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1284 lines
43 KiB
C
1284 lines
43 KiB
C
/*
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* Library: lmfit (Levenberg-Marquardt least squares fitting)
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*
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* File: lmmin.c
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*
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* Contents: Levenberg-Marquardt minimization.
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*
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* Copyright: MINPACK authors, The University of Chikago (1980-1999)
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* Joachim Wuttke, Forschungszentrum Juelich GmbH (2004-2013)
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*
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* License: see ../COPYING (FreeBSD)
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*
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* Homepage: apps.jcns.fz-juelich.de/lmfit
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*/
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#include <assert.h>
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#include <stdlib.h>
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#include <stdio.h>
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#include <math.h>
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#include <float.h>
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#include "lmmin.h"
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#define MIN(a,b) (((a)<=(b)) ? (a) : (b))
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#define MAX(a,b) (((a)>=(b)) ? (a) : (b))
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#define SQR(x) (x)*(x)
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/* Declare functions that do the heavy numerics.
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Implementions are in this source file, below lmmin.
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Dependences: lmmin calls qrfac and lmpar; lmpar calls qrsolv. */
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void lm_lmpar(
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const int n, double *const r, const int ldr, int *const ipvt,
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double *const diag, double *const qtb, double delta, double *const par,
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double *const x,
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double *const sdiag, double *const aux, double *const xdi);
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void lm_qrfac(
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const int m, const int n, double *const a, int *const ipvt,
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double *const rdiag, double *const acnorm, double *const wa );
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void lm_qrsolv(
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const int n, double *const r, const int ldr, int *const ipvt,
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double *const diag, double *const qtb, double *const x,
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double *const sdiag, double *const wa);
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/*****************************************************************************/
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/* Numeric constants */
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/*****************************************************************************/
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/* machine-dependent constants from float.h */
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#define LM_MACHEP DBL_EPSILON /* resolution of arithmetic */
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#define LM_DWARF DBL_MIN /* smallest nonzero number */
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#define LM_SQRT_DWARF sqrt(DBL_MIN) /* square should not underflow */
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#define LM_SQRT_GIANT sqrt(DBL_MAX) /* square should not overflow */
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#define LM_USERTOL 30*LM_MACHEP /* users are recommended to require this */
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/* If the above values do not work, the following seem good for an x86:
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LM_MACHEP .555e-16
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LM_DWARF 9.9e-324
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LM_SQRT_DWARF 1.e-160
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LM_SQRT_GIANT 1.e150
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LM_USER_TOL 1.e-14
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The following values should work on any machine:
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LM_MACHEP 1.2e-16
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LM_DWARF 1.0e-38
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LM_SQRT_DWARF 3.834e-20
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LM_SQRT_GIANT 1.304e19
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LM_USER_TOL 1.e-14
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*/
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const lm_control_struct lm_control_double = {
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LM_USERTOL, LM_USERTOL, LM_USERTOL, LM_USERTOL, 100., 100, 1,
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NULL, 0, -1, -1 };
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const lm_control_struct lm_control_float = {
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1.e-7, 1.e-7, 1.e-7, 1.e-7, 100., 100, 1,
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NULL, 0, -1, -1 };
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/*****************************************************************************/
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/* Message texts (indexed by status.info) */
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/*****************************************************************************/
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const char *lm_infmsg[] = {
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"found zero (sum of squares below underflow limit)",
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"converged (the relative error in the sum of squares is at most tol)",
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"converged (the relative error of the parameter vector is at most tol)",
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"converged (both errors are at most tol)",
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"trapped (by degeneracy; increasing epsilon might help)",
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"exhausted (number of function calls exceeding preset patience)",
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"failed (ftol<tol: cannot reduce sum of squares any further)",
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"failed (xtol<tol: cannot improve approximate solution any further)",
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"failed (gtol<tol: cannot improve approximate solution any further)",
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"crashed (not enough memory)",
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"exploded (fatal coding error: improper input parameters)",
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"stopped (break requested within function evaluation)",
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"found nan (function value is not-a-number or infinite)",
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"won't fit (no free parameter)"
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};
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const char *lm_shortmsg[] = {
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"found zero", // 0
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"converged (f)", // 1
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"converged (p)", // 2
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"converged (2)", // 3
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"degenerate", // 4
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"call limit", // 5
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"failed (f)", // 6
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"failed (p)", // 7
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"failed (o)", // 8
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"no memory", // 9
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"invalid input", // 10
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"user break", // 11
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"found nan", // 12
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"no free par" // 13
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};
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/*****************************************************************************/
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/* Monitoring auxiliaries. */
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/*****************************************************************************/
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void lm_print_pars(const int nout, const double *par, FILE* fout)
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{
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fprintf(fout, " pars:");
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for (int i = 0; i < nout; ++i)
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fprintf(fout, " %23.16g", par[i]);
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fprintf(fout, "\n");
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}
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/*****************************************************************************/
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/* lmmin (main minimization routine) */
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/*****************************************************************************/
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void lmmin(
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const int n, double *const x, const int m, const double* y,
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const void *const data,
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void (*const evaluate)(
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const double *const par, const int m_dat, const void *const data,
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double *const fvec, int *const userbreak),
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const lm_control_struct *const C, lm_status_struct *const S)
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{
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int j, i;
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double actred, dirder, fnorm, fnorm1, gnorm, pnorm,
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prered, ratio, step, sum, temp, temp1, temp2, temp3;
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static double p1 = 0.1, p0001 = 1.0e-4;
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int maxfev = C->patience * (n+1);
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int inner_success; /* flag for loop control */
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double lmpar = 0; /* Levenberg-Marquardt parameter */
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double delta = 0;
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double xnorm = 0;
