hybrd.c

该程序实现了非线性最小二乘问题和非线性方程组的解法

/* hybrd.f -- translated by f2c (version 20020621).
   You must link the resulting object file with the libraries:
	-lf2c -lm   (in that order)
*/

#include <math.h>
#include <cminpack.h>
#define min(a,b) ((a) <= (b) ? (a) : (b))
#define max(a,b) ((a) >= (b) ? (a) : (b))
#define TRUE_ (1)
#define FALSE_ (0)

/* Subroutine */ int hybrd(minpack_func_nn fcn, void *p, 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 *nfev, double *
	fjac, int ldfjac, double *r__, int lr, double *qtf, 
	double *wa1, double *wa2, double *wa3, double *wa4)
{
    /* Initialized data */

#define p1 .1
#define p5 .5
#define p001 .001
#define p0001 1e-4

    /* System generated locals */
    int fjac_dim1, fjac_offset, i__1, i__2;
    double d__1, d__2;

    /* Local variables */
    int i__, j, l, jm1, iwa[1];
    double sum;
    int sing;
    int iter;
    double temp;
    int msum, iflag;
    double delta;
    int jeval;
    int ncsuc;
    double ratio;
    double fnorm;
    double pnorm, xnorm, fnorm1;
    int nslow1, nslow2;
    int ncfail;
    double actred, epsmch, prered;
    int info;

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

/*     subroutine hybrd */

/*     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. */

/*     the subroutine statement is */

/*       subroutine hybrd(fcn,n,x,fvec,xtol,maxfev,ml,mu,epsfcn, */
/*                        diag,mode,factor,nprint,info,nfev,fjac, */
/*                        ldfjac,r,lr,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(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 */

/*         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. */

/*       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. */

/*     subprograms called */

/*       user-supplied ...... fcn */

/*       minpack-supplied ... dogleg,dpmpar,enorm,fdjac1, */
/*                            qform,qrfac,r1mpyq,r1updt */

/*       fortran-supplied ... dabs,dmax1,dmin1,min0,mod */

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

/*     ********** */
    /* Parameter adjustments */
    --wa4;
    --wa3;
    --wa2;
    --wa1;
    --qtf;
    --diag;
    --fvec;
    --x;
    fjac_dim1 = ldfjac;
    fjac_offset = 1 + fjac_dim1 * 1;
    fjac -= fjac_offset;
    --r__;

    /* Function Body */

/*     epsmch is the machine precision. */

    epsmch = dpmpar(1);

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

/*     check the input parameters for errors. */

    if (n <= 0 || xtol < 0. || maxfev <= 0 || ml < 0 || mu < 0 ||
	    factor <= 0. || ldfjac < n || lr < n * (n + 1) / 2) {
	goto L300;
    }
    if (mode != 2) {
	goto L20;
    }
    i__1 = n;
    for (j = 1; j <= i__1; ++j) {
	if (diag[j] <= 0.) {
	    goto L300;
	}
/* L10: */
    }
L20:

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

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

/*     determine the number of calls to fcn needed to compute */
/*     the jacobian matrix. */

/* Computing MIN */
    i__1 = ml + mu + 1;
    msum = min(i__1,n);

/*     initialize iteration counter and monitors. */

    iter = 1;
    ncsuc = 0;
    ncfail = 0;
    nslow1 = 0;
    nslow2 = 0;

/*     beginning of the outer loop. */

L30:
    jeval = TRUE_;

/*        calculate the jacobian matrix. */

    iflag = fdjac1(fcn, p, n, &x[1], &fvec[1], &fjac[fjac_offset], ldfjac,
	     ml, mu, epsfcn, &wa1[1], &wa2[1]);
    *nfev += msum;
    if (iflag < 0) {
	goto L300;
    }

/*        compute the qr factorization of the jacobian. */

    qrfac(n, n, &fjac[fjac_offset], ldfjac, FALSE_, iwa, 1, &wa1[1], &
	    wa2[1], &wa3[1]);

/*        on the first iteration and if mode is 1, scale according */
/*        to the norms of the columns of the initial jacobian. */

    if (iter != 1) {
	goto L70;
    }
    if (mode == 2) {
	goto L50;
    }
    i__1 = n;
    for (j = 1; j <= i__1; ++j) {
	diag[j] = wa2[j];
	if (wa2[j] == 0.) {
	    diag[j] = 1.;
	}
/* L40: */
    }
L50:

/*        on the first iteration, calculate the norm of the scaled x */
/*        and initialize the step bound delta. */

    i__1 = n;