eigen/unsupported/test/NonLinear.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/NonLinear>

int fcn_chkder(int /*m*/, int /*n*/, const double *x, double *fvec, double *fjac, int ldfjac, int iflag)
{
  /*      subroutine fcn for chkder example. */

  int i;
  double tmp1, tmp2, tmp3, tmp4;
  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) 
    {
      /*      insert print statements here when nprint is positive. */
      return 0;
    }

  if (iflag != 2) 

    for (i=1; i<=15; i++)
      {
	tmp1 = i;
	tmp2 = 16 - i;
	tmp3 = tmp1;
	if (i > 8) tmp3 = tmp2;
	fvec[i-1] = y[i-1] - (x[1-1] + tmp1/(x[2-1]*tmp2 + x[3-1]*tmp3));
      }
  else
    {
      for (i = 1; i <= 15; i++)
	{
	  tmp1 = i;
	  tmp2 = 16 - i;
	  
	  /* error introduced into next statement for illustration. */
	  /* corrected statement should read    tmp3 = tmp1 . */
	  
	  tmp3 = tmp2;
	  if (i > 8) tmp3 = tmp2;
	  tmp4 = (x[2-1]*tmp2 + x[3-1]*tmp3); tmp4=tmp4*tmp4;
	  fjac[i-1+ ldfjac*(1-1)] = -1.;
	  fjac[i-1+ ldfjac*(2-1)] = tmp1*tmp2/tmp4;
	  fjac[i-1+ ldfjac*(3-1)] = tmp1*tmp3/tmp4;
	}
    }
  return 0;
}

void testChkder()
{
  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;

  ei_chkder<double>(x, fvec, fjac, xp, fvecp, 1, err);
  fcn_chkder(m, n, x.data(), fvec.data(), fjac.data(), m, 1);
  fcn_chkder(m, n, x.data(), fvec.data(), fjac.data(), m, 2);
  fcn_chkder(m, n, xp.data(), fvecp.data(), fjac.data(), m, 1);
  ei_chkder<double>(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);
}


struct lmder_functor {
    static int f(int /*m*/, int /*n*/, const double *x, double *fvec, double *fjac, int ldfjac, int iflag)
    {

        /*      subroutine fcn for lmder1 example. */

        int i;
        double tmp1, tmp2, tmp3, tmp4;
        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 != 2)
        {
            for (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));
            }
        }
        else
        {
            for ( i = 0; i < 15; 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 + ldfjac*(0)] = -1.;
                fjac[i + ldfjac*(1)] = tmp1*tmp2/tmp4;
                fjac[i + ldfjac*(2)] = tmp1*tmp3/tmp4;
            }
        }
        return 0;
    }
};


void testLmder1()
{
  int m=15, n=3, info;

  VectorXd x(n), fvec(m);
  VectorXi ipvt;

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

  // do the computation
  info = ei_lmder1<lmder_functor,double>(x, fvec, ipvt);

  // check return value
  VERIFY( 1 == info);

  // check norm
  VERIFY_IS_APPROX(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, nfev=0, njev=0;
  double fnorm, covfac, covar_ftol;
  VectorXd x(n), fvec(m), diag(n);
  MatrixXd fjac;
  VectorXi ipvt;

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

  // do the computation
  info = ei_lmder<lmder_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return values
  VERIFY( 1 == info);
  VERIFY(nfev==6);
  VERIFY(njev==5);

  // check norm
  fnorm = 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
  covar_ftol = epsilon<double>();
  covfac = fnorm*fnorm/(m-n);
  VectorXd wa(n);
  ipvt.cwise()+=1; // covar() expects the fortran convention (as qrfac provides)
  ei_covar<double>(n, fjac.data(), m, ipvt.data(), covar_ftol, wa.data());

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

struct hybrj_functor {
    static int f(int n, const double *x, double *fvec, double *fjac, int ldfjac, 
            int iflag)
    {
        /*      subroutine fcn for hybrj1 example. */

        int j, k;
        double one=1, temp, temp1, temp2, three=3, two=2, zero=0, four=4;

        if (iflag != 2)
        {
            for (k = 0; k < n; k++)
            {
                temp = (three - two*x[k])*x[k];
                temp1 = zero;
                if (k) temp1 = x[k-1];
                temp2 = zero;
                if (k != n-1) temp2 = x[k+1];
                fvec[k] = temp - temp1 - two*temp2 + one;
            }
        }
        else
        {
            for (k = 0; k < n; k++)
            {
                for (j = 0; j < n; j++)
                {
                    fjac[k + ldfjac*(j)] = zero;
                }
                fjac[k + ldfjac*(k)] = three - four*x[k];
                if (k) fjac[k + ldfjac*(k-1)] = -one;
                if (k != n-1) fjac[k + ldfjac*(k+1)] = -two;
            }
        }
        return 0;
    }
};


void testHybrj1()
{
  const int n=9;
  int info;
  VectorXd x(n), fvec, diag(n);
  MatrixXd fjac;