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double eps = sqrt(MAX(C->epsilon, LM_MACHEP)); /* for forward differences */
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int nout = C->n_maxpri==-1 ? n : MIN(C->n_maxpri, n);
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/* The workaround msgfile=NULL is needed for default initialization */
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FILE* msgfile = C->msgfile ? C->msgfile : stdout;
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/* Default status info; must be set ahead of first return statements */
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S->outcome = 0; /* status code */
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S->userbreak = 0;
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S->nfev = 0; /* function evaluation counter */
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/*** Check input parameters for errors. ***/
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if ( n < 0 ) {
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fprintf(stderr, "lmmin: invalid number of parameters %i\n", n);
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S->outcome = 10; /* invalid parameter */
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return;
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}
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if (m < n) {
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fprintf(stderr, "lmmin: number of data points (%i) "
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"smaller than number of parameters (%i)\n", m, n);
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S->outcome = 10;
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return;
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}
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if (C->ftol < 0 || C->xtol < 0 || C->gtol < 0) {
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fprintf(stderr,
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"lmmin: negative tolerance (at least one of %g %g %g)\n",
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C->ftol, C->xtol, C->gtol);
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S->outcome = 10;
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return;
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}
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if (maxfev <= 0) {
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fprintf(stderr, "lmmin: nonpositive function evaluations limit %i\n",
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maxfev);
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S->outcome = 10;
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return;
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}
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if (C->stepbound <= 0) {
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fprintf(stderr, "lmmin: nonpositive stepbound %g\n", C->stepbound);
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S->outcome = 10;
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return;
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}
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if (C->scale_diag != 0 && C->scale_diag != 1) {
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fprintf(stderr, "lmmin: logical variable scale_diag=%i, "
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"should be 0 or 1\n", C->scale_diag);
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S->outcome = 10;
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return;
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}
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/*** Allocate work space. ***/
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/* Allocate total workspace with just one system call */
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char *ws;
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if ( ( ws = malloc(
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(2*m+5*n+m*n)*sizeof(double) + n*sizeof(int) ) ) == NULL ) {
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S->outcome = 9;
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return;
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}
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/* Assign workspace segments. */
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char *pws = ws;
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double *fvec = (double*) pws; pws += m * sizeof(double)/sizeof(char);
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double *diag = (double*) pws; pws += n * sizeof(double)/sizeof(char);
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double *qtf = (double*) pws; pws += n * sizeof(double)/sizeof(char);
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double *fjac = (double*) pws; pws += n*m*sizeof(double)/sizeof(char);
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double *wa1 = (double*) pws; pws += n * sizeof(double)/sizeof(char);
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double *wa2 = (double*) pws; pws += n * sizeof(double)/sizeof(char);
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double *wa3 = (double*) pws; pws += n * sizeof(double)/sizeof(char);
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double *wf = (double*) pws; pws += m * sizeof(double)/sizeof(char);
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int *ipvt = (int*) pws; pws += n * sizeof(int) /sizeof(char);
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/* Initialize diag */ // TODO: check whether this is still needed
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if (!C->scale_diag) {
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for (j = 0; j < n; j++)
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diag[j] = 1.;
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}
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/*** Evaluate function at starting point and calculate norm. ***/
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if( C->verbosity&1 )
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fprintf(msgfile, "lmmin start (ftol=%g gtol=%g xtol=%g)\n",
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C->ftol, C->gtol, C->xtol);
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if( C->verbosity&2 )
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lm_print_pars(nout, x, msgfile);
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(*evaluate)(x, m, data, fvec, &(S->userbreak));
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if( C->verbosity&8 ){
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if (y)
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for( i=0; i<m; ++i )
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fprintf(msgfile, " i, f, y-f: %4i %18.8g %18.8g\n",
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i, fvec[i], y[i]-fvec[i]);
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else
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for( i=0; i<m; ++i )
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fprintf(msgfile, " i, f: %4i %18.8g\n", i, fvec[i]);
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}
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S->nfev = 1;
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if ( S->userbreak )
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goto terminate;
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if ( n == 0 ) {
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S->outcome = 13; /* won't fit */
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goto terminate;
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}
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fnorm = lm_fnorm(m, fvec, y);
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if( C->verbosity&2 )
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fprintf(msgfile, " fnorm = %24.16g\n", fnorm);
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if( !isfinite(fnorm) ){
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if( C->verbosity )
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fprintf(msgfile, "nan case 1\n");
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S->outcome = 12; /* nan */
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goto terminate;
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} else if( fnorm <= LM_DWARF ){
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S->outcome = 0; /* sum of squares almost zero, nothing to do */
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goto terminate;
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}
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/*** The outer loop: compute gradient, then descend. ***/
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for( int outer=0; ; ++outer ) {
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/*** [outer] Calculate the Jacobian. ***/
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for (j = 0; j < n; j++) {
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temp = x[j];
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step = MAX(eps*eps, eps * fabs(temp));
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x[j] += step; /* replace temporarily */
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(*evaluate)(x, m, data, wf, &(S->userbreak));
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++(S->nfev);
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if ( S->userbreak )
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goto terminate;
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for (i = 0; i < m; i++)
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fjac[j*m+i] = (wf[i] - fvec[i]) / step;
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x[j] = temp; /* restore */
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}
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if ( C->verbosity&16 ) {
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/* print the entire matrix */
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printf("Jacobian\n");
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for (i = 0; i < m; i++) {
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printf(" ");
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for (j = 0; j < n; j++)
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printf("%.5e ", fjac[j*m+i]);
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printf("\n");
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}
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}
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/*** [outer] Compute the QR factorization of the Jacobian. ***/
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/* fjac is an m by n array. The upper n by n submatrix of fjac
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* is made to contain an upper triangular matrix R with diagonal
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* elements of nonincreasing magnitude such that
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*
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* P^T*(J^T*J)*P = R^T*R
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*
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* (NOTE: ^T stands for matrix transposition),
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*
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* where P is a permutation matrix and J is the final calculated
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* Jacobian. Column j of P is column ipvt(j) of the identity matrix.
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* The lower trapezoidal part of fjac contains information generated
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* during the computation of R.
