File: lmdif.c

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/************************lmdif*************************/

/*
 * Solves or minimizes the sum of squares of m nonlinear
 * functions of n variables.
 *
 * From public domain Fortran version
 * of Argonne National Laboratories MINPACK
 *
 * C translation by Steve Moshier
 */
#include "filter.h"
#include <float.h>

extern lmfunc fcn;

#if _MSC_VER > 1000
#pragma warning(disable: 4100) // disable unreferenced formal parameter warning
#endif

// These globals are needed by MINPACK

/* resolution of arithmetic */
double MACHEP = 1.2e-16;  	

/* smallest nonzero number */ 
double DWARF = 1.0e-38; 

int fdjac2(int,int,double*,double*,double*,int,int*,double,double*);
int qrfac(int,int,double*,int,int,int*,int,double*,double*,double*);
int lmpar(int,double*,int,int*,double*,double*,double,double*,double*,double*,double*,double*);
int qrsolv(int,double*,int,int*,double*,double*,double*,double*,double*);

static double enorm(int n, double x[]);
static double dmax1(double a, double b);
static double dmin1(double a, double b);

/*********************** lmdif.c ****************************/
#define BUG 0


extern double MACHEP;

int lmdif(int m, int n, double x[], double fvec[], 
		  double ftol, double xtol, double gtol,
		  int maxfev, double epsfcn, double diag[],
		  int mode, double factor, int nprint,
		  int *info, int *nfev, double fjac[],
		  int ldfjac, int ipvt[], double qtf[],
		  double wa1[], double wa2[], double wa3[], double wa4[])
{
/*
*     **********
*
*     subroutine lmdif
*
*     the purpose of lmdif is to minimize the sum of the squares of
*     m nonlinear functions in n variables by a modification of
*     the levenberg-marquardt algorithm. the user must provide a
*     subroutine which calculates the functions. the jacobian is
*     then calculated by a forward-difference approximation.
*
*     the subroutine statement is
*
*	subroutine lmdif(fcn,m,n,x,fvec,ftol,xtol,gtol,maxfev,epsfcn,
*			 diag,mode,factor,nprint,info,nfev,fjac,
*			 ldfjac,ipvt,qtf,wa1,wa2,wa3,wa4)
*
*     where
*
*	fcn is the name of the user-supplied subroutine which
*	  calculates the functions. fcn must be declared
*	  in an external statement in the user calling
*	  program, and should be written as follows.
*
*	  subroutine fcn(m,n,x,fvec,iflag)
*	  integer m,n,iflag
*	  double precision x(n),fvec(m)
*	  ----------
*	  calculate the functions at x and
*	  return this vector in fvec.
*	  ----------
*	  return
*	  end
*
*	  the value of iflag should not be changed by fcn unless
*	  the user wants to terminate execution of lmdif.
*	  in this case set iflag to a negative integer.
*
*	m is a positive integer input variable set to the number
*	  of functions.
*
*	n is a positive integer input variable set to the number
*	  of variables. n must not exceed m.
*
*	x is an array of length n. on input x must contain
*	  an initial estimate of the solution vector. on output x
*	  contains the final estimate of the solution vector.
*
*	fvec is an output array of length m which contains
*	  the functions evaluated at the output x.
*
*	ftol is a nonnegative input variable. termination
*	  occurs when both the actual and predicted relative
*	  reductions in the sum of squares are at most ftol.
*	  therefore, ftol measures the relative error desired
*	  in the sum of squares.
*
*	xtol is a nonnegative input variable. termination
*	  occurs when the relative error between two consecutive
*	  iterates is at most xtol. therefore, xtol measures the
*	  relative error desired in the approximate solution.
*
*	gtol is a nonnegative input variable. termination
*	  occurs when the cosine of the angle between fvec and
*	  any column of the jacobian is at most gtol in absolute
*	  value. therefore, gtol measures the orthogonality
*	  desired between the function vector and the columns
*	  of the jacobian.
*
*	maxfev is a positive integer input variable. termination
*	  occurs when the number of calls to fcn is at least
*	  maxfev by the end of an iteration.
*
*	epsfcn is an input variable used in determining a suitable
*	  step length for the forward-difference approximation. this
*	  approximation assumes that the relative errors in the
*	  functions are of the order of epsfcn. if epsfcn is less
*	  than the machine precision, it is assumed that the relative
*	  errors in the functions are of the order of the machine
*	  precision.
*
*	diag is an array of length n. if mode = 1 (see
*	  below), diag is internally set. if mode = 2, diag
*	  must contain positive entries that serve as
*	  multiplicative scale factors for the variables.
*
*	mode is an integer input variable. if mode = 1, the
*	  variables will be scaled internally. if mode = 2,
*	  the scaling is specified by the input diag. other
*	  values of mode are equivalent to mode = 1.
*
*	factor is a positive input variable used in determining the
*	  initial step bound. this bound is set to the product of
*	  factor and the euclidean norm of diag*x if nonzero, or else
*	  to factor itself. in most cases factor should lie in the
*	  interval (.1,100.). 100. is a generally recommended value.
*
*	nprint is an integer input variable that enables controlled
*	  printing of iterates if it is positive. in this case,
*	  fcn is called with iflag = 0 at the beginning of the first
*	  iteration and every nprint iterations thereafter and
*	  immediately prior to return, with x and fvec available
*	  for printing. if nprint is not positive, no special calls
*	  of fcn with iflag = 0 are made.
*
*	info is an integer output variable. if the user has
*	  terminated execution, info is set to the (negative)
*	  value of iflag. see description of fcn. otherwise,
*	  info is set as follows.
*
*	  info = 0  improper input parameters.
*
*	  info = 1  both actual and predicted relative reductions
*		    in the sum of squares are at most ftol.
*
*	  info = 2  relative error between two consecutive iterates
*		    is at most xtol.
*
*	  info = 3  conditions for info = 1 and info = 2 both hold.
*
*	  info = 4  the cosine of the angle between fvec and any
*		    column of the jacobian is at most gtol in
*		    absolute value.
*
*	  info = 5  number of calls to fcn has reached or
*		    exceeded maxfev.
*
*	  info = 6  ftol is too small. no further reduction in
*		    the sum of squares is possible.
*
*	  info = 7  xtol is too small. no further improvement in
*		    the approximate solution x is possible.
*
*	  info = 8  gtol is too small. fvec is orthogonal to the
*		    columns of the jacobian to machine precision.
*
*	nfev is an integer output variable set to the number of
*	  calls to fcn.
*
*	fjac is an output m by n array. the upper n by n submatrix
*	  of fjac contains an upper triangular matrix r with
*	  diagonal elements of nonincreasing magnitude such that
*
*		 t     t	   t
*		p *(jac *jac)*p = r *r,
*
*	  where p is a permutation matrix and jac is the final
*	  calculated jacobian. column j of p is column ipvt(j)
*	  (see below) of the identity matrix. the lower trapezoidal
*	  part of fjac contains information generated during
*	  the computation of r.
*
*	ldfjac is a positive integer input variable not less than m
*	  which specifies the leading dimension of the array fjac.
*
*	ipvt is an integer output array of length n. ipvt
*	  defines a permutation matrix p such that jac*p = q*r,
*	  where jac is the final calculated jacobian, q is
*	  orthogonal (not stored), and r is upper triangular
*	  with diagonal elements of nonincreasing magnitude.
*	  column j of p is column ipvt(j) of the identity matrix.
*
*	qtf is an output array of length n which contains
*	  the first n elements of the vector (q transpose)*fvec.
*
*	wa1, wa2, and wa3 are work arrays of length n.
*
*	wa4 is a work array of length m.
*
*     subprograms called
*
*	user-supplied ...... fcn
*
*	minpack-supplied ... dpmpar,enorm,fdjac2,lmpar,qrfac
*
*	fortran-supplied ... dabs,dmax1,dmin1,dsqrt,mod
*
*     argonne national laboratory. minpack project. march 1980.
*     burton s. garbow, kenneth e. hillstrom, jorge j. more
*
*     **********
*/
int i,iflag,ij,jj,iter,j,l;
double actred,delta = 1.0e-4,dirder,fnorm,fnorm1,gnorm;
double par,pnorm,prered,ratio;
double sum,temp,temp1,temp2,temp3,xnorm = 1.0e-4;
// int fcn();	/* user supplied function */
static double one = 1.0;
static double p1 = 0.1;
static double p5 = 0.5;
static double p25 = 0.25;
static double p75 = 0.75;
static double p0001 = 1.0e-4;
static double zero = 0.0;