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

  // do the computation
  info = ei_hybrj1<hybrj_functor, double>(x,fvec, fjac);

  // check return value
  VERIFY( 1 == info);

  // check norm
  VERIFY_IS_APPROX(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, nfev=0, njev=0, mode;
  VectorXd x(n), fvec, diag(n), R, qtf;
  MatrixXd fjac;

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

  mode = 2;
  diag.setConstant(n, 1.);

  // do the computation
  info = ei_hybrj<hybrj_functor, double>(x,fvec, nfev, njev, fjac, R, qtf, diag, mode);

  // check return value
  VERIFY( 1 == info);
  VERIFY(nfev==11);
  VERIFY(njev==1);

  // check norm
  VERIFY_IS_APPROX(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 {
    static int f(int n, const double *x, double *fvec, int /*iflag*/)
    {
        /*      subroutine fcn for hybrd1 example. */

        int k;
        double one=1, temp, temp1, temp2, three=3, two=2, zero=0;

        for (k=0; k < n; k++)
        {
            temp = (three - two*x[k])*x[k];
            temp1 = zero;
            if (k) temp1 = x[k-1];
            temp2 = zero;
            if (k != n-1) temp2 = x[k+1];
            fvec[k] = temp - temp1 - two*temp2 + one;
        }
        return 0;
    }
};

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

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

  // do the computation
  info = ei_hybrd1<hybrd_functor,double>(x, fvec);

  // check return value
  VERIFY( 1 == info);

  // check norm
  VERIFY_IS_APPROX(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, nfev=0, ml, mu, mode;
  VectorXd x(n), fvec, diag(n), R, qtf;
  MatrixXd fjac;

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

  ml = 1;
  mu = 1;
  mode = 2;
  diag.setConstant(n, 1.);

  // do the computation
  info = ei_hybrd<hybrd_functor, double>(x,fvec, nfev, fjac, R, qtf, diag, mode, ml, mu);

  // check return value
  VERIFY( 1 == info);
  VERIFY(nfev==14);

  // check norm
  VERIFY_IS_APPROX(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 {
    static int f(int /*m*/, int /*n*/, const double *x, double *fvec, double *fjrow, int iflag)
    {
        /*  subroutine fcn for lmstr1 example. */
        int i;
        double tmp1, tmp2, tmp3, tmp4;
        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 < 2)
        {
            for (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));
            }
        }
        else
        {
            i = iflag-2;
            tmp1 = i+1;
            tmp2 = 16 - i - 1;
            tmp3 = (i>=8)? tmp2 : tmp1;
            tmp4 = (x[1]*tmp2 + x[2]*tmp3); tmp4 = tmp4*tmp4;
            fjrow[0] = -1;
            fjrow[1] = tmp1*tmp2/tmp4;
            fjrow[2] = tmp1*tmp3/tmp4;
        }
        return 0;
    }
};

void testLmstr1()
{
  int m=15, n=3, info;

  VectorXd x(n), fvec(m);
  VectorXi ipvt;

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

  // do the computation
  info = ei_lmstr1<lmstr_functor,double>(x, fvec, ipvt);

  // check return value
  VERIFY( 1 == info);

  // check norm
  VERIFY_IS_APPROX(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 m=15, n=3;
  int info, nfev=0, njev=0;
  double fnorm;
  VectorXd x(n), fvec(m), diag(n);
  MatrixXd fjac;
  VectorXi ipvt;

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

  // do the computation
  info = ei_lmstr<lmstr_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);
  VectorXd wa(n);

  // check return values
  VERIFY( 1 == info);
  VERIFY(nfev==6);
  VERIFY(njev==5);

  // check norm
  fnorm = 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 {
    static int f(int /*m*/, int /*n*/, const double *x, double *fvec, int /*iflag*/)
    {
        /* function fcn for lmdif1 example */

        int i;
        double tmp1,tmp2,tmp3;
        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};

        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()
{
  int m=15, n=3, info;

  VectorXd x(n), fvec(m);