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*
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* ipvt is an integer array of length n. It defines a permutation
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* matrix P such that jac*P = Q*R, where jac is the final calculated
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* Jacobian, Q is orthogonal (not stored), and R is upper triangular
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* with diagonal elements of nonincreasing magnitude. Column j of P
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* is column ipvt(j) of the identity matrix.
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*/
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lm_qrfac(m, n, fjac, ipvt, wa1, wa2, wa3);
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/* return values are ipvt, wa1=rdiag, wa2=acnorm */
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/*** [outer] Form Q^T * fvec, and store first n components in qtf. ***/
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if (y)
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for (i = 0; i < m; i++)
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wf[i] = fvec[i] - y[i];
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else
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for (i = 0; i < m; i++)
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wf[i] = fvec[i];
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for (j = 0; j < n; j++) {
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temp3 = fjac[j*m+j];
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if (temp3 != 0) {
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sum = 0;
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for (i = j; i < m; i++)
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sum += fjac[j*m+i] * wf[i];
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temp = -sum / temp3;
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for (i = j; i < m; i++)
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wf[i] += fjac[j*m+i] * temp;
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}
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fjac[j*m+j] = wa1[j];
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qtf[j] = wf[j];
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}
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/*** [outer] Compute norm of scaled gradient and detect degeneracy. ***/
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gnorm = 0;
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for (j = 0; j < n; j++) {
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if (wa2[ipvt[j]] == 0)
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continue;
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sum = 0;
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for (i = 0; i <= j; i++)
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sum += fjac[j*m+i] * qtf[i];
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gnorm = MAX(gnorm, fabs( sum / wa2[ipvt[j]] / fnorm ));
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}
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if (gnorm <= C->gtol) {
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S->outcome = 4;
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goto terminate;
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}
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/*** [outer] Initialize / update diag and delta. ***/
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if ( !outer ) {
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/* first iteration only */
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if (C->scale_diag) {
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/* diag := norms of the columns of the initial Jacobian */
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for (j = 0; j < n; j++)
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diag[j] = wa2[j] ? wa2[j] : 1;
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/* xnorm := || D x || */
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for (j = 0; j < n; j++)
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wa3[j] = diag[j] * x[j];
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xnorm = lm_enorm(n, wa3);
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} else {
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xnorm = lm_enorm(n, x);
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}
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if( !isfinite(xnorm) ){
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if( C->verbosity )
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fprintf(msgfile, "nan case 2\n");
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S->outcome = 12; /* nan */
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goto terminate;
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}
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/* initialize the step bound delta. */
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if ( xnorm )
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delta = C->stepbound * xnorm;
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else
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delta = C->stepbound;
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/* only now print the header for the loop table */
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if( C->verbosity&2 ) {
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fprintf(msgfile, " #o #i lmpar prered actred"
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" ratio dirder delta"
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" pnorm fnorm");
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for (i = 0; i < nout; ++i)
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fprintf(msgfile, " p%i", i);
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fprintf(msgfile, "\n");
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}
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} else {
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if (C->scale_diag) {
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for (j = 0; j < n; j++)
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diag[j] = MAX( diag[j], wa2[j] );
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}
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}
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|
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/*** The inner loop. ***/
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int inner = 0;
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do {
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|
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/*** [inner] Determine the Levenberg-Marquardt parameter. ***/
|
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|
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lm_lmpar(n, fjac, m, ipvt, diag, qtf, delta, &lmpar,
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wa1, wa2, wf, wa3);
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/* used return values are fjac (partly), lmpar, wa1=x, wa3=diag*x */
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|
|
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/* predict scaled reduction */
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pnorm = lm_enorm(n, wa3);
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if( !isfinite(pnorm) ){
|
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if( C->verbosity )
|
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fprintf(msgfile, "nan case 3\n");
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S->outcome = 12; /* nan */
|
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goto terminate;
|
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}
|
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temp2 = lmpar * SQR( pnorm / fnorm );
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for (j = 0; j < n; j++) {
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wa3[j] = 0;
|
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for (i = 0; i <= j; i++)
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wa3[i] -= fjac[j*m+i] * wa1[ipvt[j]];
|
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}
|
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temp1 = SQR( lm_enorm(n, wa3) / fnorm );
|
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if( !isfinite(temp1) ){
|
|
if( C->verbosity )
|
|
fprintf(msgfile, "nan case 4\n");
|
|
S->outcome = 12; /* nan */
|
|
goto terminate;
|
|
}
|
|
prered = temp1 + 2 * temp2;
|
|
dirder = -temp1 + temp2; /* scaled directional derivative */
|
|
|
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/* at first call, adjust the initial step bound. */
|
|
if ( !outer && !inner && pnorm < delta )
|
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delta = pnorm;
|
|
|
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/*** [inner] Evaluate the function at x + p. ***/
|
|
|
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for (j = 0; j < n; j++)
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wa2[j] = x[j] - wa1[j];
|
|
|
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(*evaluate)( wa2, m, data, wf, &(S->userbreak) );
|
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++(S->nfev);
|
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if ( S->userbreak )
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goto terminate;
|
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fnorm1 = lm_fnorm(m, wf, y);
|
|
// exceptionally, for this norm we do not test for infinity
|
|
// because we can deal with it without terminating.
