APPROX(x[5], 5.0000000001E+00);

  /*
   * Second try
   */
  x << 0.5, 0.7, 3.6, 4.2, 4., 6.3;
  // do the computation
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 2);
  LM_CHECK_N_ITERS(lm, 9, 8);
  // check norm^2
  VERIFY(lm.fvec.squaredNorm() <= 1.44E-25);
  // check x
  VERIFY_IS_APPROX(x[0], 9.5100000027E-02);
  VERIFY_IS_APPROX(x[1], 1.0000000001E+00);
  VERIFY_IS_APPROX(x[2], 8.6070000013E-01);
  VERIFY_IS_APPROX(x[3], 3.0000000002E+00);
  VERIFY_IS_APPROX(x[4], 1.5575999998E+00);
  VERIFY_IS_APPROX(x[5], 5.0000000001E+00);
}

struct rat42_functor : Functor<double> {
  rat42_functor(void) : Functor<double>(3, 9) {}
  static const double x[9];
  static const double y[9];
  int operator()(const VectorXd &b, VectorXd &fvec) {
    assert(b.size() == 3);
    assert(fvec.size() == 9);
    for (int i = 0; i < 9; i++) {
      fvec[i] = b[0] / (1. + exp(b[1] - b[2] * x[i])) - y[i];
    }
    return 0;
  }

  int df(const VectorXd &b, MatrixXd &fjac) {
    assert(b.size() == 3);
    assert(fjac.rows() == 9);
    assert(fjac.cols() == 3);
    for (int i = 0; i < 9; i++) {
      double e = exp(b[1] - b[2] * x[i]);
      fjac(i, 0) = 1. / (1. + e);
      fjac(i, 1) = -b[0] * e / (1. + e) / (1. + e);
      fjac(i, 2) = +b[0] * e * x[i] / (1. + e) / (1. + e);
    }
    return 0;
  }
};
const double rat42_functor::x[9] = {9.000E0,  14.000E0, 21.000E0, 28.000E0, 42.000E0,
                                    57.000E0, 63.000E0, 70.000E0, 79.000E0};
const double rat42_functor::y[9] = {8.930E0,  10.800E0, 18.590E0, 22.330E0, 39.350E0,
                                    56.110E0, 61.730E0, 64.620E0, 67.080E0};

// http://www.itl.nist.gov/div898/strd/nls/data/ratkowsky2.shtml
void testNistRat42(void) {
  const int n = 3;
  int info;

  VectorXd x(n);

  /*
   * First try
   */
  x << 100., 1., 0.1;
  // do the computation
  rat42_functor functor;
  LevenbergMarquardt<rat42_functor> lm(functor);
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 10, 8);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 8.0565229338E+00);
  // check x
  VERIFY_IS_APPROX(x[0], 7.2462237576E+01);
  VERIFY_IS_APPROX(x[1], 2.6180768402E+00);
  VERIFY_IS_APPROX(x[2], 6.7359200066E-02);

  /*
   * Second try
   */
  x << 75., 2.5, 0.07;
  // do the computation
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 6, 5);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 8.0565229338E+00);
  // check x
  VERIFY_IS_APPROX(x[0], 7.2462237576E+01);
  VERIFY_IS_APPROX(x[1], 2.6180768402E+00);
  VERIFY_IS_APPROX(x[2], 6.7359200066E-02);
}

struct MGH10_functor : Functor<double> {
  MGH10_functor(void) : Functor<double>(3, 16) {}
  static const double x[16];
  static const double y[16];
  int operator()(const VectorXd &b, VectorXd &fvec) {
    assert(b.size() == 3);
    assert(fvec.size() == 16);
    for (int i = 0; i < 16; i++) fvec[i] = b[0] * exp(b[1] / (x[i] + b[2])) - y[i];
    return 0;
  }
  int df(const VectorXd &b, MatrixXd &fjac) {
    assert(b.size() == 3);
    assert(fjac.rows() == 16);
    assert(fjac.cols() == 3);
    for (int i = 0; i < 16; i++) {
      double factor = 1. / (x[i] + b[2]);
      double e = exp(b[1] * factor);
      fjac(i, 0) = e;
      fjac(i, 1) = b[0] * factor * e;
      fjac(i, 2) = -b[1] * b[0] * factor * factor * e;
    }
    return 0;
  }
};
const double MGH10_functor::x[16] = {5.000000E+01, 5.500000E+01, 6.000000E+01, 6.500000E+01, 7.000000E+01, 7.500000E+01,
                                     8.000000E+01, 8.500000E+01, 9.000000E+01, 9.500000E+01, 1.000000E+02, 1.050000E+02,
                                     1.100000E+02, 1.150000E+02, 1.200000E+02, 1.250000E+02};
const double MGH10_functor::y[16] = {3.478000E+04, 2.861000E+04, 2.365000E+04, 1.963000E+04, 1.637000E+04, 1.372000E+04,
                                     1.154000E+04, 9.744000E+03, 8.261000E+03, 7.030000E+03, 6.005000E+03, 5.147000E+03,
                                     4.427000E+03, 3.820000E+03, 3.307000E+03, 2.872000E+03};

