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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  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]

   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).