|
|
|
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/*** [inner] Evaluate the scaled reduction. ***/
|
|
|
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/* actual scaled reduction (supports even the case fnorm1=infty) */
|
|
if (p1 * fnorm1 < fnorm)
|
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actred = 1 - SQR(fnorm1 / fnorm);
|
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else
|
|
actred = -1;
|
|
|
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/* ratio of actual to predicted reduction */
|
|
ratio = prered ? actred/prered : 0;
|
|
|
|
if( C->verbosity&32 ){
|
|
if (y)
|
|
for( i=0; i<m; ++i )
|
|
fprintf(msgfile, " i, f, y-f: %4i %18.8g %18.8g\n",
|
|
i, fvec[i], y[i]-fvec[i]);
|
|
else
|
|
for( i=0; i<m; ++i )
|
|
fprintf(msgfile, " i, f, y-f: %4i %18.8g\n",
|
|
i, fvec[i]);
|
|
}
|
|
if( C->verbosity&2 ) {
|
|
printf("%3i %2i %9.2g %9.2g %9.2g %14.6g"
|
|
" %9.2g %10.3e %10.3e %21.15e",
|
|
outer, inner, lmpar, prered, actred, ratio,
|
|
dirder, delta, pnorm, fnorm1);
|
|
for (i = 0; i < nout; ++i)
|
|
fprintf(msgfile, " %16.9g", wa2[i]);
|
|
fprintf(msgfile, "\n");
|
|
}
|
|
|
|
/* update the step bound */
|
|
if (ratio <= 0.25) {
|
|
if (actred >= 0)
|
|
temp = 0.5;
|
|
else
|
|
temp = 0.5 * dirder / (dirder + 0.5 * actred);
|
|
if (p1 * fnorm1 >= fnorm || temp < p1)
|
|
temp = p1;
|
|
delta = temp * MIN(delta, pnorm / p1);
|
|
lmpar /= temp;
|
|
} else if (lmpar == 0 || ratio >= 0.75) {
|
|
delta = 2 * pnorm;
|
|
lmpar *= 0.5;
|
|
}
|
|
|
|
/*** [inner] On success, update solution, and test for convergence. ***/
|
|
|
|
inner_success = ratio >= p0001;
|
|
if ( inner_success ) {
|
|
|
|
/* update x, fvec, and their norms */
|
|
if (C->scale_diag) {
|
|
for (j = 0; j < n; j++) {
|
|
x[j] = wa2[j];
|
|
wa2[j] = diag[j] * x[j];
|
|
}
|
|
} else {
|
|
for (j = 0; j < n; j++)
|
|
x[j] = wa2[j];
|
|
}
|
|
for (i = 0; i < m; i++)
|
|
fvec[i] = wf[i];
|
|
xnorm = lm_enorm(n, wa2);
|
|
if( !isfinite(xnorm) ){
|
|
if( C->verbosity )
|
|
fprintf(msgfile, "nan case 6\n");
|
|
S->outcome = 12; /* nan */
|
|
goto terminate;
|
|
}
|
|
fnorm = fnorm1;
|
|
}
|
|
|
|
/* convergence tests */
|
|
S->outcome = 0;
|
|
if( fnorm<=LM_DWARF )
|
|
goto terminate; /* success: sum of squares almost zero */
|
|
/* test two criteria (both may be fulfilled) */
|
|
if (fabs(actred) <= C->ftol && prered <= C->ftol && ratio <= 2)
|
|
S->outcome = 1; /* success: x almost stable */
|
|
if (delta <= C->xtol * xnorm)
|
|
S->outcome += 2; /* success: sum of squares almost stable */
|
|
if (S->outcome != 0) {
|
|
goto terminate;
|
|
}
|
|
|
|
/*** [inner] Tests for termination and stringent tolerances. ***/
|
|
|
|
if ( S->nfev >= maxfev ){
|
|
S->outcome = 5;
|
|
goto terminate;
|
|
}
|
|
if ( fabs(actred) <= LM_MACHEP &&
|
|
prered <= LM_MACHEP && ratio <= 2 ){
|
|
S->outcome = 6;
|
|
goto terminate;
|
|
}
|
|
if ( delta <= LM_MACHEP*xnorm ){
|
|
S->outcome = 7;
|
|
goto terminate;
|
|
}
|
|
if ( gnorm <= LM_MACHEP ){
|
|
S->outcome = 8;
|
|
goto terminate;
|
|
}
|
|
|
|
/*** [inner] End of the loop. Repeat if iteration unsuccessful. ***/
|
|
|
|
++inner;
|
|
} while ( !inner_success );
|
|
|
|
/*** [outer] End of the loop. ***/
|
|
|
|
};
|
|
|
|
terminate:
|
|
S->fnorm = lm_fnorm(m, fvec, y);
|
|
if( C->verbosity&1 )
|
|
fprintf(msgfile, "lmmin terminates with outcome %i\n", S->outcome);
|
|
if( C->verbosity&2 )
|
|
lm_print_pars(nout, x, msgfile);
|
|
if( C->verbosity&8 ){
|
|
if (y)
|
|
for( i=0; i<m; ++i )
|
|
fprintf(msgfile, " i, f, y-f: %4i %18.8g %18.8g\n",
|
|
i, fvec[i], y[i]-fvec[i] );
|
|
else
|
|
for( i=0; i<m; ++i )
|
|
fprintf(msgfile, " i, f, y-f: %4i %18.8g\n", i, fvec[i]);
|
|
}
|
|
if( C->verbosity&2 )
|
|
fprintf(msgfile, " fnorm=%24.16g xnorm=%24.16g\n", S->fnorm, xnorm);
|
|
if ( S->userbreak ) /* user-requested break */
|
|
S->outcome = 11;
|
|
|
|
/*** Deallocate the workspace. ***/
|
|
free(ws);
|
|
|
|
} /*** lmmin. ***/
|
|
|
|
|
|
/*****************************************************************************/
|
|
/* lm_lmpar (determine Levenberg-Marquardt parameter) */
|
|
/*****************************************************************************/
|
|
|
|
void lm_lmpar(
|
|
const int n, double *const r, const int ldr, int *const ipvt,
|
|
double *const diag, double *const qtb, double delta, double *const par,
|
|
double *const x, double *const sdiag, double *const aux, double *const xdi)
|
|
{
|
|
/* 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 parameter value par such that if x solves the system
|
|
*
|
|
* A*x = b and sqrt(par)*D*x = 0
|
|
*
|
|
* in the least squares sense, and dxnorm is the euclidean
|
|
* norm of D*x, then either par=0 and (dxnorm-delta) < 0.1*delta,
|
|
* or par>0 and abs(dxnorm-delta) < 0.1*delta.
