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).