MACHEP 	= DBL_EPSILON;  	// machine precision, was 1.2e-16;
/* smallest nonzero number */ 
DWARF 	= DBL_MIN; // was 1.0e-38;



*info = 0;
iflag = 0;
*nfev = 0;
/*
*     check the input parameters for errors.
*/
if( (n <= 0) || (m < n) || (ldfjac < m) || (ftol < zero)
	|| (xtol < zero) || (gtol < zero) || (maxfev <= 0)
	|| (factor <= zero) )
	goto L300;

if( mode == 2 )
	{ /* scaling by diag[] */
	for( j=0; j<n; j++ )
		{
		if( diag[j] <= 0.0 )
			goto L300;
		}	
	}
#if BUG
// printf( "lmdif\n" );
#endif
/*
*     evaluate the function at the starting point
*     and calculate its norm.
*/
iflag = 1;
fcn(m,n,x,fvec,&iflag);
*nfev = 1;
if(iflag < 0)
	goto L300;
fnorm = enorm(m,fvec);
/*
*     initialize levenberg-marquardt parameter and iteration counter.
*/
par = zero;
iter = 1;
/*
*     beginning of the outer loop.
*/

L30:

/*
*	 calculate the jacobian matrix.
*/
iflag = 2;
fdjac2(m,n,x,fvec,fjac,ldfjac,&iflag,epsfcn,wa4);
*nfev += n;
if(iflag < 0)
	goto L300;
/*
*	 if requested, call fcn to enable printing of iterates.
*/
if( nprint > 0 )
	{
	iflag = 0;
	if((iter-1)%nprint == 0)
		{
		fcn(m,n,x,fvec,&iflag);
		if(iflag < 0)
			goto L300;
		// printf( "fnorm %.15e\n", enorm(m,fvec) );
		}
	}
/*
*	 compute the qr factorization of the jacobian.
*/
qrfac(m,n,fjac,ldfjac,1,ipvt,n,wa1,wa2,wa3);
/*
*	 on the first iteration and if mode is 1, scale according
*	 to the norms of the columns of the initial jacobian.
*/
if(iter == 1)
	{
	if(mode != 2)
		{
		for( j=0; j<n; j++ )
			{
			diag[j] = wa2[j];
			if( wa2[j] == zero )
				diag[j] = one;
			}
		}

/*
*	 on the first iteration, calculate the norm of the scaled x
*	 and initialize the step bound delta.
*/
	for( j=0; j<n; j++ )
		wa3[j] = diag[j] * x[j];

	xnorm = enorm(n,wa3);
	delta = factor*xnorm;
	if(delta == zero)
		delta = factor;
	}

/*
*	 form (q transpose)*fvec and store the first n components in
*	 qtf.
*/
for( i=0; i<m; i++ )
	wa4[i] = fvec[i];
jj = 0;
for( j=0; j<n; j++ )
	{
	temp3 = fjac[jj];
	if(temp3 != zero)
		{
		sum = zero;
		ij = jj;
		for( i=j; i<m; i++ )
			{
			sum += fjac[ij] * wa4[i];
			ij += 1;	/* fjac[i+m*j] */
			}
		temp = -sum / temp3;
		ij = jj;
		for( i=j; i<m; i++ )
			{
			wa4[i] += fjac[ij] * temp;
			ij += 1;	/* fjac[i+m*j] */
			}
		}
	fjac[jj] = wa1[j];
	jj += m+1;	/* fjac[j+m*j] */
	qtf[j] = wa4[j];
	}