|
|
*
|
|
* Using lm_qrsolv, 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^T*b. On output lmpar also provides an
|
|
* upper triangular matrix S such that
|
|
*
|
|
* P^T*(A^T*A + par*D*D)*P = S^T*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.
|
|
*
|
|
* Parameters:
|
|
*
|
|
* 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^T*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 array of length n needed as workspace; on OUTPUT
|
|
* it contains the diagonal elements of the upper triangular
|
|
* matrix S.
|
|
*
|
|
* aux is a multi-purpose work array of length n.
|
|
*
|
|
* xdi is a work array of length n. On OUTPUT: diag[j] * x[j].
|
|
*
|
|
*/
|
|
int i, iter, j, nsing;
|
|
double dxnorm, fp, fp_old, gnorm, parc, parl, paru;
|
|
double sum, temp;
|
|
static double p1 = 0.1;
|
|
|
|
/*** lmpar: 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++) {
|
|
aux[j] = qtb[j];
|
|
if (r[j * ldr + j] == 0 && nsing == n)
|
|
nsing = j;
|
|
if (nsing < n)
|
|
aux[j] = 0;
|
|
}
|
|
for (j = nsing - 1; j >= 0; j--) {
|
|
aux[j] = aux[j] / r[j + ldr * j];
|
|
temp = aux[j];
|
|
for (i = 0; i < j; i++)
|
|
aux[i] -= r[j * ldr + i] * temp;
|
|
}
|
|
|
|
for (j = 0; j < n; j++)
|
|
x[ipvt[j]] = aux[j];
|
|
|
|
/*** lmpar: initialize the iteration counter, evaluate the function at the
|
|
origin, and test for acceptance of the gauss-newton direction. ***/
|
|
|
|
for (j = 0; j < n; j++)
|
|
xdi[j] = diag[j] * x[j];
|
|
dxnorm = lm_enorm(n, xdi);
|
|
fp = dxnorm - delta;
|
|
if (fp <= p1 * delta) {
|
|
#ifdef LMFIT_DEBUG_MESSAGES
|
|
printf("debug lmpar nsing %d n %d, terminate (fp<p1*delta)\n",
|
|
nsing, n);
|
|
#endif
|
|
*par = 0;
|
|
return;
|
|
}
|
|
|
|
/*** lmpar: 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) {
|
|
for (j = 0; j < n; j++)
|
|
aux[j] = diag[ipvt[j]] * xdi[ipvt[j]] / dxnorm;
|
|
|
|
for (j = 0; j < n; j++) {
|
|
sum = 0;
|
|
for (i = 0; i < j; i++)
|
|
sum += r[j * ldr + i] * aux[i];
|
|
aux[j] = (aux[j] - sum) / r[j + ldr * j];
|
|
}
|
|
temp = lm_enorm(n, aux);
|
|
parl = fp / delta / temp / temp;
|
|
}
|
|
|
|
/*** lmpar: 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[j * ldr + i] * qtb[i];
|
|
aux[j] = sum / diag[ipvt[j]];
|
|
}
|
|
gnorm = lm_enorm(n, aux);
|
|
paru = gnorm / delta;
|
|
if (paru == 0)
|
|
paru = LM_DWARF / MIN(delta, p1);
|
|
|
|
/*** lmpar: 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;
|
|
|
|
/*** lmpar: iterate. ***/
|
|
|
|
for (iter=0; ; iter++) {
|
|
|
|
/** evaluate the function at the current value of par. **/
|
|
|
|
if (*par == 0)
|
|
*par = MAX(LM_DWARF, 0.001 * paru);
|
|
temp = sqrt(*par);
|
|
for (j = 0; j < n; j++)
|
|
aux[j] = temp * diag[j];
|
|
|
|
lm_qrsolv(n, r, ldr, ipvt, aux, qtb, x, sdiag, xdi);
|
|
/* return values are r, x, sdiag */
|
|
|
|
for (j = 0; j < n; j++)
|
|
xdi[j] = diag[j] * x[j]; /* used as output */
|
|
dxnorm = lm_enorm(n, xdi);
|
|
fp_old = 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 (fabs(fp) <= p1 * delta
|
|
|| (parl == 0 && fp <= fp_old && fp_old < 0)
|
|
|| iter == 10) {
|
|
#ifdef LMFIT_DEBUG_MESSAGES
|
|
printf("debug lmpar nsing %d iter %d "
|
|
"par %.4e [%.4e %.4e] delta %.4e fp %.4e\n",
|
|
nsing, iter, *par, parl, paru, delta, fp);
|
|
#endif
|
|
break; /* the only exit from the iteration. */
|
|
}
|
|
|
|
/** compute the Newton correction. **/
|
|
|
|
for (j = 0; j < n; j++)
|
|
aux[j] = diag[ipvt[j]] * xdi[ipvt[j]] / dxnorm;
|
|
|
|
for (j = 0; j < n; j++) {
|
|
aux[j] = aux[j] / sdiag[j];
|
|
for (i = j + 1; i < n; i++)
|
|
aux[i] -= r[j * ldr + i] * aux[j];
|
|
}
|
|
temp = lm_enorm(n, aux);
|
|
parc = fp / delta / temp / temp;
|
|
|
|
/** depending on the sign of the function, update parl or paru. **/
|
|
|
|
if (fp > 0)
|
|
parl = MAX(parl, *par);
|
|
else if (fp < 0)
|
|
paru = MIN(paru, *par);
|
|
/* the case fp==0 is precluded by the break condition */
|
|
|
|
/** compute an improved estimate for par. **/
|
|
|
|
*par = MAX(parl, *par + parc);
|
|
|
|
}
|
|
|
|
} /*** lm_lmpar. ***/
|
|
|
|
/*****************************************************************************/
|
|
/* lm_qrfac (QR factorization, from lapack) */
|
|
/*****************************************************************************/
|
|
|
|
void lm_qrfac(
|
|
const int m, const int n, double *const A, int *const Pivot,
|
|
double *const Rdiag, double *const Acnorm, double *const W)
|
|
{
|
|
/*
|
|
* This subroutine uses Householder transformations with column pivoting
|
|
* to compute a QR factorization of the m by n matrix A. That is, qrfac
|
|
* determines an orthogonal matrix Q, a permutation matrix P, and an
|
|
* upper trapezoidal matrix R with diagonal elements of nonincreasing
|
|
* magnitude, such that A*P = Q*R. The Householder transformation for
|
|
* column k, k = 1,2,...,n, is of the form
|
|
*
|
|
* I - 2*w*wT/|w|^2
|
|
*
|
|
* where w has zeroes in the first k-1 positions.
