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.

AUTHORS

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