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

NAME

       hybrj_, hybrj1_ - find a zero of a system of nonlinear function

SYNOPSIS

       #include <minpack.h>

       void hybrj1_ (void (*fcn)(int *n, double *x, double *fvec, double *fjac, int *ldfjac, int
                     *iflag),
           int *n, double *x, double *fvec, double *fjac,
           int *ldfjac,
           double *tol, int *info, double *wa, int *lwa);

       void hybrj_ (void (*fcn)(int *n, double *x, double *fvec, double *fjac, int *ldfjac, int
                    *iflag),
           int *n, double *x, double *fvec, double *fjac,
           int *ldfjac,
           double *xtol, int *maxfev, double *diag, int *mode, double *factor, int *nprint, int
           *info, int *nfev,
           int *njev, double *r, int *lr, double *qtf,
           double *wa1, double *wa2, double *wa3, double *wa4);

DESCRIPTION

       The purpose of hybrj_ is to find a zero  of  a  system  of  n  nonlinear  functions  in  n
       variables  by  a  modification  of  the  Powell  hybrid  method.  The  user must provide a
       subroutine which calculates the functions and a subroutine which calculates the Jacobian.

       hybrj1_ serves the same function but has a simplified calling sequence.

   Language notes
       hybrj_ and hybrj1_ 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]

   Parameters for both functions
       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(n,x,fvec,fjac,ldfjac,iflag)
       integer n,ldfjac,iflag
       double precision x(n),fvec(n),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 *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. 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 hybrj_.  In this case set iflag to a negative integer.

       n is a positive integer input variable set to the number of functions and variables.

       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.

       fjac is an output n by n array which contains the orthogonal matrix q produced by  the  qr
       factorization of the final approximate jacobian.

       ldfjac  is  a  positive integer input variable not less than n which specifies the leading
       dimension of the array fjac.

       fvec is an output array of length n which contains the functions evaluated at  the  output
       x.

   Parameters for hybrj1_
       tol  is a nonnegative input variable. Termination occurs when the algorithm estimates 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
                  between x and the solution is at most tol.

       info = 2   number of calls to fcn has reached or exceeded
                  200*(n+1).

       info = 3   tol is too small. No further improvement in
                  the approximate solution x is possible.

       info = 4   iteration is not making good progress.

       wa is a work array of length lwa.

       lwa is a positive integer input variable not less than (n*(3*n+13))/2.

   Parameters for hybrj_
       xtol  is  a nonnegative input variable. Termination occurs when the relative error between
       two consecutive iterates is at most xtol.

       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 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   relative error between two consecutive iterates
                  is at most xtol.

       info = 2   number of calls to fcn has reached or exceeded
                  maxfev.

       info = 3   xtol is too small. No further improvement in
                  the approximate solution x is possible.

       info = 4   iteration is not making good progress, as
                  measured by the improvement from the last
                  five jacobian evaluations.

       info = 5   iteration is not making good progress, as
                  measured by the improvement from the last
                  ten iterations.

       nfev is an integer output variable set to the number of calls to fcn.

       fjac  is  an output n by n array which contains the orthogonal matrix q produced by the qr
       factorization of the final approximate jacobian.

       ldfjac is a positive integer input variable not less than n which  specifies  the  leading
       dimension of the array fjac.

       r  is  an output array of length lr which contains the upper triangular matrix produced by
       the qr factorization of the final approximate Jacobian, stored rowwise.

       lr is a positive integer input variable not less than (n*(n+1))/2.

       qtf is an output array of length n which contains the vector (q transpose)*fvec.

       wa1, wa2, wa3, and wa4 are work arrays of length n.

SEE ALSO

       hybrd(3), hybrd1(3).

AUTHORS

       Burton S. Garbow, Kenneth E. Hillstrom, Jorge J. More.
       This manual page was written by Jim Van Zandt <jrv@debian.org>, for the  Debian  GNU/Linux
       system (but may be used by others).