Provided by: minpack-dev_19961126+dfsg1-5_amd64 bug

NAME

       lmdif_, lmdif1_ - minimize the sum of squares of m nonlinear functions

SYNOPSIS

       include <minpack.h>

       void lmdif1_ ( void (*fcn)(int *m, int *n, double *x, double *fvec, int *iflag),
         int *m, int * n, double *x, double *fvec,
         double *tol, int *info, int *iwa, double *wa, int *lwa);

       void lmdif_ ( void (*fcn)(int *m, int *n, double *x, double *fvec, int *iflag),
         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 );

DESCRIPTION

       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.

       lmdif1_ serves the same purpose but has a simplified calling sequence.

   Language notes
       These functions are written in FORTRAN. If calling from C, keep these points in mind:

       Name mangling.
              With gfortran, all the function names end in an underscore.

       Compile with gfortran.
              Even  if  your program is all C code, you should link with gfortran so it will pull
              in the FORTRAN libraries automatically.  It's easiest just to use  gfortran  to  do
              all the compiling.  (It handles C just fine.)

       Call by reference.
              All function parameters must be pointers.

       Column-major arrays.
              Suppose  a function returns an array with 5 rows and 3 columns in an array z and in
              the call you have declared a leading dimension of 7.  The FORTRAN and equivalent  C
              references are:

                   z(1,1)         z[0]
                   z(2,1)         z[1]
                   z(5,1)         z[4]
                   z(1,2)         z[7]
                   z(1,3)         z[14]
                   z(i,j)         z[(i-1) + (j-1)*7]
              fcn  is the name of the user-supplied subroutine which calculates the functions. In
              FORTRAN, 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

              In C, fcn should be written as follows:

                void fcn(int *m, int *n, double *x, double *fvec, int *iflag)
                {
                     /* calculate the functions at x and return
                        the values in fvec[0] through fvec[m-1] */
                }

              The  value of iflag should not be changed by fcn unless the user wants to terminate
              execution of lmdif_ (or lmdif1_). In this case set iflag to a negative integer.

   Parameters for both lmdif_ and lmdif1_
       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.

   Parameters for lmdif1_
       tol  is  a  nonnegative  input  variable.  Termination occurs when the algorithm estimates
       either that the relative error in the sum of squares is at most tol or that  the  relative
       error between x and the solution is at most tol.

       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  algorithm estimates that the relative error in the sum of squares is  at  most
       tol.

         info  =  2  algorithm estimates that the relative error between x and the solution is at
       most tol.

         info = 3  conditions for info = 1 and info = 2 both hold.

         info = 4  fvec is orthogonal to the columns of the Jacobian to machine precision.

         info = 5  number of calls to fcn has reached or exceeded 200*(n+1).

         info = 6  tol is too small. no further reduction in the sum of squares is possible.

         info = 7  tol is too small. no further improvement in  the  approximate  solution  x  is
       possible.

       iwa is an integer work array of length n.

       wa is a work array of length lwa.

       lwa is an integer input variable not less than m*n + 5*n + m.

   Parameters for lmdif_
       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 the latter is 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.

SEE ALSO

       lmder(3), lmder1(3), lmstr(3), lmstr1(3).

AUTHORS

       Jorge  More',  Burt Garbow, and Ken Hillstrom at Argonne National Laboratory.  This manual
       page was written by Jim Van Zandt <jrv@debian.org>, for the Debian GNU/Linux  system  (but
       may be used by others).