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

  // do the computation
  info = ei_lmdif1<lmdif_functor,double>(x, fvec);

  // check return value
  VERIFY( 1 == info);

  // check norm
  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, nfev=0;
  double fnorm, covfac, covar_ftol;
  VectorXd x(n), fvec(m), diag(n), qtf;
  MatrixXd fjac;
  VectorXi ipvt;

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

  // do the computation
  info = ei_lmdif<lmdif_functor, double>(x, fvec, nfev, fjac, ipvt, qtf, diag);

  // check return values
  VERIFY( 1 == info);
  VERIFY(nfev==21);

  // check norm
  fnorm = 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
  covar_ftol = epsilon<double>();
  covfac = fnorm*fnorm/(m-n);
  VectorXd wa(n);
  ipvt.cwise()+=1; // covar() expects the fortran convention (as qrfac provides)
  ei_covar<double>(n, fjac.data(), m, ipvt.data(), covar_ftol, wa.data());

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

struct chwirut2_functor {
    static int f(int m, int n, const double *b, double *fvec, double *fjac, int ldfjac, int iflag)
    {
        static const double _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};
        static const double 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  };
        int i;

        assert(m==54);
        assert(n==3);
        assert(ldfjac==54);
        if (iflag != 2) {// compute fvec at b
            for(i=0; i<54; i++) {
                double x = _x[i];
                fvec[i] = exp(-b[0]*x)/(b[1]+b[2]*x) - y[i];
            }
        }
        else { // compute fjac at b
            for(i=0; i<54; i++) {
                double x = _x[i];
                double factor = 1./(b[1]+b[2]*x);
                double e = exp(-b[0]*x);
                fjac[i+ldfjac*0] = -x*e*factor;
                fjac[i+ldfjac*1] = -e*factor*factor;
                fjac[i+ldfjac*2] = -x*e*factor*factor;
            }
        }
        return 0;
    }
};

// http://www.itl.nist.gov/div898/strd/nls/data/chwirut2.shtml
void testNistChwirut2(void)
{
  const int m=54, n=3;
  int info, nfev=0, njev=0;

  VectorXd x(n), fvec(m), diag;
  MatrixXd fjac;
  VectorXi ipvt;

  /*
   * First try
   */
  x<< 0.1, 0.01, 0.02;
  // do the computation
  info = ei_lmder<chwirut2_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 10 == nfev); 
  VERIFY( 8 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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
  info = ei_lmder<chwirut2_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag,
          1, 100., 400, 1.E6*epsilon<double>(), 1.E6*epsilon<double>());

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 7 == nfev); 
  VERIFY( 6 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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 {
    static int f(int m, int n, const double *b, double *fvec, double *fjac, int ldfjac, int iflag)
    {
        static const double 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};
        static const double 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};
        int i;

        assert(m==14);
        assert(n==2);
        assert(ldfjac==14);
        if (iflag != 2) {// compute fvec at b
            for(i=0; i<14; i++) {
                fvec[i] = b[0]*(1.-exp(-b[1]*x[i])) - y[i] ;
            }
        }
        else { // compute fjac at b
            for(i=0; i<14; i++) {
                fjac[i+ldfjac*0] = (1.-exp(-b[1]*x[i]));
                fjac[i+ldfjac*1] = (b[0]*x[i]*exp(-b[1]*x[i]));
            }
        }
        return 0;
    }
};

// http://www.itl.nist.gov/div898/strd/nls/data/misra1a.shtml
void testNistMisra1a(void)
{
  const int m=14, n=2;
  int info, nfev=0, njev=0;

  VectorXd x(n), fvec(m), diag;
  MatrixXd fjac;
  VectorXi ipvt;

  /*
   * First try
   */
  x<< 500., 0.0001;
  // do the computation
  info = ei_lmder<misra1a_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 19 == nfev); 
  VERIFY( 15 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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 = ei_lmder<misra1a_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 5 == nfev); 
  VERIFY( 4 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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 {
    static int f(int m, int n, const double *b, double *fvec, double *fjac, int ldfjac, int iflag)
    {
        static const double _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};
        static const double _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 i;