// http://www.itl.nist.gov/div898/strd/nls/data/mgh10.shtml
void testNistMGH10(void) {
  const int n = 3;
  int info;

  VectorXd x(n);

  /*
   * First try
   */
  x << 2., 400000., 25000.;
  // do the computation
  MGH10_functor functor;
  LevenbergMarquardt<MGH10_functor> lm(functor);
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 2);
  LM_CHECK_N_ITERS(lm, 284, 249);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 8.7945855171E+01);
  // check x
  VERIFY_IS_APPROX(x[0], 5.6096364710E-03);
  VERIFY_IS_APPROX(x[1], 6.1813463463E+03);
  VERIFY_IS_APPROX(x[2], 3.4522363462E+02);

  /*
   * Second try
   */
  x << 0.02, 4000., 250.;
  // do the computation
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 3);
  LM_CHECK_N_ITERS(lm, 126, 116);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 8.7945855171E+01);
  // check x
  VERIFY_IS_APPROX(x[0], 5.6096364710E-03);
  VERIFY_IS_APPROX(x[1], 6.1813463463E+03);
  VERIFY_IS_APPROX(x[2], 3.4522363462E+02);
}

struct BoxBOD_functor : Functor<double> {
  BoxBOD_functor(void) : Functor<double>(2, 6) {}
  static const double x[6];
  int operator()(const VectorXd &b, VectorXd &fvec) {
    static const double y[6] = {109., 149., 149., 191., 213., 224.};
    assert(b.size() == 2);
    assert(fvec.size() == 6);
    for (int i = 0; i < 6; i++) fvec[i] = b[0] * (1. - exp(-b[1] * x[i])) - y[i];
    return 0;
  }
  int df(const VectorXd &b, MatrixXd &fjac) {
    assert(b.size() == 2);
    assert(fjac.rows() == 6);
    assert(fjac.cols() == 2);
    for (int i = 0; i < 6; i++) {
      double e = exp(-b[1] * x[i]);
      fjac(i, 0) = 1. - e;
      fjac(i, 1) = b[0] * x[i] * e;
    }
    return 0;
  }
};
const double BoxBOD_functor::x[6] = {1., 2., 3., 5., 7., 10.};

// http://www.itl.nist.gov/div898/strd/nls/data/boxbod.shtml
void testNistBoxBOD(void) {
  const int n = 2;
  int info;

  VectorXd x(n);

  /*
   * First try
   */
  x << 1., 1.;
  // do the computation
  BoxBOD_functor functor;
  LevenbergMarquardt<BoxBOD_functor> lm(functor);
  lm.parameters.ftol = 1.E6 * NumTraits<double>::epsilon();
  lm.parameters.xtol = 1.E6 * NumTraits<double>::epsilon();
  lm.parameters.factor = 10.;
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 31, 25);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 1.1680088766E+03);
  // check x
  VERIFY_IS_APPROX(x[0], 2.1380940889E+02);
  VERIFY_IS_APPROX(x[1], 5.4723748542E-01);

  /*
   * Second try
   */
  x << 100., 0.75;
  // do the computation
  lm.resetParameters();
  lm.parameters.ftol = NumTraits<double>::epsilon();
  lm.parameters.xtol = NumTraits<double>::epsilon();
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 20, 14);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 1.1680088766E+03);
  // check x
  VERIFY_IS_APPROX(x[0], 2.1380940889E+02);
  VERIFY_IS_APPROX(x[1], 5.4723748542E-01);
}

struct MGH17_functor : Functor<double> {
  MGH17_functor(void) : Functor<double>(5, 33) {}
  static const double x[33];
  static const double y[33];
  int operator()(const VectorXd &b, VectorXd &fvec) {
    assert(b.size() == 5);
    assert(fvec.size() == 33);
    for (int i = 0; i < 33; i++) fvec[i] = b[0] + b[1] * exp(-b[3] * x[i]) + b[2] * exp(-b[4] * x[i]) - y[i];
    return 0;
  }
  int df(const VectorXd &b, MatrixXd &fjac) {
    assert(b.size() == 5);
    assert(fjac.rows() == 33);
    assert(fjac.cols() == 5);
    for (int i = 0; i < 33; i++) {
      fjac(i, 0) = 1.;
      fjac(i, 1) = exp(-b[3] * x[i]);
      fjac(i, 2) = exp(-b[4] * x[i]);
      fjac(i, 3) = -x[i] * b[1] * exp(-b[3] * x[i]);
      fjac(i, 4) = -x[i] * b[2] * exp(-b[4] * x[i]);
    }
    return 0;
  }
};
const double MGH17_functor::x[33] = {0.000000E+00, 1.000000E+01, 2.000000E+01, 3.000000E+01, 4.000000E+01, 5.000000E+01,
                                     6.000000E+01, 7.000000E+01, 8.000000E+01, 9.000000E+01, 1.000000E+02, 1.100000E+02,
                                     1.200000E+02, 1.300000E+02, 1.400000E+02, 1.500000E+02, 1.600000E+02, 1.700000E+02,
                                     1.800000E+02, 1.900000E+02, 2.000000E+02, 2.100000E+02, 2.200000E+02, 2.300000E+02,
                                     2.400000E+02, 2.500000E+02, 2.600000E+02, 2.700000E+02, 2.800000E+02, 2.900000E+02,
                                     3.000000E+02, 3.100000E+02, 3.200000E+02};
const double MGH17_functor::y[33] = {8.440000E-01, 9.080000E-01, 9.320000E-01, 9.360000E-01, 9.250000E-01, 9.080000E-01,
                                     8.810000E-01, 8.500000E-01, 8.180000E-01, 7.840000E-01, 7.510000E-01, 7.180000E-01,
                                     6.850000E-01, 6.580000E-01, 6.280000E-01, 6.030000E-01, 5.800000E-01, 5.580000E-01,
                                     5.380000E-01, 5.220000E-01, 5.060000E-01, 4.900000E-01, 4.780000E-01, 4.670000E-01,
                                     4.570000E-01, 4.480000E-01, 4.380000E-01, 4.310000E-01, 4.240000E-01, 4.200000E-01,
                                     4.140000E-01, 4.110000E-01, 4.060000E-01};