/*
*	 compute the norm of the scaled gradient.
*/
 gnorm = zero;
 if(fnorm != zero)
	{
	jj = 0;
	for( j=0; j<n; j++ )
		{
		l = ipvt[j];
		if(wa2[l] != zero)
		    	{
			sum = zero;
			ij = jj;
			for( i=0; i<=j; i++ )
				{
				sum += fjac[ij]*(qtf[i]/fnorm);
				ij += 1; /* fjac[i+m*j] */
				}
			gnorm = dmax1(gnorm,fabs(sum/wa2[l]));
			}
		jj += m;
		}
	}

/*
*	 test for convergence of the gradient norm.
*/
 if(gnorm <= gtol)
 	*info = 4;
 if( *info != 0)
 	goto L300;
/*
*	 rescale if necessary.
*/
 if(mode != 2)
	{
	for( j=0; j<n; j++ )
		diag[j] = dmax1(diag[j],wa2[j]);
	}

/*
*	 beginning of the inner loop.
*/
L200:
/*
*	    determine the levenberg-marquardt parameter.
*/
lmpar(n,fjac,ldfjac,ipvt,diag,qtf,delta,&par,wa1,wa2,wa3,wa4);
/*
*	    store the direction p and x + p. calculate the norm of p.
*/
for( j=0; j<n; j++ )
	{
       wa1[j] = -wa1[j];
       wa2[j] = x[j] + wa1[j];
       wa3[j] = diag[j]*wa1[j];
	}
pnorm = enorm(n,wa3);
/*
*	    on the first iteration, adjust the initial step bound.
*/
if(iter == 1)
	delta = dmin1(delta,pnorm);
/*
*	    evaluate the function at x + p and calculate its norm.
*/
iflag = 1;
fcn(m,n,wa2,wa4,&iflag);
*nfev += 1;
if(iflag < 0)
	goto L300;
fnorm1 = enorm(m,wa4);
#if BUG 
// printf( "pnorm %.10e  fnorm1 %.10e\n", pnorm, fnorm1 );
#endif
/*
*	    compute the scaled actual reduction.
*/
actred = -one;
if( (p1*fnorm1) < fnorm)
	{
	temp = fnorm1/fnorm;
	actred = one - temp * temp;
	}
/*
*	    compute the scaled predicted reduction and
*	    the scaled directional derivative.
*/
jj = 0;
for( j=0; j<n; j++ )
	{
	wa3[j] = zero;
	l = ipvt[j];
	temp = wa1[l];
	ij = jj;
	for( i=0; i<=j; i++ )
		{
		wa3[i] += fjac[ij]*temp;
		ij += 1; /* fjac[i+m*j] */
		}
	jj += m;
	}
temp1 = enorm(n,wa3)/fnorm;
temp2 = (sqrt(par)*pnorm)/fnorm;
prered = temp1*temp1 + (temp2*temp2)/p5;
dirder = -(temp1*temp1 + temp2*temp2);
/*
*	    compute the ratio of the actual to the predicted
*	    reduction.
*/
ratio = zero;
if(prered != zero)
	ratio = actred/prered;
/*
*	    update the step bound.
*/
if(ratio <= p25)
	{
	if(actred >= zero)
		temp = p5;
	else
		temp = p5*dirder/(dirder + p5*actred);
	if( ((p1*fnorm1) >= fnorm)
	|| (temp < p1) )
		temp = p1;
       delta = temp*dmin1(delta,pnorm/p1);
       par = par/temp;
	}
else
	{
	if( (par == zero) || (ratio >= p75) )
		{
		delta = pnorm/p5;
		par = p5*par;
		}
	}
/*
*	    test for successful iteration.
*/
if(ratio >= p0001)
	{
/*
*	    successful iteration. update x, fvec, and their norms.
*/
	for( j=0; j<n; j++ )
		{
		x[j] = wa2[j];
		wa2[j] = diag[j]*x[j];
		}
	for( i=0; i<m; i++ )
		fvec[i] = wa4[i];
	xnorm = enorm(n,wa2);
	fnorm = fnorm1;
	iter += 1;
	}
/*
*	    tests for convergence.
*/
if( (fabs(actred) <= ftol)
  && (prered <= ftol)
  && (p5*ratio <= one) )
	*info = 1;
if(delta <= xtol*xnorm)
	*info = 2;
if( (fabs(actred) <= ftol)
  && (prered <= ftol)
  && (p5*ratio <= one)
  && ( *info == 2) )
	*info = 3;
if( *info != 0)
	goto L300;
/*
*	    tests for termination and stringent tolerances.
*/
if( *nfev >= maxfev)
	*info = 5;
if( (fabs(actred) <= MACHEP)
  && (prered <= MACHEP)
  && (p5*ratio <= one) )
	*info = 6;
if(delta <= MACHEP*xnorm)
	*info = 7;
if(gnorm <= MACHEP)
	*info = 8;
if( *info != 0)
	goto L300;
/*
*	    end of the inner loop. repeat if iteration unsuccessful.
*/
if(ratio < p0001)
	goto L200;
/*
*	 end of the outer loop.
*/
goto L30;
	
L300:
/*
*     termination, either normal or user imposed.
*/
if(iflag < 0)
	*info = iflag;
iflag = 0;
if(nprint > 0)
	fcn(m,n,x,fvec,&iflag);
/*
      last card of subroutine lmdif.
*/
return 0; 
}
/************************lmpar.c*************************/

