static int gnewton_iterate (void * vstate, gsl_multiroot_function_fdf * fdf, gsl_vector * x, gsl_vector * f, gsl_matrix * J, gsl_vector * dx) { gnewton_state_t * state = (gnewton_state_t *) vstate; int signum ; double t, phi0, phi1; size_t i; size_t n = fdf->n ; gsl_matrix_memcpy (state->lu, J); gsl_linalg_LU_decomp (state->lu, state->permutation, &signum); { int status = gsl_linalg_LU_solve (state->lu, state->permutation, f, state->d); if (status) return status; } t = 1; phi0 = state->phi; new_step: for (i = 0; i < n; i++) { double di = gsl_vector_get (state->d, i); double xi = gsl_vector_get (x, i); gsl_vector_set (state->x_trial, i, xi - t*di); } { int status = GSL_MULTIROOT_FN_EVAL_F (fdf, state->x_trial, f); if (status != GSL_SUCCESS) { return GSL_EBADFUNC; } } phi1 = enorm (f); if (phi1 > phi0 && t > GSL_DBL_EPSILON) { /* full step goes uphill, take a reduced step instead */ double theta = phi1 / phi0; double u = (sqrt(1.0 + 6.0 * theta) - 1.0) / (3.0 * theta); t *= u ; goto new_step; } /* copy x_trial into x */ gsl_vector_memcpy (x, state->x_trial); for (i = 0; i < n; i++) { double di = gsl_vector_get (state->d, i); gsl_vector_set (dx, i, -t*di); } { int status = GSL_MULTIROOT_FN_EVAL_DF (fdf, x, J); if (status != GSL_SUCCESS) { return GSL_EBADFUNC; } } state->phi = phi1; return GSL_SUCCESS; }
static int iterate (void *vstate, gsl_multiroot_function_fdf * fdf, gsl_vector * x, gsl_vector * f, gsl_matrix * J, gsl_vector * dx, int scale) { hybridj_state_t *state = (hybridj_state_t *) vstate; const double fnorm = state->fnorm; gsl_matrix *q = state->q; gsl_matrix *r = state->r; gsl_vector *tau = state->tau; gsl_vector *diag = state->diag; gsl_vector *qtf = state->qtf; gsl_vector *x_trial = state->x_trial; gsl_vector *f_trial = state->f_trial; gsl_vector *df = state->df; gsl_vector *qtdf = state->qtdf; gsl_vector *rdx = state->rdx; gsl_vector *w = state->w; gsl_vector *v = state->v; double prered, actred; double pnorm, fnorm1, fnorm1p; double ratio; double p1 = 0.1, p5 = 0.5, p001 = 0.001, p0001 = 0.0001; /* Compute qtf = Q^T f */ compute_qtf (q, f, qtf); /* Compute dogleg step */ dogleg (r, qtf, diag, state->delta, state->newton, state->gradient, dx); /* Take a trial step */ compute_trial_step (x, dx, state->x_trial); pnorm = scaled_enorm (diag, dx); if (state->iter == 1) { if (pnorm < state->delta) { state->delta = pnorm; } } /* Evaluate function at x + p */ { int status = GSL_MULTIROOT_FN_EVAL_F (fdf, x_trial, f_trial); if (status != GSL_SUCCESS) { return GSL_EBADFUNC; } } /* Set df = f_trial - f */ compute_df (f_trial, f, df); /* Compute the scaled actual reduction */ fnorm1 = enorm (f_trial); actred = compute_actual_reduction (fnorm, fnorm1); /* Compute rdx = R dx */ compute_rdx (r, dx, rdx); /* Compute the scaled predicted reduction phi1p = |Q^T f + R dx| */ fnorm1p = enorm_sum (qtf, rdx); prered = compute_predicted_reduction (fnorm, fnorm1p); /* Compute the ratio of the actual to predicted reduction */ if (prered > 0) { ratio = actred / prered; } else { ratio = 0; } /* Update the step bound */ if (ratio < p1) { state->ncsuc = 0; state->ncfail++; state->delta *= p5; } else { state->ncfail = 0; state->ncsuc++; if (ratio >= p5 || state->ncsuc > 1) state->delta = GSL_MAX (state->delta, pnorm / p5); if (fabs (ratio - 1) <= p1) state->delta = pnorm / p5; } /* Test for successful iteration */ if (ratio >= p0001) { gsl_vector_memcpy (x, x_trial); gsl_vector_memcpy (f, f_trial); state->fnorm = fnorm1; state->iter++; } /* Determine the progress of the iteration */ state->nslow1++; if (actred >= p001) state->nslow1 = 0; if (actred >= p1) state->nslow2 = 0; if (state->ncfail == 2) { { int status = GSL_MULTIROOT_FN_EVAL_DF (fdf, x, J); if (status != GSL_SUCCESS) { return GSL_EBADFUNC; } } state->nslow2++; if (state->iter == 1) { if (scale) compute_diag (J, diag); state->delta = compute_delta (diag, x); } else { if (scale) update_diag (J, diag); } /* Factorize J into QR decomposition */ gsl_linalg_QR_decomp (J, tau); gsl_linalg_QR_unpack (J, tau, q, r); return GSL_SUCCESS; } /* Compute qtdf = Q^T df, w = (Q^T df - R dx)/|dx|, v = D^2 dx/|dx| */ compute_qtf (q, df, qtdf); compute_wv (qtdf, rdx, dx, diag, pnorm, w, v); /* Rank-1 update of the jacobian Q'R' = Q(R + w v^T) */ gsl_linalg_QR_update (q, r, w, v); /* No progress as measured by jacobian evaluations */ if (state->nslow2 == 5) { return GSL_ENOPROGJ; } /* No progress as measured by function evaluations */ if (state->nslow1 == 10) { return GSL_ENOPROG; } return GSL_SUCCESS; }