lmpar.c

来自「InsightToolkit-1.4.0(有大量的优化算法程序)」· C语言 代码 · 共 287 行

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

/* Table of constant values */
static integer c__2 = 2;

/* Subroutine */ void lmpar_(n, r, ldr, ipvt, diag, qtb, delta, par, x, sdiag, wa1, wa2)
integer *n;
doublereal *r;
integer *ldr, *ipvt;
doublereal *diag, *qtb, *delta, *par, *x, *sdiag, *wa1, *wa2;
{
    /* Local variables */
    static doublereal parc, parl;
    static integer iter;
    static doublereal temp, paru;
    static integer i, j, l;
    static doublereal dwarf;
    static integer nsing;
    static doublereal gnorm, fp;
    static doublereal dxnorm;
    static doublereal sum;

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

/*     subroutine lmpar */

/*     given an m by n matrix a, an n by n nonsingular diagonal */
/*     matrix d, an m-vector b, and a positive number delta, */
/*     the problem is to determine a value for the parameter */
/*     par such that if x solves the system */

/*           a*x = b ,     sqrt(par)*d*x = 0 , */

/*     in the least squares sense, and dxnorm is the euclidean */
/*     norm of d*x, then either par is zero and */

/*           (dxnorm-delta) .le. 0.1*delta , */

/*     or par is positive and */

/*           abs(dxnorm-delta) .le. 0.1*delta . */

/*     this subroutine completes the solution of the problem */
/*     if it is provided with the necessary information from the */
/*     qr factorization, with column pivoting, of a. that is, if */
/*     a*p = q*r, where p is a permutation matrix, q has orthogonal */
/*     columns, and r is an upper triangular matrix with diagonal */
/*     elements of nonincreasing magnitude, then lmpar expects */
/*     the full upper triangle of r, the permutation matrix p, */
/*     and the first n components of (q transpose)*b. on output */
/*     lmpar also provides an upper triangular matrix s such that */

/*            t   t                   t */
/*           p *(a *a + par*d*d)*p = s *s . */

/*     s is employed within lmpar and may be of separate interest. */

/*     only a few iterations are generally needed for convergence */
/*     of the algorithm. if, however, the limit of 10 iterations */
/*     is reached, then the output par will contain the best */
/*     value obtained so far. */

/*     the subroutine statement is */

/*       subroutine lmpar(n,r,ldr,ipvt,diag,qtb,delta,par,x,sdiag, */
/*                        wa1,wa2) */

/*     where */

/*       n is a positive integer input variable set to the order of r. */

/*       r is an n by n array. on input the full upper triangle */
/*         must contain the full upper triangle of the matrix r. */
/*         on output the full upper triangle is unaltered, and the */
/*         strict lower triangle contains the strict upper triangle */
/*         (transposed) of the upper triangular matrix s. */

/*       ldr is a positive integer input variable not less than n */
/*         which specifies the leading dimension of the array r. */

/*       ipvt is an integer input array of length n which defines the */
/*         permutation matrix p such that a*p = q*r. column j of p */
/*         is column ipvt(j) of the identity matrix. */

/*       diag is an input array of length n which must contain the */
/*         diagonal elements of the matrix d. */

/*       qtb is an input array of length n which must contain the first */
/*         n elements of the vector (q transpose)*b. */

/*       delta is a positive input variable which specifies an upper */
/*         bound on the euclidean norm of d*x. */

/*       par is a nonnegative variable. on input par contains an */
/*         initial estimate of the levenberg-marquardt parameter. */
/*         on output par contains the final estimate. */

/*       x is an output array of length n which contains the least */
/*         squares solution of the system a*x = b, sqrt(par)*d*x = 0, */
/*         for the output par. */

/*       sdiag is an output array of length n which contains the */
/*         diagonal elements of the upper triangular matrix s. */

/*       wa1 and wa2 are work arrays of length n. */

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

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

/*     dwarf is the smallest positive magnitude. */

    dwarf = dpmpar_(&c__2);