//        static int called=0; printf("call hahn1_functor with  iflag=%d, called=%d\n", iflag, called); if (iflag==1) called++;
        assert(m==236);
        assert(n==7);
        assert(ldfjac==236);
        if (iflag != 2) {// compute fvec at x
            for(i=0; i<236; 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];
            }
        }
        else { // compute fjac at x
            for(i=0; i<236; 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+ldfjac*0] = 1.*fact;
                fjac[i+ldfjac*1] = x*fact;
                fjac[i+ldfjac*2] = xx*fact;
                fjac[i+ldfjac*3] = xxx*fact;
                fact = - (b[0]+b[1]*x+b[2]*xx+b[3]*xxx) * fact * fact;
                fjac[i+ldfjac*4] = x*fact;
                fjac[i+ldfjac*5] = xx*fact;
                fjac[i+ldfjac*6] = xxx*fact;
            }
        }
        return 0;
    }
};

// http://www.itl.nist.gov/div898/strd/nls/data/hahn1.shtml
void testNistHahn1(void)
{
  const int m=236, n=7;
  int info, nfev=0, njev=0;

  VectorXd x(n), fvec(m), diag;
  MatrixXd fjac;
  VectorXi ipvt;

  /*
   * First try
   */
  x<< 10., -1., .05, -.00001, -.05, .001, -.000001;
  // do the computation
  info = ei_lmder<hahn1_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 11== nfev); 
  VERIFY( 10== njev); 
  // check norm^2
  VERIFY_IS_APPROX(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 = ei_lmder<hahn1_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 11 == nfev); 
  VERIFY( 10 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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 {
    static int f(int m, int n, const double *b, double *fvec, double *fjac, int ldfjac, int iflag)
    {
        static const double 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};
        static const double 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};
        int i;

        assert(m==14);
        assert(n==2);
        assert(ldfjac==14);
        if (iflag != 2) {// compute fvec at b
            for(i=0; i<14; i++) {
                fvec[i] = b[0]*b[1]*x[i]/(1.+b[1]*x[i]) - y[i];
            }
        }
        else { // compute fjac at b
            for(i=0; i<14; i++) {
                double den = 1.+b[1]*x[i];
                fjac[i+ldfjac*0] = b[1]*x[i] / den;
                fjac[i+ldfjac*1] = b[0]*x[i]*(den-b[1]*x[i])/den/den;
            }
        }
        return 0;
    }
};

// http://www.itl.nist.gov/div898/strd/nls/data/misra1d.shtml
void testNistMisra1d(void)
{
  const int m=14, n=2;
  int info, nfev=0, njev=0;

  VectorXd x(n), fvec(m), diag;
  MatrixXd fjac;
  VectorXi ipvt;

  /*
   * First try
   */
  x<< 500., 0.0001;
  // do the computation
  info = ei_lmder<misra1d_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return value
  VERIFY( 3 == info); 
  VERIFY( 9 == nfev); 
  VERIFY( 7 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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 = ei_lmder<misra1d_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 4 == nfev); 
  VERIFY( 3 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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 {
    static int f(int m, int n, const double *b, double *fvec, double *fjac, int ldfjac, int iflag)
    {
        static const double 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 };
        static const double 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 };
        int i;

        assert(m==24);
        assert(n==6);
        assert(ldfjac==24);
        if (iflag != 2) {// compute fvec at b
            for(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];
            }
        }
        else { // compute fjac at b
            for(i=0; i<24; i++) {
                fjac[i+ldfjac*0] = exp(-b[1]*x[i]);
                fjac[i+ldfjac*1] = -b[0]*x[i]*exp(-b[1]*x[i]);
                fjac[i+ldfjac*2] = exp(-b[3]*x[i]);
                fjac[i+ldfjac*3] = -b[2]*x[i]*exp(-b[3]*x[i]);
                fjac[i+ldfjac*4] = exp(-b[5]*x[i]);
                fjac[i+ldfjac*5] = -b[4]*x[i]*exp(-b[5]*x[i]);
            }
        }
        return 0;
    }
};

// http://www.itl.nist.gov/div898/strd/nls/data/lanczos1.shtml
void testNistLanczos1(void)
{
  const int m=24, n=6;
  int info, nfev=0, njev=0;

  VectorXd x(n), fvec(m), diag;
  MatrixXd fjac;
  VectorXi ipvt;