#define BUG 0

int lmpar(int n, double r[], int ldr, int ipvt[],
		  double diag[], double qtb[], double delta,
		  double *par, double x[], double sdiag[],
		  double wa1[], double wa2[])
{
/*     **********
*
*     subroutine lmpar
*
*     given an m by n matrix a, an n by n nonsingular diagonal
*     matrix d, an m-vector b, and a positive number delta,
*     the problem is to determine a value for the parameter
*     par such that if x solves the system
*
*	    a*x = b ,	  sqrt(par)*d*x = 0 ,
*
*     in the least squares sense, and dxnorm is the euclidean
*     norm of d*x, then either par is zero and
*
*	    (dxnorm-delta) .le. 0.1*delta ,
*
*     or par is positive and
*
*	    abs(dxnorm-delta) .le. 0.1*delta .
*
*     this subroutine completes the solution of the problem
*     if it is provided with the necessary information from the
*     qr factorization, with column pivoting, of a. that is, if
*     a*p = q*r, where p is a permutation matrix, q has orthogonal
*     columns, and r is an upper triangular matrix with diagonal
*     elements of nonincreasing magnitude, then lmpar expects
*     the full upper triangle of r, the permutation matrix p,
*     and the first n components of (q transpose)*b. on output
*     lmpar also provides an upper triangular matrix s such that
*
*	     t	 t		     t
*	    p *(a *a + par*d*d)*p = s *s .
*
*     s is employed within lmpar and may be of separate interest.
*
*     only a few iterations are generally needed for convergence
*     of the algorithm. if, however, the limit of 10 iterations
*     is reached, then the output par will contain the best
*     value obtained so far.
*
*     the subroutine statement is
*
*	subroutine lmpar(n,r,ldr,ipvt,diag,qtb,delta,par,x,sdiag,
*			 wa1,wa2)
*
*     where
*
*	n is a positive integer input variable set to the order of r.
*
*	r is an n by n array. on input the full upper triangle
*	  must contain the full upper triangle of the matrix r.
*	  on output the full upper triangle is unaltered, and the
*	  strict lower triangle contains the strict upper triangle
*	  (transposed) of the upper triangular matrix s.
*
*	ldr is a positive integer input variable not less than n
*	  which specifies the leading dimension of the array r.
*
*	ipvt is an integer input array of length n which defines the
*	  permutation matrix p such that a*p = q*r. column j of p
*	  is column ipvt(j) of the identity matrix.
*
*	diag is an input array of length n which must contain the
*	  diagonal elements of the matrix d.
*
*	qtb is an input array of length n which must contain the first
*	  n elements of the vector (q transpose)*b.
*
*	delta is a positive input variable which specifies an upper
*	  bound on the euclidean norm of d*x.
*
*	par is a nonnegative variable. on input par contains an
*	  initial estimate of the levenberg-marquardt parameter.
*	  on output par contains the final estimate.
*
*	x is an output array of length n which contains the least
*	  squares solution of the system a*x = b, sqrt(par)*d*x = 0,
*	  for the output par.
*
*	sdiag is an output array of length n which contains the
*	  diagonal elements of the upper triangular matrix s.
*
*	wa1 and wa2 are work arrays of length n.
*
*     subprograms called
*
*	minpack-supplied ... dpmpar,enorm,qrsolv
*
*	fortran-supplied ... dabs,dmax1,dmin1,dsqrt
*
*     argonne national laboratory. minpack project. march 1980.
*     burton s. garbow, kenneth e. hillstrom, jorge j. more
*
*     **********
*/
int i,iter,ij,jj,j,jm1,jp1,k,l,nsing;
double dxnorm,fp,gnorm,parc,parl,paru;
double sum,temp;
static double zero = 0.0;
// static double one = 1.0;
static double p1 = 0.1;
static double p001 = 0.001;
// extern double MACHEP;
extern double DWARF;

#if BUG
// printf( "lmpar\n" );
#endif
/*
*     compute and store in x the gauss-newton direction. if the
*     jacobian is rank-deficient, obtain a least squares solution.
*/
nsing = n;
jj = 0;
for( j=0; j<n; j++ )
	{
	wa1[j] = qtb[j];
	if( (r[jj] == zero) && (nsing == n) )
		nsing = j;
	if(nsing < n)
		wa1[j] = zero;
	jj += ldr+1; /* [j+ldr*j] */
	}
#if BUG
// printf( "nsing %d ", nsing );
#endif
if(nsing >= 1)
	{
	for( k=0; k<nsing; k++ )
		{
		j = nsing - k - 1;
		wa1[j] = wa1[j]/r[j+ldr*j];
		temp = wa1[j];
		jm1 = j - 1;
		if(jm1 >= 0)
			{
			ij = ldr * j;
			for( i=0; i<=jm1; i++ )
				{
				wa1[i] -= r[ij]*temp;
				ij += 1;
				}
			}
		}
	}