/*     compute and store in x the gauss-newton direction. if the */
/*     jacobian is rank-deficient, obtain a least squares solution. */

    nsing = *n;
    for (j = 0; j < *n; ++j) {
        wa1[j] = qtb[j];
        if (r[j + j * *ldr] == 0. && nsing == *n) {
            nsing = j;
        }
        if (nsing < *n) {
            wa1[j] = 0.;
        }
    }
    for (j = nsing-1; j >= 0; --j) {
        wa1[j] /= r[j + j * *ldr];
        temp = wa1[j];
        if (j < 1) {
            continue;
        }
        for (i = 0; i < j; ++i) {
            wa1[i] -= r[i + j * *ldr] * temp;
        }
    }
    for (j = 0; j < *n; ++j) {
        l = ipvt[j] - 1;
        x[l] = wa1[j];
    }

/*     initialize the iteration counter. */
/*     evaluate the function at the origin, and test */
/*     for acceptance of the gauss-newton direction. */

    iter = 0;
    for (j = 0; j < *n; ++j) {
        wa2[j] = diag[j] * x[j];
    }
    dxnorm = enorm_(n, wa2);
    fp = dxnorm - *delta;
    if (fp <= .1 * *delta) {
        goto L220;
    }

/*     if the jacobian is not rank deficient, the newton */
/*     step provides a lower bound, parl, for the zero of */
/*     the function. otherwise set this bound to zero. */

    parl = 0.;
    if (nsing < *n) {
        goto L120;
    }
    for (j = 0; j < *n; ++j) {
        l = ipvt[j] - 1;
        wa1[j] = diag[l] * (wa2[l] / dxnorm);
    }
    for (j = 0; j < *n; ++j) {
        sum = 0.;
        if (j < 2) {
            goto L100;
        }
        for (i = 0; i < j; ++i) {
            sum += r[i + j * *ldr] * wa1[i];
        }
L100:
        wa1[j] = (wa1[j] - sum) / r[j + j * *ldr];
    }
    temp = enorm_(n, wa1);
    parl = fp / *delta / temp / temp;
L120:

/*     calculate an upper bound, paru, for the zero of the function. */

    for (j = 0; j < *n; ++j) {
        sum = 0.;
        for (i = 0; i <= j; ++i) {
            sum += r[i + j * *ldr] * qtb[i];
        }
        l = ipvt[j] - 1;
        wa1[j] = sum / diag[l];
    }
    gnorm = enorm_(n, wa1);
    paru = gnorm / *delta;
    if (paru == 0.) {
        paru = dwarf / min(*delta,.1);
    }

/*     if the input par lies outside of the interval (parl,paru), */
/*     set par to the closer endpoint. */

    *par = max(*par,parl);
    *par = min(*par,paru);
    if (*par == 0.) {
        *par = gnorm / dxnorm;
    }

/*     beginning of an iteration. */

L150:
    ++iter;

/*        evaluate the function at the current value of par. */

    if (*par == 0.) {
        *par = max(dwarf,.001 * paru);
    }
    temp = sqrt(*par);
    for (j = 0; j < *n; ++j) {
        wa1[j] = temp * diag[j];
    }
    qrsolv_(n, r, ldr, ipvt, wa1, qtb, x, sdiag, wa2);
    for (j = 0; j < *n; ++j) {
        wa2[j] = diag[j] * x[j];
    }
    dxnorm = enorm_(n, wa2);
    temp = fp;
    fp = dxnorm - *delta;

/*        if the function is small enough, accept the current value */
/*        of par. also test for the exceptional cases where parl */
/*        is zero or the number of iterations has reached 10. */

    if (abs(fp) <= .1 * *delta || ( parl == 0. && fp <= temp && temp < 0. ) || iter == 10) {
        goto L220;
    }

/*        compute the newton correction. */

    for (j = 0; j < *n; ++j) {
        l = ipvt[j] - 1;
        wa1[j] = diag[l] * (wa2[l] / dxnorm);
    }
    for (j = 0; j < *n; ++j) {
        wa1[j] /= sdiag[j];
        temp = wa1[j];
        if (*n <= j+1) {
            continue;
        }
        for (i = j+1; i < *n; ++i) {
            wa1[i] -= r[i + j * *ldr] * temp;
        }
    }
    temp = enorm_(n, wa1);
    parc = fp / *delta / temp / temp;

/*        depending on the sign of the function, update parl or paru. */

    if (fp > 0.) {
        parl = max(parl,*par);
    }
    if (fp < 0.) {
        paru = min(paru,*par);
    }

/*        compute an improved estimate for par. */

    *par = max(parl,*par + parc);

/*        end of an iteration. */

    goto L150;
L220:

/*     termination. */

    if (iter == 0) {
        *par = 0.;
    }
    return;

} /* lmpar_ */