  /*
   * First try
   */
  x<< 1.2, 0.3, 5.6, 5.5, 6.5, 7.6;
  // do the computation
  info = ei_lmder<lanczos1_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return value
  VERIFY( 2 == info); 
  VERIFY( 79 == nfev); 
  VERIFY( 72 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(fvec.squaredNorm(), 1.429604433690E-25);  // should be 1.4307867721E-25, but nist results are on 128-bit floats
  // 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 = ei_lmder<lanczos1_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return value
  VERIFY( 2 == info); 
  VERIFY( 9 == nfev); 
  VERIFY( 8 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(fvec.squaredNorm(), 1.43049947737308E-25);  // should be 1.4307867721E-25, but nist results are on 128-bit floats
  // 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 {
    static int f(int m, int n, const double *b, double *fvec, double *fjac, int ldfjac, int iflag)
    {
        static const double x[9] = { 9.000E0, 14.000E0, 21.000E0, 28.000E0, 42.000E0, 57.000E0, 63.000E0, 70.000E0, 79.000E0 };
        static const double y[9] = { 8.930E0 ,10.800E0 ,18.590E0 ,22.330E0 ,39.350E0 ,56.110E0 ,61.730E0 ,64.620E0 ,67.080E0 };
        int i;

        assert(m==9);
        assert(n==3);
        assert(ldfjac==9);
        if (iflag != 2) {// compute fvec at b
            for(i=0; i<9; i++) {
                fvec[i] = b[0] / (1.+exp(b[1]-b[2]*x[i])) - y[i];
            }
        }
        else { // compute fjac at b
            for(i=0; i<9; i++) {
                double e = exp(b[1]-b[2]*x[i]);
                fjac[i+ldfjac*0] = 1./(1.+e);
                fjac[i+ldfjac*1] = -b[0]*e/(1.+e)/(1.+e);
                fjac[i+ldfjac*2] = +b[0]*e*x[i]/(1.+e)/(1.+e);
            }
        }
        return 0;
    }
};

// http://www.itl.nist.gov/div898/strd/nls/data/ratkowsky2.shtml
void testNistRat42(void)
{
  const int m=9, n=3;
  int info, nfev=0, njev=0;

  VectorXd x(n), fvec(m), diag;
  MatrixXd fjac;
  VectorXi ipvt;

  /*
   * First try
   */
  x<< 100., 1., 0.1;
  // do the computation
  info = ei_lmder<rat42_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 10 == nfev); 
  VERIFY( 8 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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 = ei_lmder<rat42_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 6 == nfev); 
  VERIFY( 5 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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 {
    static int f(int m, int n, const double *b, double *fvec, double *fjac, int ldfjac, int iflag)
    {
        static const double 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 };
        static const double 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 };
        int i;

        assert(m==16);
        assert(n==3);
        assert(ldfjac==16);
        if (iflag != 2) {// compute fvec at b
            for(i=0; i<16; i++) {
                fvec[i] =  b[0] * exp(b[1]/(x[i]+b[2])) - y[i];
            }
        }
        else { // compute fjac at b
            for(i=0; i<16; i++) {
                double factor = 1./(x[i]+b[2]);
                double e = exp(b[1]*factor);
                fjac[i+ldfjac*0] = e;
                fjac[i+ldfjac*1] = b[0]*factor*e;
                fjac[i+ldfjac*2] = -b[1]*b[0]*factor*factor*e;
            }
        }
        return 0;
    }
};

// http://www.itl.nist.gov/div898/strd/nls/data/mgh10.shtml
void testNistMGH10(void)
{
  const int m=16, n=3;
  int info, nfev=0, njev=0;

  VectorXd x(n), fvec(m), diag;
  MatrixXd fjac;
  VectorXi ipvt;

  /*
   * First try
   */
  x<< 2., 400000., 25000.;
  // do the computation
  info = ei_lmder<MGH10_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return value
  VERIFY( 2 == info); 
  VERIFY( 285 == nfev); 
  VERIFY( 250 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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 = ei_lmder<MGH10_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return value
  VERIFY( 2 == info); 
  VERIFY( 126 == nfev); 
  VERIFY( 116 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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 {
    static int f(int m, int n, const double *b, double *fvec, double *fjac, int ldfjac, int iflag)
    {
        static const double x[6] = { 1., 2., 3., 5., 7., 10. };
        static const double y[6] = { 109., 149., 149., 191., 213., 224. };
        int i;

        assert(m==6);
        assert(n==2);
        assert(ldfjac==6);
        if (iflag != 2) {// compute fvec at b
            for(i=0; i<6; i++) {
                fvec[i] =  b[0]*(1.-exp(-b[1]*x[i])) - y[i];
            }
        }
        else { // compute fjac at b
            for(i=0; i<6; i++) {
                double e = exp(-b[1]*x[i]);
                fjac[i+ldfjac*0] = 1.-e;
                fjac[i+ldfjac*1] = b[0]*x[i]*e;
            }
        }
        return 0;
    }
};

