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 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).