Provided by: minpack-dev_19961126+dfsg1-1_amd64 bug

NAME

       lmder_, lmder1_ - minimize the sum of squares of m nonlinear functions, with user supplied
       Jacobian

SYNOPSIS

       include <minpack.h>

       void lmder1_ ( void (*fcn) (int *m, int *n, double *x, double *fvec, double *fjac,
                                   int *ldfjac, int *iflag),
                      int *m, int * n, double *x, double *fvec, double *fjac, int *ldfjac,
                      double *tol, int *info, int *iwa, double *wa, int *lwa);

       void lmder_ ( void (*fcn)( int *m, int *n, double *x, double *fvec, double *fjac,
                                  int *ldfjac, int *iflag),
                     int *m, int *n, double *x, double *fvec, double *fjac, int *ldfjac,
                     double *ftol, double *xtol, double *gtol, int *maxfev, double *diag, int
                     *mode,
                     double *factor, int *nprint, int *info,
                     int *nfev, int *njev, int *ipvt,
                     double *qtf, double *wa1, double *wa2, double *wa3, double *wa4 );

DESCRIPTION

       The  purpose of lmder_ 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
       function which calculates the functions and the Jacobian.

       lmder1_ performs the same function as lmder_ but has a simplified calling sequence.

       lmstr and lmstr1 also perform the same function but use minimal storage.

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

       Name mangling.
              With  g77  version  2.95 or 3.0, all the function names end in an underscore.  This
              may change with future versions of g77.

       Compile with g77.
              Even if your program is all C code, you should link with g77 so it will pull in the
              FORTRAN  libraries  automatically.   It's  easiest  just  to  use g77 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]

   User-supplied Function
       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,fjac,ldfjac,iflag)
         integer m,n,iflag
         double precision x(n),fvec(m),fjac(ldfjac,n)
         ----------
         if iflag = 1 calculate the functions at x and
         return this vector in fvec. do not alter fjac.
         if iflag = 2 calculate the jacobian at x and
         return this matrix in fjac. do not alter fvec.
         ----------
         return
         end

       In C, fcn should be written as follows:

         void fcn(int m, int n, double *x, double *fvec, double *fjac,
                  int *ldfjac, int *iflag)
         {
           /* if iflag = 1 calculate the functions at x and return this
              vector in fvec[0] through fvec[m-1]. do not alter fjac.
              if iflag = 2 calculate the jacobian at x and return this
              matrix in fjac. do not alter fvec. */
         }

       The value of iflag should not be changed  by  fcn  unless  the  user  wants  to  terminate
       execution of lmder_ (or lmder1_). In this case set iflag to a negative integer.

   Parameters for both lmder_ and lmder1_
       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.

       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.

   Parameters for lmder1_
       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 for lmder1_.

   Parameters for lmder_
       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.

       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 with iflag = 1.

       njev is an integer output variable set to the number of calls to fcn with iflag = 2.

       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

       lmdif(3), lmdif1(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).