HYBRJ_(3)						     Library Functions Manual							 HYBRJ_(3)

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 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,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 nega-
       tive 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 itera-
       tion.

       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  eu-
       clidean 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 avail-
       able 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 Ja-
       cobian, 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).

Minpack 							   March 8, 2002							 HYBRJ_(3)