for( j=0; j<n; j++ )
	{
	l = ipvt[j];
	x[l] = wa1[j];
	}
/*
*     initialize the iteration counter.
*     evaluate the function at the origin, and test
*     for acceptance of the gauss-newton direction.
*/
iter = 0;
for( j=0; j<n; j++ )
	wa2[j] = diag[j]*x[j];
dxnorm = enorm(n,wa2);
fp = dxnorm - delta;
if(fp <= p1*delta)
	{
#if BUG
	// printf( "going to L220\n" );
#endif
	goto L220;
	}
/*
*     if the jacobian is not rank deficient, the newton
*     step provides a lower bound, parl, for the zero of
*     the function. otherwise set this bound to zero.
*/
parl = zero;
if(nsing >= n)
	{
	for( j=0; j<n; j++ )
		{
		l = ipvt[j];
		wa1[j] = diag[l]*(wa2[l]/dxnorm);
		}
	jj = 0;
	for( j=0; j<n; j++ )
		{
		sum = zero;
		jm1 = j - 1;
		if(jm1 >= 0)
			{
			ij = jj;
			for( i=0; i<=jm1; i++ )
				{
				sum += r[ij]*wa1[i];
				ij += 1;
				}
			}
		wa1[j] = (wa1[j] - sum)/r[j+ldr*j];
		jj += ldr; /* [i+ldr*j] */
		}
	temp = enorm(n,wa1);
	parl = ((fp/delta)/temp)/temp;
	}
/*
*     calculate an upper bound, paru, for the zero of the function.
*/
jj = 0;
for( j=0; j<n; j++ )
	{
	sum = zero;
	ij = jj;
	for( i=0; i<=j; i++ )
		{
		sum += r[ij]*qtb[i];
		ij += 1;
		}
	l = ipvt[j];
	wa1[j] = sum/diag[l];
	jj += ldr; /* [i+ldr*j] */
	}
gnorm = enorm(n,wa1);
paru = gnorm/delta;
if(paru == zero)
	paru = DWARF/dmin1(delta,p1);
/*
*     if the input par lies outside of the interval (parl,paru),
*     set par to the closer endpoint.
*/
*par = dmax1( *par,parl);
*par = dmin1( *par,paru);
if( *par == zero)
	*par = gnorm/dxnorm;
#if BUG
// printf( "parl %.4e  par %.4e  paru %.4e\n", parl, *par, paru );
#endif
/*
*     beginning of an iteration.
*/
L150:
iter += 1;
/*
*	 evaluate the function at the current value of par.
*/
if( *par == zero)
	*par = dmax1(DWARF,p001*paru);
temp = sqrt( *par );
for( j=0; j<n; j++ )
	wa1[j] = temp*diag[j];
qrsolv(n,r,ldr,ipvt,wa1,qtb,x,sdiag,wa2);
for( j=0; j<n; j++ )
	wa2[j] = diag[j]*x[j];
dxnorm = enorm(n,wa2);
temp = fp;
fp = dxnorm - delta;
/*
*	 if the function is small enough, accept the current value
*	 of par. also test for the exceptional cases where parl
*	 is zero or the number of iterations has reached 10.
*/
if( (fabs(fp) <= p1*delta)
 || ((parl == zero) && (fp <= temp) && (temp < zero))
 || (iter == 10) )
	goto L220;
/*
*	 compute the newton correction.
*/
for( j=0; j<n; j++ )
	{
	l = ipvt[j];
	wa1[j] = diag[l]*(wa2[l]/dxnorm);
	}
jj = 0;
for( j=0; j<n; j++ )
	{
	wa1[j] = wa1[j]/sdiag[j];
	temp = wa1[j];
	jp1 = j + 1;
	if(jp1 < n)
		{
		ij = jp1 + jj;
		for( i=jp1; i<n; i++ )
			{
			wa1[i] -= r[ij]*temp;
			ij += 1; /* [i+ldr*j] */
			}
		}
	jj += ldr; /* ldr*j */
	}
temp = enorm(n,wa1);
parc = ((fp/delta)/temp)/temp;
/*
*	 depending on the sign of the function, update parl or paru.
*/
if(fp > zero)
	parl = dmax1(parl, *par);
if(fp < zero)
	paru = dmin1(paru, *par);
/*
*	 compute an improved estimate for par.
*/
*par = dmax1(parl, *par + parc);
/*
*	 end of an iteration.
*/
goto L150;

L220:
/*
*     termination.
*/
if(iter == 0)
	*par = zero;
/*
*     last card of subroutine lmpar.
*/
return 0;
}
/************************qrfac.c*************************/

#define BUG 0

int qrfac(int m, int n, double a[], int lda PT_UNUSED, int pivot,
		  int ipvt[], int lipvt PT_UNUSED, double rdiag[], 
		  double acnorm[], double wa[])
{
/*
*     **********
*
*     subroutine qrfac
*
*     this subroutine uses householder transformations with column
*     pivoting (optional) to compute a qr factorization of the
*     m by n matrix a. that is, qrfac determines an orthogonal
*     matrix q, a permutation matrix p, and an upper trapezoidal
*     matrix r with diagonal elements of nonincreasing magnitude,
*     such that a*p = q*r. the householder transformation for
*     column k, k = 1,2,...,min(m,n), is of the form
*
*			    t
*	    i - (1/u(k))*u*u
*
*     where u has zeros in the first k-1 positions. the form of
*     this transformation and the method of pivoting first
*     appeared in the corresponding linpack subroutine.
*
*     the subroutine statement is
*
*	subroutine qrfac(m,n,a,lda,pivot,ipvt,lipvt,rdiag,acnorm,wa)
*
*     where
*
*	m is a positive integer input variable set to the number
*	  of rows of a.
*
*	n is a positive integer input variable set to the number
*	  of columns of a.
*
*	a is an m by n array. on input a contains the matrix for
*	  which the qr factorization is to be computed. on output
*	  the strict upper trapezoidal part of a contains the strict
*	  upper trapezoidal part of r, and the lower trapezoidal
*	  part of a contains a factored form of q (the non-trivial
*	  elements of the u vectors described above).
*
*	lda is a positive integer input variable not less than m
*	  which specifies the leading dimension of the array a.
*
*	pivot is a logical input variable. if pivot is set true,
*	  then column pivoting is enforced. if pivot is set false,
*	  then no column pivoting is done.
*
*	ipvt is an integer output array of length lipvt. ipvt
*	  defines the permutation matrix p such that a*p = q*r.
*	  column j of p is column ipvt(j) of the identity matrix.
*	  if pivot is false, ipvt is not referenced.
*
*	lipvt is a positive integer input variable. if pivot is false,
*	  then lipvt may be as small as 1. if pivot is true, then
*	  lipvt must be at least n.
*
*	rdiag is an output array of length n which contains the
*	  diagonal elements of r.
*
*	acnorm is an output array of length n which contains the
*	  norms of the corresponding columns of the input matrix a.
*	  if this information is not needed, then acnorm can coincide
*	  with rdiag.
*
*	wa is a work array of length n. if pivot is false, then wa
*	  can coincide with rdiag.
*
*     subprograms called
*
*	minpack-supplied ... dpmpar,enorm
*
*	fortran-supplied ... dmax1,dsqrt
*
*     argonne national laboratory. minpack project. march 1980.
*     burton s. garbow, kenneth e. hillstrom, jorge j. more
*
*     **********
*/
int i,ij,jj,j,jp1,k,kmax,minmn;
double ajnorm,sum,temp;
static double zero = 0.0;
static double one = 1.0;
static double p05 = 0.05;
extern double MACHEP;
/*
*     compute the initial column norms and initialize several arrays.
*/
ij = 0;
for( j=0; j<n; j++ )
	{
	acnorm[j] = enorm(m,&a[ij]);
	rdiag[j] = acnorm[j];
	wa[j] = rdiag[j];
	if(pivot != 0)
		ipvt[j] = j;
	ij += m; /* m*j */
	}
#if BUG
// printf( "qrfac\n" );
#endif
/*
*     reduce a to r with householder transformations.
*/
minmn = m<=n?m:n;
for( j=0; j<minmn; j++ )
{
if(pivot == 0)
	goto L40;
/*
*	 bring the column of largest norm into the pivot position.
*/
kmax = j;
for( k=j; k<n; k++ )
	{
	if(rdiag[k] > rdiag[kmax])
		kmax = k;
	}
if(kmax == j)
	goto L40;