|
|
*
|
|
* Parameters:
|
|
*
|
|
* m is an INPUT parameter set to the number of rows of A.
|
|
*
|
|
* n is an INPUT parameter set to the number of columns of A.
|
|
*
|
|
* A is an m by n array. On INPUT, A contains the matrix for
|
|
* which the QR factorization is to be computed. On OUTPUT
|
|
* the strict upper trapezoidal part of A contains the strict
|
|
* upper trapezoidal part of R, and the lower trapezoidal
|
|
* part of A contains a factored form of Q (the non-trivial
|
|
* elements of the vectors w described above).
|
|
*
|
|
* Pivot is an integer OUTPUT array of length n that describes the
|
|
* permutation matrix P:
|
|
* Column j of P is column ipvt(j) of the identity matrix.
|
|
*
|
|
* Rdiag is an OUTPUT array of length n which contains the
|
|
* diagonal elements of R.
|
|
*
|
|
* Acnorm is an OUTPUT array of length n which contains the norms
|
|
* of the corresponding columns of the input matrix A. If this
|
|
* information is not needed, then Acnorm can share storage with Rdiag.
|
|
*
|
|
* W is a work array of length n.
|
|
*
|
|
*/
|
|
int i, j, k, kmax;
|
|
double ajnorm, sum, temp;
|
|
|
|
#ifdef LMFIT_DEBUG_MESSAGES
|
|
printf("debug qrfac\n");
|
|
#endif
|
|
|
|
/** Compute initial column norms;
|
|
initialize Pivot with identity permutation. ***/
|
|
|
|
for (j = 0; j < n; j++) {
|
|
W[j] = Rdiag[j] = Acnorm[j] = lm_enorm(m, &A[j*m]);
|
|
Pivot[j] = j;
|
|
}
|
|
|
|
/** Loop over columns of A. **/
|
|
|
|
assert( n <= m );
|
|
for (j = 0; j < n; j++) {
|
|
|
|
/** Bring the column of largest norm into the pivot position. **/
|
|
|
|
kmax = j;
|
|
for (k = j+1; k < n; k++)
|
|
if (Rdiag[k] > Rdiag[kmax])
|
|
kmax = k;
|
|
|
|
if (kmax != j) {
|
|
/* Swap columns j and kmax. */
|
|
k = Pivot[j];
|
|
Pivot[j] = Pivot[kmax];
|
|
Pivot[kmax] = k;
|
|
for (i = 0; i < m; i++) {
|
|
temp = A[j*m+i];
|
|
A[j*m+i] = A[kmax*m+i];
|
|
A[kmax*m+i] = temp;
|
|
}
|
|
/* Half-swap: Rdiag[j], W[j] won't be needed any further. */
|
|
Rdiag[kmax] = Rdiag[j];
|
|
W[kmax] = W[j];
|
|
}
|
|
|
|
/** Compute the Householder reflection vector w_j to reduce the
|
|
j-th column of A to a multiple of the j-th unit vector. **/
|
|
|
|
ajnorm = lm_enorm(m-j, &A[j*m+j]);
|
|
if (ajnorm == 0) {
|
|
Rdiag[j] = 0;
|
|
continue;
|
|
}
|
|
|
|
/* Let the partial column vector A[j][j:] contain w_j := e_j+-a_j/|a_j|,
|
|
where the sign +- is chosen to avoid cancellation in w_jj. */
|
|
if (A[j*m+j] < 0)
|
|
ajnorm = -ajnorm;
|
|
for (i = j; i < m; i++)
|
|
A[j*m+i] /= ajnorm;
|
|
A[j*m+j] += 1;
|
|
|
|
/** Apply the Householder transformation U_w := 1 - 2*w_j.w_j/|w_j|^2
|
|
to the remaining columns, and update the norms. **/
|
|
|
|
for (k = j+1; k < n; k++) {
|
|
/* Compute scalar product w_j * a_j. */
|
|
sum = 0;
|
|
for (i = j; i < m; i++)
|
|
sum += A[j*m+i] * A[k*m+i];
|
|
|
|
/* Normalization is simplified by the coincidence |w_j|^2=2w_jj. */
|
|
temp = sum / A[j*m+j];
|
|
|
|
/* Carry out transform U_w_j * a_k. */
|
|
for (i = j; i < m; i++)
|
|
A[k*m+i] -= temp * A[j*m+i];
|
|
|
|
/* No idea what happens here. */
|
|
if (Rdiag[k] != 0) {
|
|
temp = A[m*k+j] / Rdiag[k];
|
|
if ( fabs(temp)<1 ) {
|
|
Rdiag[k] *= sqrt(1-SQR(temp));
|
|
temp = Rdiag[k] / W[k];
|
|
} else
|
|
temp = 0;
|
|
if ( temp == 0 || 0.05 * SQR(temp) <= LM_MACHEP ) {
|
|
Rdiag[k] = lm_enorm(m-j-1, &A[m*k+j+1]);
|
|
W[k] = Rdiag[k];
|
|
}
|
|
}
|
|
}
|
|
|
|
Rdiag[j] = -ajnorm;
|
|
}
|
|
} /*** lm_qrfac. ***/
|
|
|
|
|
|
/*****************************************************************************/
|
|
/* lm_qrsolv (linear least-squares) */
|
|
/*****************************************************************************/
|
|
|
|
void lm_qrsolv(
|
|
const int n, double *const r, const int ldr, int *const ipvt,
|
|
double *const diag, double *const qtb, double *const x,
|
|
double *const sdiag, double *const wa)
|
|
{
|
|
/*
|
|
* Given an m by n matrix A, an n by n diagonal matrix D, and an
|
|
* m-vector b, the problem is to determine an x which solves the
|
|
* system
|
|
*
|
|
* A*x = b and D*x = 0
|
|
*
|
|
* in the least squares sense.
