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lmfit.c
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#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include <float.h>
#include "lmfit.h"
#define MIN(a,b) (((a)<=(b)) ? (a) : (b))
#define MAX(a,b) (((a)>=(b)) ? (a) : (b))
#define SQR(x) (x)*(x)
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 );
void lm_qrfac(const int m, const int n, double *const a, int *const ipvt,
double *const rdiag, double *const acnorm, double *const wa );
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 );
/* Numeric constants */
/* machine-dependent constants from float.h */
#define LM_MACHEP DBL_EPSILON /* resolution of arithmetic */
#define LM_DWARF DBL_MIN /* smallest nonzero number */
#define LM_SQRT_DWARF sqrt(DBL_MIN) /* square should not underflow */
#define LM_SQRT_GIANT sqrt(DBL_MAX) /* square should not overflow */
#define LM_USERTOL 30*LM_MACHEP /* users are recommended to require this */
/* If the above values do not work, the following seem good for an x86:
LM_MACHEP .555e-16
LM_DWARF 9.9e-324
LM_SQRT_DWARF 1.e-160
LM_SQRT_GIANT 1.e150
LM_USER_TOL 1.e-14
The following values should work on any machine:
LM_MACHEP 1.2e-16
LM_DWARF 1.0e-38
LM_SQRT_DWARF 3.834e-20
LM_SQRT_GIANT 1.304e19
LM_USER_TOL 1.e-14
*/
const lm_control_struct lm_control_double = {
LM_USERTOL, LM_USERTOL, LM_USERTOL, LM_USERTOL, 100., 100, 1,
NULL, 0, -1, -1
};
const lm_control_struct lm_control_float = {
1.e-7, 1.e-7, 1.e-7, 1.e-7, 100., 100, 1,
NULL, 0, -1, -1
};
/* Message texts (indexed by status.info) */
const char *lm_infmsg[] = {
"found zero (sum of squares below underflow limit)",
"converged (the relative error in the sum of squares is at most tol)",
"converged (the relative error of the parameter vector is at most tol)",
"converged (both errors are at most tol)",
"trapped (by degeneracy; increasing epsilon might help)",
"exhausted (number of function calls exceeding preset patience)",
"failed (ftol<tol: cannot reduce sum of squares any further)",
"failed (xtol<tol: cannot improve approximate solution any further)",
"failed (gtol<tol: cannot improve approximate solution any further)",
"crashed (not enough memory)",
"exploded (fatal coding error: improper input parameters)",
"stopped (break requested within function evaluation)",
"found nan (function value is not-a-number or infinite)"
};
const char *lm_shortmsg[] = {
"found zero",
"converged (f)",
"converged (p)",
"converged (2)",
"degenerate",
"call limit",
"failed (f)",
"failed (p)",
"failed (o)",
"no memory",
"invalid input",
"user break",
"found nan"
};
void lm_print_pars(const int nout, const double *par, FILE* fout)
{
for (int i = 0; i < nout; ++i)
fprintf( fout, " %16.9g", par[i] );
fprintf( fout, "\n" );
}
/**
* Main minimization routine
*/
void lmmin(const int n, double* x, const int m, const void* data,
void (*evaluate)(const double* par, const int m_dat, const void* data, double* fvec, int* userbreak),
const lm_control_struct* C, lm_status_struct* S)
{
int j, i;
double actred, dirder, fnorm, fnorm1, gnorm, pnorm, prered, ratio, sum, temp, temp1, temp2, temp3;
static double p0001 = 1.0e-4;
int maxfev = C->patience * (n + 1);
int inner_success; /* flag for loop control */
double lmpar = 0; /* Levenberg-Marquardt parameter */
double delta = 0;
double xnorm = 0;
double eps = sqrt(MAX(C->epsilon, LM_MACHEP)); /* for forward differences */
int nout = C->n_maxpri == -1 ? n : MIN( C->n_maxpri, n );
/* The workaround msgfile=NULL is needed for default initialization */
FILE* msgfile = (FILE*)C->msgfile ? C->msgfile : stdout;
/* Default status info; must be set ahead of first return statements */
S->outcome = 0; /* status code */
S->userbreak = 0;
S->nfev = 0; /* function evaluation counter */
/* Check input parameters for errors. */
if ( n <= 0 ) {
fprintf( stderr, "lmmin: invalid number of parameters %i\n", n );
S->outcome = 10; /* invalid parameter */
return;
}
if (m < n) {