// http://www.itl.nist.gov/div898/strd/nls/data/mgh17.shtml
void testNistMGH17(void) {
  const int n = 5;
  int info;

  VectorXd x(n);

  /*
   * First try
   */
  x << 50., 150., -100., 1., 2.;
  // do the computation
  MGH17_functor functor;
  LevenbergMarquardt<MGH17_functor> lm(functor);
  lm.parameters.ftol = NumTraits<double>::epsilon();
  lm.parameters.xtol = NumTraits<double>::epsilon();
  lm.parameters.maxfev = 1000;
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 5.4648946975E-05);
  // check x
  VERIFY_IS_APPROX(x[0], 3.7541005211E-01);
  VERIFY_IS_APPROX(x[1], 1.9358469127E+00);
  VERIFY_IS_APPROX(x[2], -1.4646871366E+00);
  VERIFY_IS_APPROX(x[3], 1.2867534640E-02);
  VERIFY_IS_APPROX(x[4], 2.2122699662E-02);

  // check return value
  // VERIFY_IS_EQUAL(info, 2);
  LM_CHECK_N_ITERS(lm, 602, 545);

  /*
   * Second try
   */
  x << 0.5, 1.5, -1, 0.01, 0.02;
  // do the computation
  lm.resetParameters();
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 18, 15);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 5.4648946975E-05);
  // check x
  VERIFY_IS_APPROX(x[0], 3.7541005211E-01);
  VERIFY_IS_APPROX(x[1], 1.9358469127E+00);
  VERIFY_IS_APPROX(x[2], -1.4646871366E+00);
  VERIFY_IS_APPROX(x[3], 1.2867534640E-02);
  VERIFY_IS_APPROX(x[4], 2.2122699662E-02);
}

struct MGH09_functor : Functor<double> {
  MGH09_functor(void) : Functor<double>(4, 11) {}
  static const double _x[11];
  static const double y[11];
  int operator()(const VectorXd &b, VectorXd &fvec) {
    assert(b.size() == 4);
    assert(fvec.size() == 11);
    for (int i = 0; i < 11; i++) {
      double x = _x[i], xx = x * x;
      fvec[i] = b[0] * (xx + x * b[1]) / (xx + x * b[2] + b[3]) - y[i];
    }
    return 0;
  }
  int df(const VectorXd &b, MatrixXd &fjac) {
    assert(b.size() == 4);
    assert(fjac.rows() == 11);
    assert(fjac.cols() == 4);
    for (int i = 0; i < 11; i++) {
      double x = _x[i], xx = x * x;
      double factor = 1. / (xx + x * b[2] + b[3]);
      fjac(i, 0) = (xx + x * b[1]) * factor;
      fjac(i, 1) = b[0] * x * factor;
      fjac(i, 2) = -b[0] * (xx + x * b[1]) * x * factor * factor;
      fjac(i, 3) = -b[0] * (xx + x * b[1]) * factor * factor;
    }
    return 0;
  }
};
const double MGH09_functor::_x[11] = {4.,           2.,     1.,           5.E-1,        2.5E-01,     1.670000E-01,
                                      1.250000E-01, 1.E-01, 8.330000E-02, 7.140000E-02, 6.250000E-02};
const double MGH09_functor::y[11] = {1.957000E-01, 1.947000E-01, 1.735000E-01, 1.600000E-01, 8.440000E-02, 6.270000E-02,
                                     4.560000E-02, 3.420000E-02, 3.230000E-02, 2.350000E-02, 2.460000E-02};

// http://www.itl.nist.gov/div898/strd/nls/data/mgh09.shtml
void testNistMGH09(void) {
  const int n = 4;
  int info;

  VectorXd x(n);

