lmdif.c

InsightToolkit-1.4.0(有大量的优化算法程序)

#include "f2c.h"
#include "netlib.h"
extern double sqrt(double); /* #include <math.h> */

/* Table of constant values */
static integer c__1 = 1;

/* ====================================================================== */
/* NIST Guide to Available Math Software. */
/* Fullsource for module LMDIF from package MINPACK. */
/* Retrieved from NETLIB on Wed Jul  3 14:18:04 1996. */
/* ====================================================================== */
/* Subroutine */ void lmdif_(fcn, m, n, x, fvec, ftol, xtol, gtol, maxfev,
        epsfcn, diag, mode, factor, nprint, info, nfev, fjac, ldfjac, ipvt,
        qtf, wa1, wa2, wa3, wa4, errors)
void (*fcn) (integer*,integer*,doublereal*,doublereal*,integer*);
integer *m, *n;
doublereal *x, *fvec, *ftol, *xtol, *gtol;
integer *maxfev;
doublereal *epsfcn, *diag;
integer *mode;
doublereal *factor;
integer *nprint, *info, *nfev;
doublereal *fjac;
integer *ldfjac, *ipvt;
doublereal *qtf, *wa1, *wa2, *wa3, *wa4;
doublereal *errors;
{
    /* Local variables */
    static integer iter;
    static doublereal temp, temp1, temp2;
    static integer i, j, l, iflag;
    static doublereal delta;
    static doublereal ratio;
    static doublereal fnorm, gnorm;
    static doublereal pnorm, xnorm, fnorm1, actred, dirder, epsmch, prered;
    static doublereal par, sum;

/*     ********** */

/*     subroutine lmdif */

/*     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. */

/*     the subroutine statement is */

/*       subroutine lmdif(fcn,m,n,x,fvec,ftol,xtol,gtol,maxfev,epsfcn, */
/*                        diag,mode,factor,nprint,info,nfev,fjac, */
/*                        ldfjac,ipvt,qtf,wa1,wa2,wa3,wa4) */

/*     where */

/*       fcn is the name of the user-supplied subroutine which */
/*         calculates the functions. 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 */

/*         the value of iflag should not be changed by fcn unless */
/*         the user wants to terminate execution of lmdif. */
/*         in this case set iflag to a negative integer. */

/*       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. */

/*       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 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. */

/*     argonne national laboratory. minpack project. march 1980. */
/*     burton s. garbow, kenneth e. hillstrom, jorge j. more */

/*     ********** */

/*     epsmch is the machine precision. */

    epsmch = dpmpar_(&c__1);

    *info = 0;
    iflag = 0;
    *nfev = 0;

/*     check the input parameters for errors. */

    if (*n <= 0 || *m < *n || *ldfjac < *m || *ftol < 0. || *xtol < 0. || *gtol < 0. || *maxfev <= 0 || *factor <= 0.) {
        goto L300;
    }
    if (*mode == 2)
    for (j = 0; j < *n; ++j) {
        if (diag[j] <= 0.) {
            goto L300;
        }
    }

/*     evaluate the function at the starting point */
/*     and calculate its norm. */

    iflag = 1;
    (*fcn)(m, n, x, fvec, &iflag);
    *nfev = 1;
    if (iflag < 0) {
        goto L300;
    }
    fnorm = enorm_(m, fvec);
    errors[0] = fnorm;

/*     initialize levenberg-marquardt parameter and iteration counter. */

    par = 0.;
    iter = 1;

/*     beginning of the outer loop. */

L30:

/*        calculate the jacobian matrix. */

    iflag = 2;
    fdjac2_(fcn, m, n, x, fvec, fjac, ldfjac, &iflag, epsfcn, wa4);
    *nfev += *n;
    if (iflag < 0) {
        goto L300;
    }

/*        if requested, call fcn to enable printing of iterates. */