fprintf( stderr, "lmmin: number of data points (%i) "
"smaller than number of parameters (%i)\n", m, n );
S->outcome = 10;
return;
}
if (C->ftol < 0 || C->xtol < 0 || C->gtol < 0) {
fprintf( stderr, "lmmin: negative tolerance (at least one of %g %g %g)\n",
C->ftol, C->xtol, C->gtol );
S->outcome = 10;
return;
}
if (maxfev <= 0) {
fprintf( stderr, "lmmin: nonpositive function evaluations limit %i\n",
maxfev );
S->outcome = 10;
return;
}
if (C->stepbound <= 0) {
fprintf( stderr, "lmmin: nonpositive stepbound %g\n", C->stepbound );
S->outcome = 10;
return;
}
if (C->scale_diag != 0 && C->scale_diag != 1) {
fprintf( stderr, "lmmin: logical variable scale_diag=%i, should be 0 or 1\n", C->scale_diag );
S->outcome = 10;
return;
}
/* Allocate work space. */
/* Allocate total workspace with just one system call */
char *ws;
if ( ( ws = malloc((2 * m + 5 * n + m * n) * sizeof(double) + n * sizeof(int) ) ) == NULL ) {
S->outcome = 9;
return;
}
/*
where to store the params for all p + step
*/
double* par2 = (double*)malloc(n * n * sizeof(double));
double* steps = (double*)malloc(n * sizeof(double));
double* wfs = (double*)malloc(m * n * sizeof(double));
/* Assign workspace segments. */
char *pws = ws;
double *fvec = (double*) pws;
pws += m * sizeof(double) / sizeof(char);
double *diag = (double*) pws;
pws += n * sizeof(double) / sizeof(char);
double *qtf = (double*) pws;
pws += n * sizeof(double) / sizeof(char);
double *fjac = (double*) pws;
pws += n * m * sizeof(double) / sizeof(char);
double *wa1 = (double*) pws;
pws += n * sizeof(double) / sizeof(char);
double *wa2 = (double*) pws;
pws += n * sizeof(double) / sizeof(char);
double *wa3 = (double*) pws;
pws += n * sizeof(double) / sizeof(char);
double *wf = (double*) pws;
pws += m * sizeof(double) / sizeof(char);
int *ipvt = (int*) pws;
pws += n * sizeof(int) / sizeof(char);
/* Initialize diag */ // TODO: check whether this is still needed
if (!C->scale_diag) {
for (j = 0; j < n; j++)
diag[j] = 1.;
}
/* Evaluate function at starting point and calculate norm. */
if( C->verbosity ) {
fprintf( msgfile, "lmmin start " );
lm_print_pars( nout, x, msgfile );
}
//printf("evaluate: starting\n");
(*evaluate)( x, m, data, fvec, &(S->userbreak) );
if( C->verbosity > 4 )
for( i = 0; i < m; ++i )
fprintf( msgfile, " fvec[%4i] = %18.8g\n", i, fvec[i] );
S->nfev = 1;
if ( S->userbreak )
goto terminate;
fnorm = lm_enorm(m, fvec);
if( C->verbosity )
fprintf( msgfile, " fnorm = %18.8g\n", fnorm );
if( !isfinite(fnorm) ) {
if( C->verbosity )
fprintf( msgfile, "nan case 1\n" );
S->outcome = 12; /* nan */
goto terminate;
} else if( fnorm <= LM_DWARF ) {
S->outcome = 0; /* sum of squares almost zero, nothing to do */
goto terminate;
}
/* The outer loop: compute gradient, then descend. */
for( int outer = 0; ; ++outer ) {
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
steps[j] = MAX(eps * eps, eps * fabs(x[j]));
par2[i * n + j] = x[j] + ((i == j) ? steps[j] : 0.0);
}
}
/* [outer] Calculate the Jacobian. */
#ifdef LMFIT_OPENMP
#pragma omp parallel for
#endif
for (j = 0; j < n; j++) {
(*evaluate)(par2 + n * j, m, data, wfs + m * j, &(S->userbreak));
#ifdef LMFIT_OPENMP
#pragma omp critical
#endif
for (i = 0; i < m; i++)
fjac[m * j + i] = (wfs[m * j + i] - fvec[i]) / steps[j];
}
S->nfev += n;
if (S->userbreak)
goto terminate;
if (C->verbosity > 6) {
/* print the entire matrix */
printf("\nlmmin Jacobian\n");
for (i = 0; i < m; i++) {
printf(" ");
for (j = 0; j < n; j++)
printf("%.5e ", fjac[j * m + i]);
printf("\n");
}
}
/* [outer] Compute the QR factorization of the Jacobian. */
/* fjac is an m by n array. The upper n by n submatrix of fjac
* is made to contain an upper triangular matrix R with diagonal
* elements of nonincreasing magnitude such that
*
* P^T*(J^T*J)*P = R^T*R
*
* (NOTE: ^T stands for matrix transposition),
*
* where P is a permutation matrix and J is the final calculated
* Jacobian. Column j of P is column ipvt(j) of the identity matrix.