  /*
   * First try
   */
  x << 25., 39, 41.5, 39.;
  // do the computation
  MGH09_functor functor;
  LevenbergMarquardt<MGH09_functor> lm(functor);
  lm.parameters.maxfev = 1000;
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 490, 376);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 3.0750560385E-04);
  // check x
  VERIFY_IS_APPROX(x[0], 0.1928077089);   // should be 1.9280693458E-01
  VERIFY_IS_APPROX(x[1], 0.19126423573);  // should be 1.9128232873E-01
  VERIFY_IS_APPROX(x[2], 0.12305309914);  // should be 1.2305650693E-01
  VERIFY_IS_APPROX(x[3], 0.13605395375);  // should be 1.3606233068E-01

  /*
   * Second try
   */
  x << 0.25, 0.39, 0.415, 0.39;
  // do the computation
  lm.resetParameters();
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 18, 16);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 3.0750560385E-04);
  // check x
  VERIFY_IS_APPROX(x[0], 0.19280781);  // should be 1.9280693458E-01
  VERIFY_IS_APPROX(x[1], 0.19126265);  // should be 1.9128232873E-01
  VERIFY_IS_APPROX(x[2], 0.12305280);  // should be 1.2305650693E-01
  VERIFY_IS_APPROX(x[3], 0.13605322);  // should be 1.3606233068E-01
}

struct Bennett5_functor : Functor<double> {
  Bennett5_functor(void) : Functor<double>(3, 154) {}
  static const double x[154];
  static const double y[154];
  int operator()(const VectorXd &b, VectorXd &fvec) {
    assert(b.size() == 3);
    assert(fvec.size() == 154);
    for (int i = 0; i < 154; i++) fvec[i] = b[0] * pow(b[1] + x[i], -1. / b[2]) - y[i];
    return 0;
  }
  int df(const VectorXd &b, MatrixXd &fjac) {
    assert(b.size() == 3);
    assert(fjac.rows() == 154);
    assert(fjac.cols() == 3);
    for (int i = 0; i < 154; i++) {
      double e = pow(b[1] + x[i], -1. / b[2]);
      fjac(i, 0) = e;
      fjac(i, 1) = -b[0] * e / b[2] / (b[1] + x[i]);
      fjac(i, 2) = b[0] * e * log(b[1] + x[i]) / b[2] / b[2];
    }
    return 0;
  }
};
const double Bennett5_functor::x[154] = {
    7.447168E0,  8.102586E0,  8.452547E0,  8.711278E0,  8.916774E0,  9.087155E0,  9.232590E0,  9.359535E0,  9.472166E0,
    9.573384E0,  9.665293E0,  9.749461E0,  9.827092E0,  9.899128E0,  9.966321E0,  10.029280E0, 10.088510E0, 10.144430E0,
    10.197380E0, 10.247670E0, 10.295560E0, 10.341250E0, 10.384950E0, 10.426820E0, 10.467000E0, 10.505640E0, 10.542830E0,
    10.578690E0, 10.613310E0, 10.646780E0, 10.679150E0, 10.710520E0, 10.740920E0, 10.770440E0, 10.799100E0, 10.826970E0,
    10.854080E0, 10.880470E0, 10.906190E0, 10.931260E0, 10.955720E0, 10.979590E0, 11.002910E0, 11.025700E0, 11.047980E0,
    11.069770E0, 11.091100E0, 11.111980E0, 11.132440E0, 11.152480E0, 11.172130E0, 11.191410E0, 11.210310E0, 11.228870E0,
    11.247090E0, 11.264980E0, 11.282560E0, 11.299840E0, 11.316820E0, 11.333520E0, 11.349940E0, 11.366100E0, 11.382000E0,
    11.397660E0, 11.413070E0, 11.428240E0, 11.443200E0, 11.457930E0, 11.472440E0, 11.486750E0, 11.500860E0, 11.514770E0,
    11.528490E0, 11.542020E0, 11.555380E0, 11.568550E0, 11.581560E0, 11.594420E0, 11.607121E0, 11.619640E0, 11.632000E0,
    11.644210E0, 11.656280E0, 11.668200E0, 11.679980E0, 11.691620E0, 11.703130E0, 11.714510E0, 11.725760E0, 11.736880E0,