ij = m * j;
jj = m * kmax;
for( i=0; i<m; i++ )
	{
	temp = a[ij]; /* [i+m*j] */
	a[ij] = a[jj]; /* [i+m*kmax] */
	a[jj] = temp;
	ij += 1;
	jj += 1;
	}
rdiag[kmax] = rdiag[j];
wa[kmax] = wa[j];
k = ipvt[j];
ipvt[j] = ipvt[kmax];
ipvt[kmax] = k;

L40:
/*
*	 compute the householder transformation to reduce the
*	 j-th column of a to a multiple of the j-th unit vector.
*/
jj = j + m*j;
ajnorm = enorm(m-j,&a[jj]);
if(ajnorm == zero)
	goto L100;
if(a[jj] < zero)
	ajnorm = -ajnorm;
ij = jj;
for( i=j; i<m; i++ )
	{
	a[ij] /= ajnorm;
	ij += 1; /* [i+m*j] */
	}
a[jj] += one;
/*
*	 apply the transformation to the remaining columns
*	 and update the norms.
*/
jp1 = j + 1;
if(jp1 < n )
{
for( k=jp1; k<n; k++ )
	{
	sum = zero;
	ij = j + m*k;
	jj = j + m*j;
	for( i=j; i<m; i++ )
		{
		sum += a[jj]*a[ij];
		ij += 1; /* [i+m*k] */
		jj += 1; /* [i+m*j] */
		}
	temp = sum/a[j+m*j];
	ij = j + m*k;
	jj = j + m*j;
	for( i=j; i<m; i++ )
		{
		a[ij] -= temp*a[jj];
		ij += 1; /* [i+m*k] */
		jj += 1; /* [i+m*j] */
		}
	if( (pivot != 0) && (rdiag[k] != zero) )
		{
		temp = a[j+m*k]/rdiag[k];
		temp = dmax1( zero, one-temp*temp );
		rdiag[k] *= sqrt(temp);
		temp = rdiag[k]/wa[k];
		if( (p05*temp*temp) <= MACHEP)
			{
			rdiag[k] = enorm(m-j-1,&a[jp1+m*k]);
			wa[k] = rdiag[k];
			}
		}
	}
}

L100:
	rdiag[j] = -ajnorm;
}
/*
*     last card of subroutine qrfac.
*/
return 0;
}
/************************qrsolv.c*************************/

#define BUG 0

int qrsolv(int n, double r[], int ldr, int ipvt[], double diag[],
		   double qtb[], double x[], double sdiag[], double wa[])
{
/*
*     **********
*
*     subroutine qrsolv
*
*     given an m by n matrix a, an n by n diagonal matrix d,
*     and an m-vector b, the problem is to determine an x which
*     solves the system
*
*	    a*x = b ,	  d*x = 0 ,
*
*     in the least squares sense.
*
*     this subroutine completes the solution of the problem
*     if it is provided with the necessary information from the
*     qr factorization, with column pivoting, of a. that is, if
*     a*p = q*r, where p is a permutation matrix, q has orthogonal
*     columns, and r is an upper triangular matrix with diagonal
*     elements of nonincreasing magnitude, then qrsolv expects
*     the full upper triangle of r, the permutation matrix p,
*     and the first n components of (q transpose)*b. the system
*     a*x = b, d*x = 0, is then equivalent to
*
*		   t	   t
*	    r*z = q *b ,  p *d*p*z = 0 ,
*
*     where x = p*z. if this system does not have full rank,
*     then a least squares solution is obtained. on output qrsolv
*     also provides an upper triangular matrix s such that
*
*	     t	 t		 t
*	    p *(a *a + d*d)*p = s *s .
*
*     s is computed within qrsolv and may be of separate interest.
*
*     the subroutine statement is
*
*	subroutine qrsolv(n,r,ldr,ipvt,diag,qtb,x,sdiag,wa)
*
*     where
*
*	n is a positive integer input variable set to the order of r.
*
*	r is an n by n array. on input the full upper triangle
*	  must contain the full upper triangle of the matrix r.
*	  on output the full upper triangle is unaltered, and the
*	  strict lower triangle contains the strict upper triangle
*	  (transposed) of the upper triangular matrix s.
*
*	ldr is a positive integer input variable not less than n
*	  which specifies the leading dimension of the array r.
*
*	ipvt is an integer input array of length n which defines the
*	  permutation matrix p such that a*p = q*r. column j of p
*	  is column ipvt(j) of the identity matrix.
*
*	diag is an input array of length n which must contain the
*	  diagonal elements of the matrix d.
*
*	qtb is an input array of length n which must contain the first
*	  n elements of the vector (q transpose)*b.
*
*	x is an output array of length n which contains the least
*	  squares solution of the system a*x = b, d*x = 0.
*
*	sdiag is an output array of length n which contains the
*	  diagonal elements of the upper triangular matrix s.
*
*	wa is a work array of length n.
*
*     subprograms called
*
*	fortran-supplied ... dabs,dsqrt
*
*     argonne national laboratory. minpack project. march 1980.
*     burton s. garbow, kenneth e. hillstrom, jorge j. more
*
*     **********
*/
int i,ij,ik,kk,j,jp1,k,kp1,l,nsing;
double cos,cotan,qtbpj,sin,sum,tan,temp;
static double zero = 0.0;
static double p25 = 0.25;
static double p5 = 0.5;
double fabs(), sqrt();