* The lower trapezoidal part of fjac contains information generated
* during the computation of R.
*
* ipvt is an integer array of length n. It 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.
*/
lm_qrfac(m, n, fjac, ipvt, wa1, wa2, wa3);
/* return values are ipvt, wa1=rdiag, wa2=acnorm */
/* [outer] Form Q^T * fvec, and store first n components in qtf. */
for (i = 0; i < m; i++)
wf[i] = fvec[i];
for (j = 0; j < n; j++) {
temp3 = fjac[j * m + j];
if (temp3 != 0) {
sum = 0;
for (i = j; i < m; i++)
sum += fjac[j * m + i] * wf[i];
temp = -sum / temp3;
for (i = j; i < m; i++)
wf[i] += fjac[j * m + i] * temp;
}
fjac[j * m + j] = wa1[j];
qtf[j] = wf[j];
}
/* [outer] Compute norm of scaled gradient and detect degeneracy. */
gnorm = 0;
for (j = 0; j < n; j++) {
if (wa2[ipvt[j]] == 0)
continue;
sum = 0;
for (i = 0; i <= j; i++)
sum += fjac[j * m + i] * qtf[i];
gnorm = MAX( gnorm, fabs( sum / wa2[ipvt[j]] / fnorm ) );
}
if (gnorm <= C->gtol) {
S->outcome = 4;
goto terminate;
}
/* [outer] Initialize / update diag and delta. */
if ( !outer ) {
/* first iteration only */
if (C->scale_diag) {
/* diag := norms of the columns of the initial Jacobian */
for (j = 0; j < n; j++)
diag[j] = wa2[j] ? wa2[j] : 1;
/* xnorm := || D x || */
for (j = 0; j < n; j++)
wa3[j] = diag[j] * x[j];
xnorm = lm_enorm(n, wa3);
if( C->verbosity >= 2 ) {
fprintf( msgfile, "lmmin diag " );
lm_print_pars( nout, x, msgfile ); // xnorm
fprintf( msgfile, " xnorm = %18.8g\n", xnorm );
}
/* only now print the header for the loop table */
if( C->verbosity >= 3 ) {
fprintf( msgfile, " #o #i lmpar prered ratio dirder delta pnorm fnorm" );
for (i = 0; i < nout; ++i)
fprintf( msgfile, " p%i", i );
fprintf( msgfile, "\n" );
}
} else {
xnorm = lm_enorm(n, x);
}
if( !isfinite(xnorm) ) {
if( C->verbosity )
fprintf( msgfile, "nan case 2\n" );
S->outcome = 12; /* nan */
goto terminate;
}
/* initialize the step bound delta. */
if ( xnorm )
delta = C->stepbound * xnorm;
else
delta = C->stepbound;
} else {
if (C->scale_diag) {
for (j = 0; j < n; j++)
diag[j] = MAX( diag[j], wa2[j] );
}
}
/* The inner loop. */
int inner = 0;
do {
/* [inner] Determine the Levenberg-Marquardt parameter. */
lm_lmpar( n, fjac, m, ipvt, diag, qtf, delta, &lmpar,
wa1, wa2, wf, wa3 );
/* used return values are fjac (partly), lmpar, wa1=x, wa3=diag*x */
/* predict scaled reduction */
pnorm = lm_enorm(n, wa3);
if( !isfinite(pnorm) ) {
if( C->verbosity )
fprintf( msgfile, "nan case 3\n" );
S->outcome = 12; /* nan */
goto terminate;
}
temp2 = lmpar * SQR( pnorm / fnorm );
for (j = 0; j < n; j++) {
wa3[j] = 0;
for (i = 0; i <= j; i++)
wa3[i] -= fjac[j * m + i] * wa1[ipvt[j]];
}
temp1 = SQR( lm_enorm(n, wa3) / fnorm );
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 */
/* at first call, adjust the initial step bound. */
if ( !outer && pnorm < delta )
delta = pnorm;
/* [inner] Evaluate the function at x + p. */
for (j = 0; j < n; j++)
wa2[j] = x[j] - wa1[j];
//printf("evaluate: inner\n");
(*evaluate)( wa2, m, data, wf, &(S->userbreak) );
++(S->nfev);
if ( S->userbreak )
goto terminate;
fnorm1 = lm_enorm(m, wf);
// exceptionally, for this norm we do not test for infinity
// because we can deal with it without terminating.