    11.747890E0, 11.758780E0, 11.769550E0, 11.780200E0, 11.790730E0, 11.801160E0, 11.811480E0, 11.821700E0, 11.831810E0,
    11.841820E0, 11.851730E0, 11.861550E0, 11.871270E0, 11.880890E0, 11.890420E0, 11.899870E0, 11.909220E0, 11.918490E0,
    11.927680E0, 11.936780E0, 11.945790E0, 11.954730E0, 11.963590E0, 11.972370E0, 11.981070E0, 11.989700E0, 11.998260E0,
    12.006740E0, 12.015150E0, 12.023490E0, 12.031760E0, 12.039970E0, 12.048100E0, 12.056170E0, 12.064180E0, 12.072120E0,
    12.080010E0, 12.087820E0, 12.095580E0, 12.103280E0, 12.110920E0, 12.118500E0, 12.126030E0, 12.133500E0, 12.140910E0,
    12.148270E0, 12.155570E0, 12.162830E0, 12.170030E0, 12.177170E0, 12.184270E0, 12.191320E0, 12.198320E0, 12.205270E0,
    12.212170E0, 12.219030E0, 12.225840E0, 12.232600E0, 12.239320E0, 12.245990E0, 12.252620E0, 12.259200E0, 12.265750E0,
    12.272240E0};
const double Bennett5_functor::y[154] = {
    -34.834702E0, -34.393200E0, -34.152901E0, -33.979099E0, -33.845901E0, -33.732899E0, -33.640301E0, -33.559200E0,
    -33.486801E0, -33.423100E0, -33.365101E0, -33.313000E0, -33.260899E0, -33.217400E0, -33.176899E0, -33.139198E0,
    -33.101601E0, -33.066799E0, -33.035000E0, -33.003101E0, -32.971298E0, -32.942299E0, -32.916302E0, -32.890202E0,
    -32.864101E0, -32.841000E0, -32.817799E0, -32.797501E0, -32.774300E0, -32.757000E0, -32.733799E0, -32.716400E0,
    -32.699100E0, -32.678799E0, -32.661400E0, -32.644001E0, -32.626701E0, -32.612202E0, -32.597698E0, -32.583199E0,
    -32.568699E0, -32.554298E0, -32.539799E0, -32.525299E0, -32.510799E0, -32.499199E0, -32.487598E0, -32.473202E0,
    -32.461601E0, -32.435501E0, -32.435501E0, -32.426800E0, -32.412300E0, -32.400799E0, -32.392101E0, -32.380501E0,
    -32.366001E0, -32.357300E0, -32.348598E0, -32.339901E0, -32.328400E0, -32.319698E0, -32.311001E0, -32.299400E0,
    -32.290699E0, -32.282001E0, -32.273300E0, -32.264599E0, -32.256001E0, -32.247299E0, -32.238602E0, -32.229900E0,
    -32.224098E0, -32.215401E0, -32.203800E0, -32.198002E0, -32.189400E0, -32.183601E0, -32.174900E0, -32.169102E0,
    -32.163300E0, -32.154598E0, -32.145901E0, -32.140099E0, -32.131401E0, -32.125599E0, -32.119801E0, -32.111198E0,
    -32.105400E0, -32.096699E0, -32.090900E0, -32.088001E0, -32.079300E0, -32.073502E0, -32.067699E0, -32.061901E0,
    -32.056099E0, -32.050301E0, -32.044498E0, -32.038799E0, -32.033001E0, -32.027199E0, -32.024300E0, -32.018501E0,
    -32.012699E0, -32.004002E0, -32.001099E0, -31.995300E0, -31.989500E0, -31.983700E0, -31.977900E0, -31.972099E0,
    -31.969299E0, -31.963501E0, -31.957701E0, -31.951900E0, -31.946100E0, -31.940300E0, -31.937401E0, -31.931601E0,
    -31.925800E0, -31.922899E0, -31.917101E0, -31.911301E0, -31.908400E0, -31.902599E0, -31.896900E0, -31.893999E0,
    -31.888201E0, -31.885300E0, -31.882401E0, -31.876600E0, -31.873699E0, -31.867901E0, -31.862101E0, -31.859200E0,
    -31.856300E0, -31.850500E0, -31.844700E0, -31.841801E0, -31.838900E0, -31.833099E0, -31.830200E0, -31.827299E0,
    -31.821600E0, -31.818701E0, -31.812901E0, -31.809999E0, -31.807100E0, -31.801300E0, -31.798401E0, -31.795500E0,
    -31.789700E0, -31.786800E0};

