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lmdif.cpp

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

/*
The original Fortran version is Copyright (C) 1999 University of Chicago.
All rights reserved.

Redistribution and use in source and binary forms, with or
without modification, are permitted provided that the
following conditions are met:

1. Redistributions of source code must retain the above
copyright notice, this list of conditions and the following
disclaimer.

2. Redistributions in binary form must reproduce the above
copyright notice, this list of conditions and the following
disclaimer in the documentation and/or other materials
provided with the distribution.

3. The end-user documentation included with the
redistribution, if any, must include the following
acknowledgment:

   "This product includes software developed by the
   University of Chicago, as Operator of Argonne National
   Laboratory.

Alternately, this acknowledgment may appear in the software
itself, if and wherever such third-party acknowledgments
normally appear.

4. WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
BE CORRECTED.

5. LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
(INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
POSSIBILITY OF SUCH LOSS OR DAMAGES.


C translation Copyright (C) Steve Moshier

What you see here may be used freely but it comes with no support
or guarantee.
*/

#include <ql/math/optimization/levenbergmarquardt.hpp>
#include <cmath>
#include <cstdio>

namespace QuantLib {
  namespace MINPACK {
#define BUG 0
/* resolution of arithmetic */
double MACHEP = 1.2e-16;
/* smallest nonzero number */
double DWARF = 1.0e-38;





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;

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

for( i=0; i<n; i++ )
{
xabs = std::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*std::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 = std::sqrt(temp);
    }
else
    {
    ans = x3max*std::sqrt(s3);
    }
return(ans);
/*
*     last card of function enorm.
*/
}
/************************lmmisc.c*************************/

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

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

int min0(int a,int b)

{
if( a <= b )
    return(a);
else
    return(b);
}

int mod( int k, int m )
{
return( k % m );
}

#if BUG
void pmat( int m, int n, double* y  )
{
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" );
    }
}
#endif

/***********Sample of user supplied function****************
 * m = number of functions
 * n = number of variables
 * x = vector of function arguments
 * fvec = vector of function values
 * iflag = error return variable
 */
void fcn(int m,int n, double* x, double* fvec,int *iflag)
{
  QuantLib::LevenbergMarquardt::fcn(m, n, x, fvec, iflag);
}

void
fdjac2(int m,int n,double* x,double* fvec,double* fjac,int,
       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;


temp = dmax1(epsfcn,MACHEP);
eps = std::sqrt(temp);
#if BUG
printf( "fdjac2\n" );
#endif
ij = 0;
for( j=0; j<n; j++ )
    {
    temp = x[j];
    h = eps * std::fabs(temp);
    if(h == zero)
        h = eps;
    x[j] = temp + h;
    fcn(m,n,x,wa,iflag);
    if( *iflag < 0)
        return;
    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.
*/
}
/************************qrfac.c*************************/


void
qrfac(int m,int n,double* a,int,int pivot,int* ipvt,
      int,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,min0
*
*     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;

/*
*     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 = min0(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] *= std::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.
*/
}

/************************qrsolv.c*************************/


void
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;

/*
*     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(std::fabs(r[kk]) < std::fabs(sdiag[k]))
        {
        cotan = r[kk]/sdiag[k];
        sin = p5/std::sqrt(p25+p25*cotan*cotan);
        cos = sin*cotan;
        }
    else
        {
        tan = sdiag[k]/r[kk];
        cos = p5/std::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.
*/
}

/************************lmpar.c*************************/


void
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 p1 = 0.1;
static double p001 = 0.001;

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 = std::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( (std::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.
*/
}

/*********************** lmdif.c ****************************/




void 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=0,dirder,fnorm,fnorm1,gnorm;
double par,pnorm,prered,ratio;
double sum,temp,temp1,temp2,temp3,xnorm=0;
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;

*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(mod(iter-1,nprint) == 0)
        {
        fcn(m,n,x,fvec,&iflag);
        if(iflag < 0)
            goto L300;
        #if BUG
        printf( "fnorm %.15e\n", enorm(m,fvec) );
        #endif
        }
    }
/*
*    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,std::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 = (std::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( (std::fabs(actred) <= ftol)
  && (prered <= ftol)
  && (p5*ratio <= one) )
    *info = 1;
if(delta <= xtol*xnorm)
    *info = 2;
if( (std::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( (std::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.
*/
}
}
}

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


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