/*
*     copy r and (q transpose)*b to preserve input and initialize s.
*     in particular, save the diagonal elements of r in x.
*/
kk = 0;
for( j=0; j<n; j++ )
	{
	ij = kk;
	ik = kk;
	for( i=j; i<n; i++ )
		{
		r[ij] = r[ik];
		ij += 1;   /* [i+ldr*j] */
		ik += ldr; /* [j+ldr*i] */
		}
	x[j] = r[kk];
	wa[j] = qtb[j];
	kk += ldr+1; /* j+ldr*j */
	}
#if BUG
// printf( "qrsolv\n" );
#endif
/*
*     eliminate the diagonal matrix d using a givens rotation.
*/
for( j=0; j<n; j++ )
{
/*
*	 prepare the row of d to be eliminated, locating the
*	 diagonal element using p from the qr factorization.
*/
l = ipvt[j];
if(diag[l] == zero)
	goto L90;
for( k=j; k<n; k++ )
	sdiag[k] = zero;
sdiag[j] = diag[l];
/*
*	 the transformations to eliminate the row of d
*	 modify only a single element of (q transpose)*b
*	 beyond the first n, which is initially zero.
*/
qtbpj = zero;
for( k=j; k<n; k++ )
	{
/*
*	    determine a givens rotation which eliminates the
*	    appropriate element in the current row of d.
*/
	if(sdiag[k] == zero)
		continue;
	kk = k + ldr * k;
	if(fabs(r[kk]) < fabs(sdiag[k]))
		{
		cotan = r[kk]/sdiag[k];
		sin = p5/sqrt(p25+p25*cotan*cotan);
		cos = sin*cotan;
		}
	else
		{
		tan = sdiag[k]/r[kk];
		cos = p5/sqrt(p25+p25*tan*tan);
		sin = cos*tan;
		}
/*
*	    compute the modified diagonal element of r and
*	    the modified element of ((q transpose)*b,0).
*/
	r[kk] = cos*r[kk] + sin*sdiag[k];
	temp = cos*wa[k] + sin*qtbpj;
	qtbpj = -sin*wa[k] + cos*qtbpj;
	wa[k] = temp;
/*
*	    accumulate the tranformation in the row of s.
*/
	kp1 = k + 1;
	if( n > kp1 )
		{
		ik = kk + 1;
		for( i=kp1; i<n; i++ )
			{
			temp = cos*r[ik] + sin*sdiag[i];
			sdiag[i] = -sin*r[ik] + cos*sdiag[i];
			r[ik] = temp;
			ik += 1; /* [i+ldr*k] */
			}
		}
	}
L90:
/*
*	 store the diagonal element of s and restore
*	 the corresponding diagonal element of r.
*/
	kk = j + ldr*j;
	sdiag[j] = r[kk];
	r[kk] = x[j];
}
/*
*     solve the triangular system for z. if the system is
*     singular, then obtain a least squares solution.
*/
nsing = n;
for( j=0; j<n; j++ )
	{
	if( (sdiag[j] == zero) && (nsing == n) )
		nsing = j;
	if(nsing < n)
		wa[j] = zero;
	}
if(nsing < 1)
	goto L150;

for( k=0; k<nsing; k++ )
	{
	j = nsing - k - 1;
	sum = zero;
	jp1 = j + 1;
	if(nsing > jp1)
		{
		ij = jp1 + ldr * j;
		for( i=jp1; i<nsing; i++ )
			{
			sum += r[ij]*wa[i];
			ij += 1; /* [i+ldr*j] */
			}
		}
	wa[j] = (wa[j] - sum)/sdiag[j];
	}
L150:
/*
*     permute the components of z back to components of x.
*/
for( j=0; j<n; j++ )
	{
	l = ipvt[j];
	x[l] = wa[j];
	}
/*
*     last card of subroutine qrsolv.
*/
return 0;
}
/************************enorm.c*************************/
 
static double enorm(int n, double x[])
{
/*
*     **********
*
*     function enorm
*
*     given an n-vector x, this function calculates the
*     euclidean norm of x.
*
*     the euclidean norm is computed by accumulating the sum of
*     squares in three different sums. the sums of squares for the
*     small and large components are scaled so that no overflows
*     occur. non-destructive underflows are permitted. underflows
*     and overflows do not occur in the computation of the unscaled
*     sum of squares for the intermediate components.
*     the definitions of small, intermediate and large components
*     depend on two constants, rdwarf and rgiant. the main
*     restrictions on these constants are that rdwarf**2 not
*     underflow and rgiant**2 not overflow. the constants
*     given here are suitable for every known computer.
*
*     the function statement is
*
*	double precision function enorm(n,x)
*
*     where
*
*	n is a positive integer input variable.
*
*	x is an input array of length n.
*
*     subprograms called
*
*	fortran-supplied ... dabs,dsqrt
*
*     argonne national laboratory. minpack project. march 1980.
*     burton s. garbow, kenneth e. hillstrom, jorge j. more
*
*     **********
*/
int i;
double agiant,floatn,s1,s2,s3,xabs,x1max,x3max;
double ans, temp;
static double rdwarf = 3.834e-20;
static double rgiant = 1.304e19;
static double zero = 0.0;
static double one = 1.0;
double fabs(), sqrt();