// http://www.itl.nist.gov/div898/strd/nls/data/bennett5.shtml
void testNistBennett5(void) {
  const int n = 3;
  int info;

  VectorXd x(n);

  /*
   * First try
   */
  x << -2000., 50., 0.8;
  // do the computation
  Bennett5_functor functor;
  LevenbergMarquardt<Bennett5_functor> lm(functor);
  lm.parameters.maxfev = 1000;
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 758, 744);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 5.2404744073E-04);
  // check x
  VERIFY_IS_APPROX(x[0], -2.5235058043E+03);
  VERIFY_IS_APPROX(x[1], 4.6736564644E+01);
  VERIFY_IS_APPROX(x[2], 9.3218483193E-01);
  /*
   * Second try
   */
  x << -1500., 45., 0.85;
  // do the computation
  lm.resetParameters();
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 203, 192);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 5.2404744073E-04);
  // check x
  VERIFY_IS_APPROX(x[0], -2523.3007865);  // should be -2.5235058043E+03
  VERIFY_IS_APPROX(x[1], 46.735705771);   // should be 4.6736564644E+01);
  VERIFY_IS_APPROX(x[2], 0.93219881891);  // should be 9.3218483193E-01);
}

struct thurber_functor : Functor<double> {
  thurber_functor(void) : Functor<double>(7, 37) {}
  static const double _x[37];
  static const double _y[37];
  int operator()(const VectorXd &b, VectorXd &fvec) {
    //        int called=0; printf("call hahn1_functor with  iflag=%d, called=%d\n", iflag, called); if (iflag==1)
    //        called++;
    assert(b.size() == 7);
    assert(fvec.size() == 37);
    for (int i = 0; i < 37; i++) {
      double x = _x[i], xx = x * x, xxx = xx * x;
      fvec[i] = (b[0] + b[1] * x + b[2] * xx + b[3] * xxx) / (1. + b[4] * x + b[5] * xx + b[6] * xxx) - _y[i];
    }
    return 0;
  }
  int df(const VectorXd &b, MatrixXd &fjac) {
    assert(b.size() == 7);
    assert(fjac.rows() == 37);
    assert(fjac.cols() == 7);
    for (int i = 0; i < 37; i++) {
      double x = _x[i], xx = x * x, xxx = xx * x;
      double fact = 1. / (1. + b[4] * x + b[5] * xx + b[6] * xxx);
      fjac(i, 0) = 1. * fact;
      fjac(i, 1) = x * fact;
      fjac(i, 2) = xx * fact;
      fjac(i, 3) = xxx * fact;
      fact = -(b[0] + b[1] * x + b[2] * xx + b[3] * xxx) * fact * fact;
      fjac(i, 4) = x * fact;
      fjac(i, 5) = xx * fact;
      fjac(i, 6) = xxx * fact;
    }
    return 0;
  }
};
const double thurber_functor::_x[37] = {-3.067E0, -2.981E0, -2.921E0, -2.912E0, -2.840E0, -2.797E0, -2.702E0, -2.699E0,
                                        -2.633E0, -2.481E0, -2.363E0, -2.322E0, -1.501E0, -1.460E0, -1.274E0, -1.212E0,
                                        -1.100E0, -1.046E0, -0.915E0, -0.714E0, -0.566E0, -0.545E0, -0.400E0, -0.309E0,
                                        -0.109E0, -0.103E0, 0.010E0,  0.119E0,  0.377E0,  0.790E0,  0.963E0,  1.006E0,
                                        1.115E0,  1.572E0,  1.841E0,  2.047E0,  2.200E0};
const double thurber_functor::_y[37] = {
    80.574E0,   84.248E0,   87.264E0,   87.195E0,   89.076E0,   89.608E0,   89.868E0,   90.101E0,
    92.405E0,   95.854E0,   100.696E0,  101.060E0,  401.672E0,  390.724E0,  567.534E0,  635.316E0,
    733.054E0,  759.087E0,  894.206E0,  990.785E0,  1090.109E0, 1080.914E0, 1122.643E0, 1178.351E0,
    1260.531E0, 1273.514E0, 1288.339E0, 1327.543E0, 1353.863E0, 1414.509E0, 1425.208E0, 1421.384E0,
    1442.962E0, 1464.350E0, 1468.705E0, 1447.894E0, 1457.628E0};

// http://www.itl.nist.gov/div898/strd/nls/data/thurber.shtml
void testNistThurber(void) {
  const int n = 7;
  int info;

  VectorXd x(n);

  /*
   * First try
   */
  x << 1000, 1000, 400, 40, 0.7, 0.3, 0.0;
  // do the computation
  thurber_functor functor;
  LevenbergMarquardt<thurber_functor> lm(functor);
  lm.parameters.ftol = 1.E4 * NumTraits<double>::epsilon();
  lm.parameters.xtol = 1.E4 * NumTraits<double>::epsilon();
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 39, 36);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 5.6427082397E+03);
  // check x
  VERIFY_IS_APPROX(x[0], 1.2881396800E+03);
  VERIFY_IS_APPROX(x[1], 1.4910792535E+03);
  VERIFY_IS_APPROX(x[2], 5.8323836877E+02);
  VERIFY_IS_APPROX(x[3], 7.5416644291E+01);
  VERIFY_IS_APPROX(x[4], 9.6629502864E-01);
  VERIFY_IS_APPROX(x[5], 3.9797285797E-01);
  VERIFY_IS_APPROX(x[6], 4.9727297349E-02);

  /*
   * Second try
   */
  x << 1300, 1500, 500, 75, 1, 0.4, 0.05;
  // do the computation
  lm.resetParameters();
  lm.parameters.ftol = 1.E4 * NumTraits<double>::epsilon();
  lm.parameters.xtol = 1.E4 * NumTraits<double>::epsilon();
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 29, 28);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 5.6427082397E+03);
  // check x
  VERIFY_IS_APPROX(x[0], 1.2881396800E+03);
  VERIFY_IS_APPROX(x[1], 1.4910792535E+03);
  VERIFY_IS_APPROX(x[2], 5.8323836877E+02);
  VERIFY_IS_APPROX(x[3], 7.5416644291E+01);
  VERIFY_IS_APPROX(x[4], 9.6629502864E-01);
  VERIFY_IS_APPROX(x[5], 3.9797285797E-01);
  VERIFY_IS_APPROX(x[6], 4.9727297349E-02);
}