// http://www.itl.nist.gov/div898/strd/nls/data/boxbod.shtml
void testNistBoxBOD(void)
{
  const int m=6, n=2;
  int info, nfev=0, njev=0;

  VectorXd x(n), fvec(m), diag;
  MatrixXd fjac;
  VectorXi ipvt;

  /*
   * First try
   */
  x<< 1., 1.;
  // do the computation
  info = ei_lmder<BoxBOD_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag,
          1, 10., 400, 1E6*epsilon<double>(), 1E6*epsilon<double>());

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 31 == nfev); 
  VERIFY( 25  == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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
  info = ei_lmder<BoxBOD_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag,
          1, 100., 14000, epsilon<double>(), epsilon<double>());

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 15 == nfev); 
  VERIFY( 14 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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 {
    static int f(int m, int n, const double *b, double *fvec, double *fjac, int ldfjac, int iflag)
    {
        static const double 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 };
        static const double 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 };
        int i;

        assert(m==33);
        assert(n==5);
        assert(ldfjac==33);
        if (iflag != 2) {// compute fvec at b
            for(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];
            }
        }
        else { // compute fjac at b
            for(i=0; i<33; i++) {
                fjac[i+ldfjac*0] = 1.;
                fjac[i+ldfjac*1] = exp(-b[3]*x[i]);
                fjac[i+ldfjac*2] = exp(-b[4]*x[i]);
                fjac[i+ldfjac*3] = -x[i]*b[1]*exp(-b[3]*x[i]);
                fjac[i+ldfjac*4] = -x[i]*b[2]*exp(-b[4]*x[i]);
            }
        }
        return 0;
    }
};

// http://www.itl.nist.gov/div898/strd/nls/data/mgh17.shtml
void testNistMGH17(void)
{
  const int m=33, n=5;
  int info, nfev=0, njev=0;

  VectorXd x(n), fvec(m), diag;
  MatrixXd fjac;
  VectorXi ipvt;

#if 1
  /*
   * First try
   */
  x<< 50., 150., -100., 1., 2.;
  // do the computation
  info = ei_lmder<MGH17_functor, double>(
          x, fvec, nfev, njev, fjac, ipvt, diag,
          1, 100., 5000, epsilon<double>(), epsilon<double>());

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 599 == nfev); 
  VERIFY( 544 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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);
#endif

  /*
   * Second try
   */
  x<< 0.5  ,1.5  ,-1   ,0.01 ,0.02;
  // do the computation
  info = ei_lmder<MGH17_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 18 == nfev); 
  VERIFY( 15 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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 {
    static int f(int m, int n, const double *b, double *fvec, double *fjac, int ldfjac, int iflag)
    {
        static const double _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 };
        static const double 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 };
        int i;

        assert(m==11);
        assert(n==4);
        assert(ldfjac==11);
        if (iflag != 2) {// compute fvec at b
            for(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];
            }
        }
        else { // compute fjac at b
            for(i=0; i<11; i++) {
                double x = _x[i], xx=x*x;
                double factor = 1./(xx+x*b[2]+b[3]);
                fjac[i+ldfjac*0] = (xx+x*b[1]) * factor;
                fjac[i+ldfjac*1] = b[0]*x* factor;
                fjac[i+ldfjac*2] = - b[0]*(xx+x*b[1]) * x * factor * factor;
                fjac[i+ldfjac*3] = - b[0]*(xx+x*b[1]) * factor * factor;
            }
        }
        return 0;
    }
};

// http://www.itl.nist.gov/div898/strd/nls/data/mgh09.shtml
void testNistMGH09(void)
{
  const int m=11, n=4;
  int info, nfev=0, njev=0;

  VectorXd x(n), fvec(m), diag;
  MatrixXd fjac;
  VectorXi ipvt;

  /*
   * First try
   */
  x<< 25., 39, 41.5, 39.;
  // do the computation
  info = ei_lmder<MGH09_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag,
          1, 100., 5000);
//          1, 100., 5000, epsilon<double>(), epsilon<double>());

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 503== nfev); 
  VERIFY( 385 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(fvec.squaredNorm(), 3.0750560385E-04);
  // check x
  VERIFY_IS_APPROX(x[0], 0.19280624); // should be 1.9280693458E-01
  VERIFY_IS_APPROX(x[1], 0.19129774); // should be 1.9128232873E-01
  VERIFY_IS_APPROX(x[2], 0.12305940); // should be 1.2305650693E-01
  VERIFY_IS_APPROX(x[3], 0.13606946); // should be 1.3606233068E-01

