www.pudn.com > TSAI30B3.rar > LMDIF.C
/* lmdif.f -- translated by f2c (version of 17 January 1992 0:17:58).
You must link the resulting object file with the libraries:
-lf77 -li77 -lm -lc (in that order)
*/
#include "f2c.h"
/* Table of constant values */
static integer c__1 = 1;
/* Subroutine */ int 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)
/* Subroutine */ int (*fcn) ();
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;
{
/* Initialized data */
static doublereal one = 1.;
static doublereal p1 = .1;
static doublereal p5 = .5;
static doublereal p25 = .25;
static doublereal p75 = .75;
static doublereal p0001 = 1e-4;
static doublereal zero = 0.;
/* System generated locals */
integer fjac_dim1, fjac_offset, i__1, i__2;
doublereal d__1, d__2, d__3;
/* Builtin functions */
double sqrt();
/* Local variables */
static integer iter;
static doublereal temp, temp1, temp2;
static integer i, j, l, iflag;
static doublereal delta;
extern /* Subroutine */ int qrfac_(), lmpar_();
static doublereal ratio;
extern doublereal enorm_();
static doublereal fnorm, gnorm;
extern /* Subroutine */ int fdjac2_();
static doublereal pnorm, xnorm, fnorm1, actred, dirder, epsmch, prered;
extern doublereal dpmpar_();
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. */
/* subprograms called */
/* user-supplied ...... fcn */
/* minpack-supplied ... dpmpar,enorm,fdjac2,lmpar,qrfac */
/* fortran-supplied ... dabs,dmax1,dmin1,dsqrt,mod */
/* argonne national laboratory. minpack project. march 1980. */
/* burton s. garbow, kenneth e. hillstrom, jorge j. more */
/* ********** */
/* Parameter adjustments */
--wa4;
--wa3;
--wa2;
--wa1;
--qtf;
--ipvt;
fjac_dim1 = *ldfjac;
fjac_offset = fjac_dim1 + 1;
fjac -= fjac_offset;
--diag;
--fvec;
--x;
/* Function Body */
/* 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 < zero || *xtol < zero ||
*gtol < zero || *maxfev <= 0 || *factor <= zero) {
goto L300;
}
if (*mode != 2) {
goto L20;
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
if (diag[j] <= zero) {
goto L300;
}
/* L10: */
}
L20:
/* evaluate the function at the starting point */
/* and calculate its norm. */
iflag = 1;
(*fcn)(m, n, &x[1], &fvec[1], &iflag);
*nfev = 1;
if (iflag < 0) {
goto L300;
}
fnorm = enorm_(m, &fvec[1]);
/* initialize levenberg-marquardt parameter and iteration counter. */
par = zero;
iter = 1;
/* beginning of the outer loop. */
L30:
/* calculate the jacobian matrix. */
iflag = 2;
fdjac2_(fcn, m, n, &x[1], &fvec[1], &fjac[fjac_offset], ldfjac, &iflag,
epsfcn, &wa4[1]);
*nfev += *n;
if (iflag < 0) {
goto L300;
}
/* if requested, call fcn to enable printing of iterates. */
if (*nprint <= 0) {
goto L40;
}
iflag = 0;
if ((iter - 1) % *nprint == 0) {
(*fcn)(m, n, &x[1], &fvec[1], &iflag);
}
if (iflag < 0) {
goto L300;
}
L40:
/* compute the qr factorization of the jacobian. */
qrfac_(m, n, &fjac[fjac_offset], ldfjac, &c__1, &ipvt[1], n, &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 L80;
}
if (*mode == 2) {
goto L60;
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
diag[j] = wa2[j];
if (wa2[j] == zero) {
diag[j] = one;
}
/* L50: */
}
L60:
/* on the first iteration, calculate the norm of the scaled x */
/* and initialize the step bound delta. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
wa3[j] = diag[j] * x[j];
/* L70: */
}
xnorm = enorm_(n, &wa3[1]);
delta = *factor * xnorm;
if (delta == zero) {
delta = *factor;
}
L80:
/* form (q transpose)*fvec and store the first n components in */
/* qtf. */
i__1 = *m;
for (i = 1; i <= i__1; ++i) {
wa4[i] = fvec[i];
/* L90: */
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
if (fjac[j + j * fjac_dim1] == zero) {
goto L120;
}
sum = zero;
i__2 = *m;
for (i = j; i <= i__2; ++i) {
sum += fjac[i + j * fjac_dim1] * wa4[i];
/* L100: */
}
temp = -sum / fjac[j + j * fjac_dim1];
i__2 = *m;
for (i = j; i <= i__2; ++i) {
wa4[i] += fjac[i + j * fjac_dim1] * temp;
/* L110: */
}
L120:
fjac[j + j * fjac_dim1] = wa1[j];
qtf[j] = wa4[j];
/* L130: */
}
/* compute the norm of the scaled gradient. */
gnorm = zero;
