NAME
lmdif_, lmdif1_ - minimize the sum of squares of m nonlinear functions
SYNOPSIS
include <minpack.h>
void lmdif1_ ( void (*fcn)(int *m, int *n, double *x, double *fvec, int
*iflag),
int *m, int * n, double *x, double *fvec,
double *tol, int *info, int *iwa, double *wa, int *lwa);
void lmdif_ ( void (*fcn)(int *m, int *n, double *x, double *fvec, int
*iflag),
int *m, int *n, double *x, double *fvec,
double *ftol, double *xtol, double *gtol, int *maxfev, double
*epsfcn, double *diag, int *mode, double *factor, int *nprint, int
*info, int *nfev, double *fjac,
int *ldfjac, int *ipvt, double *qtf,
double *wa1, double *wa2, double *wa3, double *wa4 );
DESCRIPTION
The purpose of lmdif_ is to minimize the sum of the squares of m
nonlinear functions in n variables by a modification of the Levenberg-
Marquardt algorithm. The user must provide a subroutine which
calculates the functions. The Jacobian is then calculated by a forward-
difference approximation.
lmdif1_ serves the same purpose but has a simplified calling sequence.
Language notes
These functions 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]
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(m,n,x,fvec,iflag)
integer m,n,iflag
double precision x(n),fvec(m)
----------
calculate the functions at x and
return this vector in fvec.
----------
return
end
In C, fcn should be written as follows:
void fcn(int m, int n, double *x, double *fvec, int *iflag)
{
/* calculate the functions at x and return
the values in fvec[0] through fvec[m-1] */
}
The value of iflag should not be changed by fcn unless the user
wants to terminate execution of lmdif_ (or lmdif1_). In this
case set iflag to a negative integer.
Parameters for both lmdif_ and lmdif1_
m is a positive integer input variable set to the number of functions.
n is a positive integer input variable set to the number of variables.
n must not exceed m.
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 m which contains the functions
evaluated at the output x.
Parameters for lmdif1_
tol is a nonnegative input variable. Termination occurs when the
algorithm estimates either that the relative error in the sum of
squares is at most tol or 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 in the sum of
squares is at most tol.
info = 2 algorithm estimates that the relative error between x and
the solution is at most tol.
info = 3 conditions for info = 1 and info = 2 both hold.
info = 4 fvec is orthogonal to the columns of the Jacobian to
machine precision.
info = 5 number of calls to fcn has reached or exceeded 200*(n+1).
info = 6 tol is too small. no further reduction in the sum of
squares is possible.
info = 7 tol is too small. no further improvement in the approximate
solution x is possible.
iwa is an integer work array of length n.
wa is a work array of length lwa.
lwa is an integer input variable not less than m*n + 5*n + m.
Parameters for lmdif_
ftol is a nonnegative input variable. Termination occurs when both the
actual and predicted relative reductions in the sum of squares are at
most ftol. Therefore, ftol measures the relative error desired in the
sum of squares.
xtol is a nonnegative input variable. Termination occurs when the
relative error between two consecutive iterates is at most xtol.
Therefore, xtol measures the relative error desired in the approximate
solution.
gtol is a nonnegative input variable. Termination occurs when the
cosine of the angle between fvec and any column of the Jacobian is at
most gtol in absolute value. Therefore, gtol measures the orthogonality
desired between the function vector and the columns of the Jacobian.
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.
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 the latter is 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 both actual and predicted relative reductions in the sum of
squares are at most ftol.
info = 2 relative error between two consecutive iterates is at most
xtol.
info = 3 conditions for info = 1 and info = 2 both hold.
info = 4 the cosine of the angle between fvec and any column of the
Jacobian is at most gtol in absolute value.
info = 5 number of calls to fcn has reached or exceeded maxfev.
info = 6 ftol is too small. No further reduction in the sum of
squares is possible.
info = 7 xtol is too small. No further improvement in the
approximate solution x is possible.
info = 8 gtol is too small. fvec is orthogonal to the columns of the
Jacobian to machine precision.
nfev is an integer output variable set to the number of calls to fcn.
fjac is an output m by n array. The upper n by n submatrix of fjac
contains an upper triangular matrix r with diagonal elements of
nonincreasing magnitude such that
t t t
p *(jac *jac)*p = r *r,
where p is a permutation matrix and jac is the final calculated
Jacobian. column j of p is column ipvt(j) (see below) of the identity
matrix. The lower trapezoidal part of fjac contains information
generated during the computation of r.
ldfjac is a positive integer input variable not less than m which
specifies the leading dimension of the array fjac.
ipvt is an integer output array of length n. ipvt defines a permutation
matrix p such that jac*p = q*r, where jac is the final calculated
Jacobian, q is orthogonal (not stored), and r is upper triangular with
diagonal elements of nonincreasing magnitude. Column j of p is column
ipvt(j) of the identity matrix.
qtf is an output array of length n which contains the first n elements
of the vector (q transpose)*fvec.
wa1, wa2, and wa3 are work arrays of length n.
wa4 is a work array of length m.
SEE ALSO
lmder(3), lmder1(3), lmstr(3), lmstr1(3).
AUTHORS
Jorge More’, Burt Garbow, and Ken Hillstrom at Argonne National
Laboratory. This manual page was written by Jim Van Zandt
<jrv@debian.org>, for the Debian GNU/Linux system (but may be used by
others).