|
|
*
|
|
* 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 qrsolv expects the full upper triangle of R, the permutation
|
|
* matrix P, and the first n components of Q^T*b. The system
|
|
* A*x = b, D*x = 0, is then equivalent to
|
|
*
|
|
* R*z = Q^T*b, P^T*D*P*z = 0,
|
|
*
|
|
* where x = P*z. If this system does not have full rank, then a least
|
|
* squares solution is obtained. On output qrsolv also provides an upper
|
|
* triangular matrix S such that
|
|
*
|
|
* P^T*(A^T*A + D*D)*P = S^T*S.
|
|
*
|
|
* S is computed within qrsolv and may be of separate interest.
|
|
*
|
|
* Parameters:
|
|
*
|
|
* 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^T*b.
|
|
*
|
|
* x is an OUTPUT array of length n which contains the least
|
|
* squares solution of the system A*x = b, D*x = 0.
|
|
*
|
|
* sdiag is an OUTPUT array of length n which contains the
|
|
* diagonal elements of the upper triangular matrix S.
|
|
*
|
|
* wa is a work array of length n.
|
|
*
|
|
*/
|
|
int i, kk, j, k, nsing;
|
|
double qtbpj, sum, temp;
|
|
double _sin, _cos, _tan, _cot; /* local variables, not functions */
|
|
|
|
/*** qrsolv: copy R and Q^T*b to preserve input and initialize S.
|
|
In particular, save the diagonal elements of R in x. ***/
|
|
|
|
for (j = 0; j < n; j++) {
|
|
for (i = j; i < n; i++)
|
|
r[j * ldr + i] = r[i * ldr + j];
|
|
x[j] = r[j * ldr + j];
|
|
wa[j] = qtb[j];
|
|
}
|
|
|
|
/*** qrsolv: eliminate the diagonal matrix D using a Givens rotation. ***/
|
|
|
|
for (j = 0; j < n; j++) {
|
|
|
|
/*** qrsolv: prepare the row of D to be eliminated, locating the
|
|
diagonal element using P from the QR factorization. ***/
|
|
|
|
if (diag[ipvt[j]] == 0)
|
|
goto L90;
|
|
for (k = j; k < n; k++)
|
|
sdiag[k] = 0;
|
|
sdiag[j] = diag[ipvt[j]];
|
|
|
|
/*** qrsolv: the transformations to eliminate the row of D modify only
|
|
a single element of Q^T*b beyond the first n, which is initially 0. ***/
|
|
|
|
qtbpj = 0;
|
|
for (k = j; k < n; k++) {
|
|
|
|
/** determine a Givens rotation which eliminates the
|
|
appropriate element in the current row of D. **/
|
|
|
|
if (sdiag[k] == 0)
|
|
continue;
|
|
kk = k + ldr * k;
|
|
if (fabs(r[kk]) < fabs(sdiag[k])) {
|
|
_cot = r[kk] / sdiag[k];
|
|
_sin = 1 / sqrt(1 + SQR(_cot));
|
|
_cos = _sin * _cot;
|
|
} else {
|
|
_tan = sdiag[k] / r[kk];
|
|
_cos = 1 / sqrt(1 + SQR(_tan));
|
|
_sin = _cos * _tan;
|
|
}
|
|
|
|
/** compute the modified diagonal element of R and
|
|
the modified element of (Q^T*b,0). **/
|
|
|
|
r[kk] = _cos * r[kk] + _sin * sdiag[k];
|
|
temp = _cos * wa[k] + _sin * qtbpj;
|
|
qtbpj = -_sin * wa[k] + _cos * qtbpj;
|
|
wa[k] = temp;
|
|
|
|
/** accumulate the tranformation in the row of S. **/
|
|
|
|
for (i = k + 1; i < n; i++) {
|
|
temp = _cos * r[k * ldr + i] + _sin * sdiag[i];
|
|
sdiag[i] = -_sin * r[k * ldr + i] + _cos * sdiag[i];
|
|
r[k * ldr + i] = temp;
|
|
}
|
|
}
|
|
|
|
L90:
|
|
/** store the diagonal element of S and restore
|
|
the corresponding diagonal element of R. **/
|
|
|
|
sdiag[j] = r[j * ldr + j];
|
|
r[j * ldr + j] = x[j];
|
|
}
|
|
|
|
/*** qrsolv: solve the triangular system for z. If the system is
|
|
singular, then obtain a least squares solution. ***/
|
|
|
|
nsing = n;
|
|
for (j = 0; j < n; j++) {
|
|
if (sdiag[j] == 0 && nsing == n)
|
|
nsing = j;
|
|
if (nsing < n)
|
|
wa[j] = 0;
|
|
}
|
|
|
|
for (j = nsing - 1; j >= 0; j--) {
|
|
sum = 0;
|
|
for (i = j + 1; i < nsing; i++)
|
|
sum += r[j * ldr + i] * wa[i];
|
|
wa[j] = (wa[j] - sum) / sdiag[j];
|
|
}
|
|
|
|
/*** qrsolv: permute the components of z back to components of x. ***/
|
|
|
|
for (j = 0; j < n; j++)
|
|
x[ipvt[j]] = wa[j];
|
|
|
|
} /*** lm_qrsolv. ***/
|
|
|
|
|
|
/*****************************************************************************/
|
|
/* lm_enorm (Euclidean norm) */
|
|
/*****************************************************************************/
|
|
|
|
double lm_enorm(const int n, const double *const x)
|
|
{
|
|
/* This function calculates the Euclidean norm of an n-vector x.