  /*
   * Second try
   */
  x<< 0.25, 0.39, 0.415, 0.39;
  // do the computation
  info = ei_lmder<MGH09_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 18 == nfev); 
  VERIFY( 16 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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 {
    static int f(int m, int n, const double *b, double *fvec, double *fjac, int ldfjac, int iflag)
    {
        static const double 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 };
        static const double 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 };

        int i;

        assert(m==154);
        assert(n==3);
        assert(ldfjac==154);
        if (iflag != 2) {// compute fvec at b
            for(i=0; i<154; i++) {
                fvec[i] = b[0]* pow(b[1]+x[i],-1./b[2]) - y[i];
            }
        }
        else { // compute fjac at b
            for(i=0; i<154; i++) {
                double e = pow(b[1]+x[i],-1./b[2]);
                fjac[i+ldfjac*0] = e;
                fjac[i+ldfjac*1] = - b[0]*e/b[2]/(b[1]+x[i]);
                fjac[i+ldfjac*2] = b[0]*e*log(b[1]+x[i])/b[2]/b[2];
            }
        }
        return 0;
    }
};

// http://www.itl.nist.gov/div898/strd/nls/data/bennett5.shtml
void testNistBennett5(void)
{
  const int m=154, n=3;
  int info, nfev=0, njev=0;

  VectorXd x(n), fvec(m), diag;
  MatrixXd fjac;
  VectorXi ipvt;

  /*
   * First try
   */
  x<< -2000., 50., 0.8;
  // do the computation
  info = ei_lmder<Bennett5_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag,
          1, 100., 5000);

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 758 == nfev); 
  VERIFY( 744 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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
  info = ei_lmder<Bennett5_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 203 == nfev); 
  VERIFY( 192 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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 {
    static int f(int m, int n, const double *b, double *fvec, double *fjac, int ldfjac, int iflag)
    {
        static const double _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 };
        static const double _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};
        int i;

//        static int called=0; printf("call hahn1_functor with  iflag=%d, called=%d\n", iflag, called); if (iflag==1) called++;
        assert(m==37);
        assert(n==7);
        assert(ldfjac==37);
        if (iflag != 2) {// compute fvec at x
            for(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];
            }
        }
        else { // compute fjac at x
            for(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+ldfjac*0] = 1.*fact;
                fjac[i+ldfjac*1] = x*fact;
                fjac[i+ldfjac*2] = xx*fact;
                fjac[i+ldfjac*3] = xxx*fact;
                fact = - (b[0]+b[1]*x+b[2]*xx+b[3]*xxx) * fact * fact;
                fjac[i+ldfjac*4] = x*fact;
                fjac[i+ldfjac*5] = xx*fact;
                fjac[i+ldfjac*6] = xxx*fact;
            }
        }
        return 0;
    }
};

// http://www.itl.nist.gov/div898/strd/nls/data/thurber.shtml
void testNistThurber(void)
{
  const int m=37, n=7;
  int info, nfev=0, njev=0;

  VectorXd x(n), fvec(m), diag;
  MatrixXd fjac;
  VectorXi ipvt;

  /*
   * First try
   */
  x<< 1000 ,1000 ,400 ,40 ,0.7,0.3,0.0 ;
  // do the computation
  info = ei_lmder<thurber_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag,
          1, 100., 400, 1.E4*epsilon<double>(), 1.E4*epsilon<double>());

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 39 == nfev); 
  VERIFY( 36== njev); 
  // check norm^2
  VERIFY_IS_APPROX(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
  info = ei_lmder<thurber_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag,
          1, 100., 400, 1.E4*epsilon<double>(), 1.E4*epsilon<double>());

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 29 == nfev); 
  VERIFY( 28 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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 {
    static int f(int m, int n, const double *b, double *fvec, double *fjac, int ldfjac, int iflag)
    {
        static const double x[15] = { 1., 2., 3., 4., 5., 6., 7., 8., 9., 10., 11., 12., 13., 14., 15. };
        static const double 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 };
        int i;

        assert(m==15);
        assert(n==4);
        assert(ldfjac==15);
        if (iflag != 2) {// compute fvec at b
            for(i=0; i<15; i++) {
                fvec[i] = b[0] * pow(1.+exp(b[1]-b[2]*x[i]),-1./b[3]) - y[i];
            }
        }
        else { // compute fjac at b
            for(i=0; i<15; i++) {
                double e = exp(b[1]-b[2]*x[i]);
                double power = -1./b[3];
                fjac[i+ldfjac*0] = pow(1.+e, power);
                fjac[i+ldfjac*1] = power*b[0]*e*pow(1.+e, power-1.);
                fjac[i+ldfjac*2] = -power*b[0]*e*x[i]*pow(1.+e, power-1.);
                fjac[i+ldfjac*3] = b[0]*power*power*log(1.+e)*pow(1.+e, power);
            }
        }
        return 0;
    }
};