/* [inner] Evaluate the scaled reduction. */
/* actual scaled reduction (supports even the case fnorm1=infty) */
actred = fnorm1 < 10 * fnorm ? 1 - SQR(fnorm1 / fnorm) : -1;
/* ratio of actual to predicted reduction */
ratio = prered ? actred / prered : 0;
if (C->verbosity == 1)
{
printf("%21.15g %g\n", fnorm1, lmpar);
}
if( C->verbosity == 2 ) {
fprintf( msgfile, "lmmin (%i:%i) ", outer, inner );
lm_print_pars( nout, wa2, msgfile ); // fnorm1,
} else if( C->verbosity >= 3 ) {
printf("%3i %2i %9.2g %9.2g %14.6g %9.2g %10.3e %10.3e %21.15e",
outer, inner, lmpar, prered, 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 if ( actred > -99 ) { /* -99 = 1-1/0.1^2 */
temp = MAX( dirder / (2 * dirder + actred), 0.1 );
} else {
temp = 0.1;
}
delta = temp * MIN(delta, pnorm / 0.1);
lmpar /= temp;
} else if ( ratio >= 0.75 ) {
delta = 2 * pnorm;
lmpar *= 0.5;
} else if ( !lmpar ) {
delta = 2 * pnorm;
}
/* [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_enorm(m, fvec);
if ( C->verbosity >= 2 )
printf("lmmin outcome (%i) xnorm %g ftol %g xtol %g\n",
S->outcome, xnorm, C->ftol, C->xtol );
if( C->verbosity & 1 ) {
fprintf( msgfile, "lmmin final " );
lm_print_pars( nout, x, msgfile ); // S->fnorm,
fprintf( msgfile, " fnorm = %18.8g\n", S->fnorm );
}
if ( S->userbreak ) /* user-requested break */
S->outcome = 11;
free(ws);
free(steps);
free(par2);
}
/**
* Determine Levenberg-Marquardt parameter.
* 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.
*
* @param n positive integer INPUT variable set to the order of r.
*
* @param r 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.
*
* @param ldr positive integer INPUT variable not less than n
* which specifies the leading dimension of the array R.
*
* @param ipvt 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.
*
* @param diag INPUT array of length n which must contain the
* diagonal elements of the matrix D.
*
* @param qtb INPUT array of length n which must contain the first
* n elements of the vector Q^T*b.
*
* @param delta positive INPUT variable which specifies an upper
* bound on the euclidean norm of D*x.
*
* @param par nonnegative variable. On INPUT par contains an
* initial estimate of the Levenberg-Marquardt parameter.
* On OUTPUT par contains the final estimate.
*
* @param x 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.
*
* @param sdiag array of length n needed as workspace; on OUTPUT
* it contains the diagonal elements of the upper triangular
* matrix S.
*
* @param aux multi-purpose work array of length n.
*
* @param xdi work array of length n. On OUTPUT: diag[j] * x[j].
*/
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)
{
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);
}
}
/**
* QR factorization, from lapack.
* 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.
*
* @param m INPUT parameter set to the number of rows of A.
*
* @param n INPUT parameter set to the number of columns of A.
*
* @param A 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).
*
* @param Pivot integer OUTPUT array of length n that describes the
* permutation matrix P:
* Column j of P is column ipvt(j) of the identity matrix.
*
* @param Rdiag OUTPUT array of length n which contains the
* diagonal elements of R.
*
* @param Acnorm 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.
*
* @param W work array of length n.
*/
void lm_qrfac(const int m, const int n, double *const A, int *const Pivot,
double *const Rdiag, double *const Acnorm, double *const W)
{
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;
}
}
/**
* Linear least-squares.
* 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.
*
* @param n a positive integer INPUT variable set to the order of R.
*
* @param r 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.
*
* @param ldr a positive integer INPUT variable not less than n
* which specifies the leading dimension of the array R.
*
* @param ipvt 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.
*
* @param diag an INPUT array of length n which must contain the
* diagonal elements of the matrix D.
*
* @param qtb an INPUT array of length n which must contain the first
* n elements of the vector Q^T*b.
*
* @param x an OUTPUT array of length n which contains the least
* squares solution of the system A*x = b, D*x = 0.
*
* @param sdiag an OUTPUT array of length n which contains the
* diagonal elements of the upper triangular matrix S.
*
* @param wa a work array of length n.
*/
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)
{
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])) {