|
|
*
|
|
* The Euclidean norm is computed by accumulating the sum of
|
|
* squares in three different sums. The sums of squares for the
|
|
* small and large components are scaled so that no overflows
|
|
* occur. Non-destructive underflows are permitted. Underflows
|
|
* and overflows do not occur in the computation of the unscaled
|
|
* sum of squares for the intermediate components.
|
|
* The definitions of small, intermediate and large components
|
|
* depend on two constants, LM_SQRT_DWARF and LM_SQRT_GIANT. The main
|
|
* restrictions on these constants are that LM_SQRT_DWARF**2 not
|
|
* underflow and LM_SQRT_GIANT**2 not overflow.
|
|
*
|
|
* Parameters:
|
|
*
|
|
* n is a positive integer INPUT variable.
|
|
*
|
|
* x is an INPUT array of length n.
|
|
*/
|
|
int i;
|
|
double agiant, s1, s2, s3, xabs, x1max, x3max, temp;
|
|
|
|
s1 = 0;
|
|
s2 = 0;
|
|
s3 = 0;
|
|
x1max = 0;
|
|
x3max = 0;
|
|
agiant = LM_SQRT_GIANT / n;
|
|
|
|
/** sum squares. **/
|
|
|
|
for (i = 0; i < n; i++) {
|
|
xabs = fabs(x[i]);
|
|
if (xabs > LM_SQRT_DWARF) {
|
|
if ( xabs < agiant ) {
|
|
s2 += xabs * xabs;
|
|
} else if ( xabs > x1max ) {
|
|
temp = x1max / xabs;
|
|
s1 = 1 + s1 * SQR(temp);
|
|
x1max = xabs;
|
|
} else {
|
|
temp = xabs / x1max;
|
|
s1 += SQR(temp);
|
|
}
|
|
} else if ( xabs > x3max ) {
|
|
temp = x3max / xabs;
|
|
s3 = 1 + s3 * SQR(temp);
|
|
x3max = xabs;
|
|
} else if (xabs != 0) {
|
|
temp = xabs / x3max;
|
|
s3 += SQR(temp);
|
|
}
|
|
}
|
|
|
|
/** calculation of norm. **/
|
|
|
|
if (s1 != 0)
|
|
return x1max * sqrt(s1 + (s2 / x1max) / x1max);
|
|
else if (s2 != 0) {
|
|
if (s2 >= x3max)
|
|
return sqrt(s2 * (1 + (x3max / s2) * (x3max * s3)));
|
|
else
|
|
return sqrt(x3max * ((s2 / x3max) + (x3max * s3)));
|
|
} else
|
|
return x3max * sqrt(s3);
|
|
|
|
} /*** lm_enorm. ***/
|
|
|
|
|
|
/*****************************************************************************/
|
|
/* lm_fnorm (Euclidean norm of difference) */
|
|
/*****************************************************************************/
|
|
|
|
double lm_fnorm(const int n, const double *const x, const double *const y)
|
|
{
|
|
/* This function calculates the Euclidean norm of an n-vector x-y.
|
|
*
|
|
* The Euclidean norm is computed by accumulating the sum of
|
|
* squares in three different sums. The sums of squares for the
|
|
* small and large components are scaled so that no overflows
|
|
* occur. Non-destructive underflows are permitted. Underflows
|
|
* and overflows do not occur in the computation of the unscaled
|
|
* sum of squares for the intermediate components.
|
|
* The definitions of small, intermediate and large components
|
|
* depend on two constants, LM_SQRT_DWARF and LM_SQRT_GIANT. The main
|
|
* restrictions on these constants are that LM_SQRT_DWARF**2 not
|
|
* underflow and LM_SQRT_GIANT**2 not overflow.
|
|
*
|
|
* Parameters:
|
|
*
|
|
* n is a positive integer INPUT variable.
|
|
*
|
|
* x, y are INPUT arrays of length n.
|
|
*/
|
|
if (!y)
|
|
return lm_enorm(n, x);
|
|
int i;
|
|
double agiant, s1, s2, s3, xabs, x1max, x3max, temp;
|
|
|
|
s1 = 0;
|
|
s2 = 0;
|
|
s3 = 0;
|
|
x1max = 0;
|
|
x3max = 0;
|
|
agiant = LM_SQRT_GIANT / n;
|
|
|
|
/** sum squares. **/
|
|
|
|
for (i = 0; i < n; i++) {
|
|
xabs = fabs(x[i]-y[i]);
|
|
if (xabs > LM_SQRT_DWARF) {
|
|
if ( xabs < agiant ) {
|
|
s2 += xabs * xabs;
|
|
} else if ( xabs > x1max ) {
|
|
temp = x1max / xabs;
|
|
s1 = 1 + s1 * SQR(temp);
|
|
x1max = xabs;
|
|
} else {
|
|
temp = xabs / x1max;
|
|
s1 += SQR(temp);
|
|
}
|
|
} else if ( xabs > x3max ) {
|
|
temp = x3max / xabs;
|
|
s3 = 1 + s3 * SQR(temp);
|
|
x3max = xabs;
|
|
} else if (xabs != 0) {
|
|
temp = xabs / x3max;
|
|
s3 += SQR(temp);
|
|
}
|
|
}
|
|
|
|
/** calculation of norm. **/
|
|
|
|
if (s1 != 0)
|
|
return x1max * sqrt(s1 + (s2 / x1max) / x1max);
|
|
else if (s2 != 0) {
|
|
if (s2 >= x3max)
|
|
return sqrt(s2 * (1 + (x3max / s2) * (x3max * s3)));
|
|
else
|
|
return sqrt(x3max * ((s2 / x3max) + (x3max * s3)));
|
|
} else
|
|
return x3max * sqrt(s3);
|
|
|
|
} /*** lm_fnorm. ***/
|