// http://www.itl.nist.gov/div898/strd/nls/data/ratkowsky3.shtml
void testNistRat43(void)
{
  const int m=15, n=4;
  int info, nfev=0, njev=0;

  VectorXd x(n), fvec(m), diag;
  MatrixXd fjac;
  VectorXi ipvt;

  /*
   * First try
   */
  x<< 100., 10., 1., 1.;
  // do the computation
  info = ei_lmder<rat43_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag,
          1, 100., 400, 1.E6*epsilon<double>(), 1.E6*epsilon<double>());

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 27 == nfev); 
  VERIFY( 20 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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
  info = ei_lmder<rat43_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag,
          1, 100., 400, 1.E5*epsilon<double>(), 1.E5*epsilon<double>());

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 9 == nfev); 
  VERIFY( 8 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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 {
    static int f(int m, int n, const double *b, double *fvec, double *fjac, int ldfjac, int iflag)
    {
        static const double 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};
        static const double 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 };

        int i;

        assert(m==35);
        assert(n==3);
        assert(ldfjac==35);
        if (iflag != 2) {// compute fvec at b
            for(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];
            }
        }
        else { // compute fjac at b
            for(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+ldfjac*0] = e / b[1];
                fjac[i+ldfjac*1] = ((x[i]-b[2])*(x[i]-b[2])/b12-1.) * b[0]*e/b12;
                fjac[i+ldfjac*2] = (x[i]-b[2])*e*b[0]/b[1]/b12;
            }
        }
        return 0;
    }
};

// http://www.itl.nist.gov/div898/strd/nls/data/eckerle4.shtml
void testNistEckerle4(void)
{
  const int m=35, n=3;
  int info, nfev=0, njev=0;

  VectorXd x(n), fvec(m), diag;
  MatrixXd fjac;
  VectorXi ipvt;

  /*
   * First try
   */
  x<< 1., 10., 500.;
  // do the computation
  info = ei_lmder<eckerle4_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 18 == nfev); 
  VERIFY( 15 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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 = ei_lmder<eckerle4_functor, double>(x, fvec, nfev, njev, fjac, ipvt, diag);

  // check return value
  VERIFY( 1 == info); 
  VERIFY( 7 == nfev); 
  VERIFY( 6 == njev); 
  // check norm^2
  VERIFY_IS_APPROX(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);
}

void test_NonLinear()
{
    // Tests using the examples provided by (c)minpack
    CALL_SUBTEST(testChkder());
    CALL_SUBTEST(testLmder1());
    CALL_SUBTEST(testLmder());
    CALL_SUBTEST(testHybrj1());
    CALL_SUBTEST(testHybrj());
    CALL_SUBTEST(testHybrd1());
    CALL_SUBTEST(testHybrd());
    CALL_SUBTEST(testLmstr1());
    CALL_SUBTEST(testLmstr());
    CALL_SUBTEST(testLmdif1());
    CALL_SUBTEST(testLmdif());

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

    // NIST tests, level of difficulty = "Average"
    CALL_SUBTEST(testNistHahn1());
    CALL_SUBTEST(testNistMisra1d());
    CALL_SUBTEST(testNistMGH17());
    CALL_SUBTEST(testNistLanczos1());

    // NIST tests, level of difficulty = "Higher"
    CALL_SUBTEST(testNistRat42());
    CALL_SUBTEST(testNistMGH10());
    CALL_SUBTEST(testNistBoxBOD());
    CALL_SUBTEST(testNistMGH09());
    CALL_SUBTEST(testNistBennett5());
    CALL_SUBTEST(testNistThurber());
    CALL_SUBTEST(testNistRat43());
    CALL_SUBTEST(testNistEckerle4());
}

/*
 * Can be useful for debugging...
  printf("info, nfev, njev : %d, %d, %d\n", info, nfev, njev);
  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.squaredNorm() : %.32g\n", fvec.squaredNorm());
*/