if (fnorm == zero) {
goto L170;
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
l = ipvt[j];
if (wa2[l] == zero) {
goto L150;
}
sum = zero;
i__2 = j;
for (i = 1; i <= i__2; ++i) {
sum += fjac[i + j * fjac_dim1] * (qtf[i] / fnorm);
/* L140: */
}
/* Computing MAX */
d__2 = gnorm, d__3 = (d__1 = sum / wa2[l], abs(d__1));
gnorm = max(d__2,d__3);
L150:
/* L160: */
;
}
L170:
/* test for convergence of the gradient norm. */
if (gnorm <= *gtol) {
*info = 4;
}
if (*info != 0) {
goto L300;
}
/* rescale if necessary. */
if (*mode == 2) {
goto L190;
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/* Computing MAX */
d__1 = diag[j], d__2 = wa2[j];
diag[j] = max(d__1,d__2);
/* L180: */
}
L190:
/* beginning of the inner loop. */
L200:
/* determine the levenberg-marquardt parameter. */
lmpar_(n, &fjac[fjac_offset], ldfjac, &ipvt[1], &diag[1], &qtf[1], &delta,
&par, &wa1[1], &wa2[1], &wa3[1], &wa4[1]);
/* store the direction p and x + p. calculate the norm of p. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
wa1[j] = -wa1[j];
wa2[j] = x[j] + wa1[j];
wa3[j] = diag[j] * wa1[j];
/* L210: */
}
pnorm = enorm_(n, &wa3[1]);
/* on the first iteration, adjust the initial step bound. */
if (iter == 1) {
delta = min(delta,pnorm);
}
/* evaluate the function at x + p and calculate its norm. */
iflag = 1;
(*fcn)(m, n, &wa2[1], &wa4[1], &iflag);
++(*nfev);
if (iflag < 0) {
goto L300;
}
fnorm1 = enorm_(m, &wa4[1]);
/* compute the scaled actual reduction. */
actred = -one;
if (p1 * fnorm1 < fnorm) {
/* Computing 2nd power */
d__1 = fnorm1 / fnorm;
actred = one - d__1 * d__1;
}
/* compute the scaled predicted reduction and */
/* the scaled directional derivative. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
wa3[j] = zero;
l = ipvt[j];
temp = wa1[l];
i__2 = j;
for (i = 1; i <= i__2; ++i) {
wa3[i] += fjac[i + j * fjac_dim1] * temp;
/* L220: */
}
/* L230: */
}
temp1 = enorm_(n, &wa3[1]) / fnorm;
temp2 = sqrt(par) * pnorm / fnorm;
/* Computing 2nd power */
d__1 = temp1;
/* Computing 2nd power */
d__2 = temp2;
prered = d__1 * d__1 + d__2 * d__2 / p5;
/* Computing 2nd power */
d__1 = temp1;
/* Computing 2nd power */
d__2 = temp2;
dirder = -(d__1 * d__1 + d__2 * d__2);
/* compute the ratio of the actual to the predicted */
/* reduction. */
ratio = zero;
if (prered != zero) {
ratio = actred / prered;
}
/* update the step bound. */
if (ratio > p25) {
goto L240;
}
if (actred >= zero) {
temp = p5;
}
if (actred < zero) {
temp = p5 * dirder / (dirder + p5 * actred);
}
if (p1 * fnorm1 >= fnorm || temp < p1) {
temp = p1;
}
/* Computing MIN */
d__1 = delta, d__2 = pnorm / p1;
delta = temp * min(d__1,d__2);
par /= temp;
goto L260;
L240:
if (par != zero && ratio < p75) {
goto L250;
}
delta = pnorm / p5;
par = p5 * par;
L250:
L260:
/* test for successful iteration. */
if (ratio < p0001) {
goto L290;
}
/* successful iteration. update x, fvec, and their norms. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
x[j] = wa2[j];
wa2[j] = diag[j] * x[j];
/* L270: */
}
i__1 = *m;
for (i = 1; i <= i__1; ++i) {
fvec[i] = wa4[i];
/* L280: */
}
xnorm = enorm_(n, &wa2[1]);
fnorm = fnorm1;
++iter;
L290:
/* tests for convergence. */
if (abs(actred) <= *ftol && prered <= *ftol && p5 * ratio <= one) {
*info = 1;
}
if (delta <= *xtol * xnorm) {
*info = 2;
}
if (abs(actred) <= *ftol && prered <= *ftol && p5 * ratio <= one && *info
== 2) {
*info = 3;
}
if (*info != 0) {
goto L300;
}
/* tests for termination and stringent tolerances. */
if (*nfev >= *maxfev) {
*info = 5;
}
if (abs(actred) <= epsmch && prered <= epsmch && p5 * ratio <= one) {
*info = 6;
}
if (delta <= epsmch * xnorm) {
*info = 7;
}
if (gnorm <= epsmch) {
*info = 8;
}
if (*info != 0) {
goto L300;
}
/* end of the inner loop. repeat if iteration unsuccessful. */
if (ratio < p0001) {
goto L200;
}
/* end of the outer loop. */
goto L30;
L300:
/* termination, either normal or user imposed. */
if (iflag < 0) {
*info = iflag;
}
iflag = 0;
if (*nprint > 0) {
(*fcn)(m, n, &x[1], &fvec[1], &iflag);
}
return 0;
/* last card of subroutine lmdif. */
} /* lmdif_ */