s1 = zero;
s2 = zero;
s3 = zero;
x1max = zero;
x3max = zero;
floatn = n;
agiant = rgiant/floatn;

for( i=0; i<n; i++ )
{
xabs = fabs(x[i]);
if( (xabs > rdwarf) && (xabs < agiant) )
	{
/*
*	    sum for intermediate components.
*/
	s2 += xabs*xabs;
	continue;
	}

if(xabs > rdwarf)
	{
/*
*	       sum for large components.
*/
	if(xabs > x1max)
		{
		temp = x1max/xabs;
		s1 = one + s1*temp*temp;
		x1max = xabs;
		}
	else
		{
		temp = xabs/x1max;
		s1 += temp*temp;
		}
	continue;
	}
/*
*	       sum for small components.
*/
if(xabs > x3max)
	{
	temp = x3max/xabs;
	s3 = one + s3*temp*temp;
	x3max = xabs;
	}
else	
	{
	if(xabs != zero)
		{
		temp = xabs/x3max;
		s3 += temp*temp;
		}
	}
}
/*
*     calculation of norm.
*/
if(s1 != zero)
	{
	temp = s1 + (s2/x1max)/x1max;
	ans = x1max*sqrt(temp);
	return(ans);
	}
if(s2 != zero)
	{
	if(s2 >= x3max)
		temp = s2*(one+(x3max/s2)*(x3max*s3));
	else
		temp = x3max*((s2/x3max)+(x3max*s3));
	ans = sqrt(temp);
	}
else
	{
	ans = x3max*sqrt(s3);
	}
return(ans);
/*
*     last card of function enorm.
*/
}

/************************fdjac2.c*************************/

#define BUG 0

int fdjac2(int m, int n, double x[], double fvec[], double fjac[],
		   int ldfjac PT_UNUSED, int *iflag, double epsfcn, double wa[])
{
/*
*     **********
*
*     subroutine fdjac2
*
*     this subroutine computes a forward-difference approximation
*     to the m by n jacobian matrix associated with a specified
*     problem of m functions in n variables.
*
*     the subroutine statement is
*
*	subroutine fdjac2(fcn,m,n,x,fvec,fjac,ldfjac,iflag,epsfcn,wa)
*
*     where
*
*	fcn is the name of the user-supplied subroutine which
*	  calculates the functions. fcn must be declared
*	  in an external statement in the user calling
*	  program, and should be written as follows.
*
*	  subroutine fcn(m,n,x,fvec,iflag)
*	  integer m,n,iflag
*	  double precision x(n),fvec(m)
*	  ----------
*	  calculate the functions at x and
*	  return this vector in fvec.
*	  ----------
*	  return
*	  end
*
*	  the value of iflag should not be changed by fcn unless
*	  the user wants to terminate execution of fdjac2.
*	  in this case set iflag to a negative integer.
*
*	m is a positive integer input variable set to the number
*	  of functions.
*
*	n is a positive integer input variable set to the number
*	  of variables. n must not exceed m.
*
*	x is an input array of length n.
*
*	fvec is an input array of length m which must contain the
*	  functions evaluated at x.
*
*	fjac is an output m by n array which contains the
*	  approximation to the jacobian matrix evaluated at x.
*
*	ldfjac is a positive integer input variable not less than m
*	  which specifies the leading dimension of the array fjac.
*
*	iflag is an integer variable which can be used to terminate
*	  the execution of fdjac2. see description of fcn.
*
*	epsfcn is an input variable used in determining a suitable
*	  step length for the forward-difference approximation. this
*	  approximation assumes that the relative errors in the
*	  functions are of the order of epsfcn. if epsfcn is less
*	  than the machine precision, it is assumed that the relative
*	  errors in the functions are of the order of the machine
*	  precision.
*
*	wa is a work array of length m.
*
*     subprograms called
*
*	user-supplied ...... fcn
*
*	minpack-supplied ... dpmpar
*
*	fortran-supplied ... dabs,dmax1,dsqrt
*
*     argonne national laboratory. minpack project. march 1980.
*     burton s. garbow, kenneth e. hillstrom, jorge j. more
*
      **********
*/
int i,j,ij;
double eps,h,temp;
static double zero = 0.0;
extern double MACHEP;

temp = dmax1(epsfcn,MACHEP);
eps = sqrt(temp);
#if BUG
// printf( "fdjac2\n" );
#endif
ij = 0;
for( j=0; j<n; j++ )
	{
	temp = x[j];
	h = eps * fabs(temp);
	if(h == zero)
		h = eps;
	x[j] = temp + h;
	fcn(m,n,x,wa,iflag);
	if( *iflag < 0)
		return 0;
	x[j] = temp;
	for( i=0; i<m; i++ )
		{
		fjac[ij] = (wa[i] - fvec[i])/h;
		ij += 1;	/* fjac[i+m*j] */
		}
	}
#if BUG
pmat( m, n, fjac );
#endif
/*
*     last card of subroutine fdjac2.
*/
return 0;
}
/************************lmmisc.c*************************/

static double dmax1(double a, double b)
{
if( a >= b )
	return(a);
else
	return(b);
}

static double dmin1(double a, double b)
{
if( a <= b )
	return(a);
else
	return(b);
}

static int PT_UNUSED pmat( int m, int n, double y[] PT_UNUSED)
{
int i, j, k;

k = 0;
for( i=0; i<m; i++ )
	{
	for( j=0; j<n; j++ )
		{
		// printf( "%.5e ", y[k] );
		k += 1;
		}
	// printf( "\n" );
	}
return 0;
}