struct rat43_functor : Functor<double> {
  rat43_functor(void) : Functor<double>(4, 15) {}
  static const double x[15];
  static const double y[15];
  int operator()(const VectorXd &b, VectorXd &fvec) {
    assert(b.size() == 4);
    assert(fvec.size() == 15);
    for (int i = 0; i < 15; i++) fvec[i] = b[0] * pow(1. + exp(b[1] - b[2] * x[i]), -1. / b[3]) - y[i];
    return 0;
  }
  int df(const VectorXd &b, MatrixXd &fjac) {
    assert(b.size() == 4);
    assert(fjac.rows() == 15);
    assert(fjac.cols() == 4);
    for (int i = 0; i < 15; i++) {
      double e = exp(b[1] - b[2] * x[i]);
      double power = -1. / b[3];
      fjac(i, 0) = pow(1. + e, power);
      fjac(i, 1) = power * b[0] * e * pow(1. + e, power - 1.);
      fjac(i, 2) = -power * b[0] * e * x[i] * pow(1. + e, power - 1.);
      fjac(i, 3) = b[0] * power * power * log(1. + e) * pow(1. + e, power);
    }
    return 0;
  }
};
const double rat43_functor::x[15] = {1., 2., 3., 4., 5., 6., 7., 8., 9., 10., 11., 12., 13., 14., 15.};
const double rat43_functor::y[15] = {16.08,  33.83,  65.80,  97.20,  191.55, 326.20, 386.87, 520.53,
                                     590.03, 651.92, 724.93, 699.56, 689.96, 637.56, 717.41};

// http://www.itl.nist.gov/div898/strd/nls/data/ratkowsky3.shtml
void testNistRat43(void) {
  const int n = 4;
  int info;

  VectorXd x(n);

  /*
   * First try
   */
  x << 100., 10., 1., 1.;
  // do the computation
  rat43_functor functor;
  LevenbergMarquardt<rat43_functor> lm(functor);
  lm.parameters.ftol = 1.E6 * NumTraits<double>::epsilon();
  lm.parameters.xtol = 1.E6 * NumTraits<double>::epsilon();
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 27, 20);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 8.7864049080E+03);
  // check x
  VERIFY_IS_APPROX(x[0], 6.9964151270E+02);
  VERIFY_IS_APPROX(x[1], 5.2771253025E+00);
  VERIFY_IS_APPROX(x[2], 7.5962938329E-01);
  VERIFY_IS_APPROX(x[3], 1.2792483859E+00);

  /*
   * Second try
   */
  x << 700., 5., 0.75, 1.3;
  // do the computation
  lm.resetParameters();
  lm.parameters.ftol = 1.E5 * NumTraits<double>::epsilon();
  lm.parameters.xtol = 1.E5 * NumTraits<double>::epsilon();
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 9, 8);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 8.7864049080E+03);
  // check x
  VERIFY_IS_APPROX(x[0], 6.9964151270E+02);
  VERIFY_IS_APPROX(x[1], 5.2771253025E+00);
  VERIFY_IS_APPROX(x[2], 7.5962938329E-01);
  VERIFY_IS_APPROX(x[3], 1.2792483859E+00);
}

struct eckerle4_functor : Functor<double> {
  eckerle4_functor(void) : Functor<double>(3, 35) {}
  static const double x[35];
  static const double y[35];
  int operator()(const VectorXd &b, VectorXd &fvec) {
    assert(b.size() == 3);
    assert(fvec.size() == 35);
    for (int i = 0; i < 35; i++)
      fvec[i] = b[0] / b[1] * exp(-0.5 * (x[i] - b[2]) * (x[i] - b[2]) / (b[1] * b[1])) - y[i];
    return 0;
  }
  int df(const VectorXd &b, MatrixXd &fjac) {
    assert(b.size() == 3);
    assert(fjac.rows() == 35);
    assert(fjac.cols() == 3);
    for (int i = 0; i < 35; i++) {
      double b12 = b[1] * b[1];
      double e = exp(-0.5 * (x[i] - b[2]) * (x[i] - b[2]) / b12);
      fjac(i, 0) = e / b[1];
      fjac(i, 1) = ((x[i] - b[2]) * (x[i] - b[2]) / b12 - 1.) * b[0] * e / b12;
      fjac(i, 2) = (x[i] - b[2]) * e * b[0] / b[1] / b12;
    }
    return 0;
  }
};
const double eckerle4_functor::x[35] = {400.0, 405.0, 410.0, 415.0, 420.0, 425.0, 430.0, 435.0, 436.5,
                                        438.0, 439.5, 441.0, 442.5, 444.0, 445.5, 447.0, 448.5, 450.0,
                                        451.5, 453.0, 454.5, 456.0, 457.5, 459.0, 460.5, 462.0, 463.5,
                                        465.0, 470.0, 475.0, 480.0, 485.0, 490.0, 495.0, 500.0};
const double eckerle4_functor::y[35] = {0.0001575, 0.0001699, 0.0002350, 0.0003102, 0.0004917, 0.0008710, 0.0017418,
                                        0.0046400, 0.0065895, 0.0097302, 0.0149002, 0.0237310, 0.0401683, 0.0712559,
                                        0.1264458, 0.2073413, 0.2902366, 0.3445623, 0.3698049, 0.3668534, 0.3106727,
                                        0.2078154, 0.1164354, 0.0616764, 0.0337200, 0.0194023, 0.0117831, 0.0074357,
                                        0.0022732, 0.0008800, 0.0004579, 0.0002345, 0.0001586, 0.0001143, 0.0000710};

