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

NAME

       hybrd_, hybrd1_ - find a zero of a system of nonlinear function

SYNOPSIS

       #include <minpack.h>

       void hybrd1_ ( void (*fcn)( int *n, double *x,
                                   double *fvec, int *iflag),
                      int *n, double *x, double *fvec,
                      double *tol, int *info, double *wa,
                      int *lwa);

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

DESCRIPTION

       The purpose of hybrd_ 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. The Jacobian is then calculated by  a  forward-
       difference approximation.

       hybrd1_ serves the same function but has a simplified calling sequence.

   Language notes
       hybrd_ and hybrd1_ 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,iflag)
       integer n,iflag
       double precision x(n),fvec(n)
       ----------
       calculate the functions at x and
       return this vector in fvec.
       ---------
       return
       end

       In C, fcn should be written as follows:

         void fcn(int *n, double *x, double *fvec, int *iflag)
         {
           /* calculate the functions at x and
              return this vector in fvec. */
         }

       The  value  of  iflag  should  not  be  changed  by fcn unless the user wants to terminate
       execution of hybrd_.  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.

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

   Parameters for hybrd1_
       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 hybrd_
       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.

       ml is a nonnegative integer input variable which  specifies  the  number  of  subdiagonals
       within  the band of the jacobian matrix. If the Jacobian is not banded, set ml to at least
       n - 1.

       mu is a nonnegative integer input variable which specifies the  number  of  superdiagonals
       within  the band of the jacobian matrix. If the jacobian is not banded, set mu to at least
       n - 1.

       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   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

       hybrj(3), hybrj1(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).