// http://www.itl.nist.gov/div898/strd/nls/data/eckerle4.shtml
void testNistEckerle4(void) {
  const int n = 3;
  int info;

  VectorXd x(n);

  /*
   * First try
   */
  x << 1., 10., 500.;
  // do the computation
  eckerle4_functor functor;
  LevenbergMarquardt<eckerle4_functor> lm(functor);
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 18, 15);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 1.4635887487E-03);
  // check x
  VERIFY_IS_APPROX(x[0], 1.5543827178);
  VERIFY_IS_APPROX(x[1], 4.0888321754);
  VERIFY_IS_APPROX(x[2], 4.5154121844E+02);

  /*
   * Second try
   */
  x << 1.5, 5., 450.;
  // do the computation
  info = lm.minimize(x);
  EIGEN_UNUSED_VARIABLE(info)

  // check return value
  // VERIFY_IS_EQUAL(info, 1);
  LM_CHECK_N_ITERS(lm, 7, 6);
  // check norm^2
  VERIFY_IS_APPROX(lm.fvec.squaredNorm(), 1.4635887487E-03);
  // check x
  VERIFY_IS_APPROX(x[0], 1.5543827178);
  VERIFY_IS_APPROX(x[1], 4.0888321754);
  VERIFY_IS_APPROX(x[2], 4.5154121844E+02);
}

EIGEN_DECLARE_TEST(NonLinearOptimization) {
  // Tests using the examples provided by (c)minpack
  CALL_SUBTEST /*_1*/ (testChkder());
  CALL_SUBTEST /*_1*/ (testLmder1());
  CALL_SUBTEST /*_1*/ (testLmder());
  CALL_SUBTEST /*_2*/ (testHybrj1());
  CALL_SUBTEST /*_2*/ (testHybrj());
  CALL_SUBTEST /*_2*/ (testHybrd1());
  CALL_SUBTEST /*_2*/ (testHybrd());
  CALL_SUBTEST /*_3*/ (testLmstr1());
  CALL_SUBTEST /*_3*/ (testLmstr());
  CALL_SUBTEST /*_3*/ (testLmdif1());
  CALL_SUBTEST /*_3*/ (testLmdif());

  // NIST tests, level of difficulty = "Lower"
  CALL_SUBTEST /*_4*/ (testNistMisra1a());
  CALL_SUBTEST /*_4*/ (testNistChwirut2());

  // NIST tests, level of difficulty = "Average"
  CALL_SUBTEST /*_5*/ (testNistHahn1());
  CALL_SUBTEST /*_6*/ (testNistMisra1d());
  CALL_SUBTEST /*_7*/ (testNistMGH17());
  CALL_SUBTEST /*_8*/ (testNistLanczos1());

  //     // NIST tests, level of difficulty = "Higher"
  CALL_SUBTEST /*_9*/ (testNistRat42());
  //     CALL_SUBTEST/*_10*/(testNistMGH10());
  CALL_SUBTEST /*_11*/ (testNistBoxBOD());
  //     CALL_SUBTEST/*_12*/(testNistMGH09());
  CALL_SUBTEST /*_13*/ (testNistBennett5());
  CALL_SUBTEST /*_14*/ (testNistThurber());
  CALL_SUBTEST /*_15*/ (testNistRat43());
  CALL_SUBTEST /*_16*/ (testNistEckerle4());
}

/*
 * Can be useful for debugging...
  printf("info, nfev : %d, %d\n", info, lm.nfev);
  printf("info, nfev, njev : %d, %d, %d\n", info, solver.nfev, solver.njev);
  printf("info, nfev : %d, %d\n", info, solver.nfev);
  printf("x[0] : %.32g\n", x[0]);
  printf("x[1] : %.32g\n", x[1]);
  printf("x[2] : %.32g\n", x[2]);
  printf("x[3] : %.32g\n", x[3]);
  printf("fvec.blueNorm() : %.32g\n", solver.fvec.blueNorm());
  printf("fvec.blueNorm() : %.32g\n", lm.fvec.blueNorm());

  printf("info, nfev, njev : %d, %d, %d\n", info, lm.nfev, lm.njev);
  printf("fvec.squaredNorm() : %.13g\n", lm.fvec.squaredNorm());
  std::cout << x << std::endl;
  std::cout.precision(9);
  std::cout << x[0] << std::endl;
  std::cout << x[1] << std::endl;
  std::cout << x[2] << std::endl;
  std::cout << x[3] << std::endl;
*/