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- /* monte/vegas.c
- *
- * Copyright (C) 1996, 1997, 1998, 1999, 2000 Michael Booth
- *
- * This program is free software; you can redistribute it and/or modify
- * it under the terms of the GNU General Public License as published by
- * the Free Software Foundation; either version 3 of the License, or (at
- * your option) any later version.
- *
- * This program is distributed in the hope that it will be useful, but
- * WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
- * General Public License for more details.
- *
- * You should have received a copy of the GNU General Public License
- * along with this program; if not, write to the Free Software
- * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
- */
- /* Author: MJB */
- /* Modified by: Brian Gough, 12/2000 */
- /* This is an implementation of the adaptive Monte-Carlo algorithm
- of G. P. Lepage, originally described in J. Comp. Phys. 27, 192(1978).
- The current version of the algorithm was described in the Cornell
- preprint CLNS-80/447 of March, 1980.
- This code follows most closely the c version by D.R.Yennie, coded
- in 1984.
- The input coordinates are x[j], with upper and lower limits xu[j]
- and xl[j]. The integration length in the j-th direction is
- delx[j]. Each coordinate x[j] is rescaled to a variable y[j] in
- the range 0 to 1. The range is divided into bins with boundaries
- xi[i][j], where i=0 corresponds to y=0 and i=bins to y=1. The grid
- is refined (ie, bins are adjusted) using d[i][j] which is some
- variation on the squared sum. A third parameter used in defining
- the real coordinate using random numbers is called z. It ranges
- from 0 to bins. Its integer part gives the lower index of the bin
- into which a call is to be placed, and the remainder gives the
- location inside the bin.
- When stratified sampling is used the bins are grouped into boxes,
- and the algorithm allocates an equal number of function calls to
- each box.
- The variable alpha controls how "stiff" the rebinning algorithm is.
- alpha = 0 means never change the grid. Alpha is typically set between
- 1 and 2.
- */
- /* configuration headers */
- #include "gsl__config.h"
- /* standard headers */
- #include <math.h>
- #include <stdio.h>
- /* gsl headers */
- #include "gsl_math.h"
- #include "gsl_errno.h"
- #include "gsl_rng.h"
- #include "gsl_monte_vegas.h"
- /* lib-specific headers */
- #define BINS_MAX 50 /* even integer, will be divided by two */
- /* A separable grid with coordinates and values */
- #define COORD(s,i,j) ((s)->xi[(i)*(s)->dim + (j)])
- #define NEW_COORD(s,i) ((s)->xin[(i)])
- #define VALUE(s,i,j) ((s)->d[(i)*(s)->dim + (j)])
- /* predeclare functions */
- typedef int coord;
- static void init_grid (gsl_monte_vegas_state * s, double xl[], double xu[],
- size_t dim);
- static void reset_grid_values (gsl_monte_vegas_state * s);
- static void init_box_coord (gsl_monte_vegas_state * s, coord box[]);
- static int change_box_coord (gsl_monte_vegas_state * s, coord box[]);
- static void accumulate_distribution (gsl_monte_vegas_state * s, coord bin[],
- double y);
- static void random_point (double x[], coord bin[], double *bin_vol,
- const coord box[],
- const double xl[], const double xu[],
- gsl_monte_vegas_state * s, gsl_rng * r);
- static void resize_grid (gsl_monte_vegas_state * s, unsigned int bins);
- static void refine_grid (gsl_monte_vegas_state * s);
- static void print_lim (gsl_monte_vegas_state * state,
- double xl[], double xu[], unsigned long dim);
- static void print_head (gsl_monte_vegas_state * state,
- unsigned long num_dim, unsigned long calls,
- unsigned int it_num,
- unsigned int bins, unsigned int boxes);
- static void print_res (gsl_monte_vegas_state * state,
- unsigned int itr, double res, double err,
- double cum_res, double cum_err,
- double chi_sq);
- static void print_dist (gsl_monte_vegas_state * state, unsigned long dim);
- static void print_grid (gsl_monte_vegas_state * state, unsigned long dim);
- int
- gsl_monte_vegas_integrate (gsl_monte_function * f,
- double xl[], double xu[],
- size_t dim, size_t calls,
- gsl_rng * r,
- gsl_monte_vegas_state * state,
- double *result, double *abserr)
- {
- double cum_int, cum_sig;
- size_t i, k, it;
- if (dim != state->dim)
- {
- GSL_ERROR ("number of dimensions must match allocated size", GSL_EINVAL);
- }
- for (i = 0; i < dim; i++)
- {
- if (xu[i] <= xl[i])
- {
- GSL_ERROR ("xu must be greater than xl", GSL_EINVAL);
- }
- if (xu[i] - xl[i] > GSL_DBL_MAX)
- {
- GSL_ERROR ("Range of integration is too large, please rescale",
- GSL_EINVAL);
- }
- }
- if (state->stage == 0)
- {
- init_grid (state, xl, xu, dim);
- if (state->verbose >= 0)
- {
- print_lim (state, xl, xu, dim);
- }
- }
- if (state->stage <= 1)
- {
- state->wtd_int_sum = 0;
- state->sum_wgts = 0;
- state->chi_sum = 0;
- state->it_num = 1;
- state->samples = 0;
- }
- if (state->stage <= 2)
- {
- unsigned int bins = state->bins_max;
- unsigned int boxes = 1;
- if (state->mode != GSL_VEGAS_MODE_IMPORTANCE_ONLY)
- {
- /* shooting for 2 calls/box */
- boxes = floor (pow (calls / 2.0, 1.0 / dim));
- state->mode = GSL_VEGAS_MODE_IMPORTANCE;
- if (2 * boxes >= state->bins_max)
- {
- /* if bins/box < 2 */
- int box_per_bin = GSL_MAX (boxes / state->bins_max, 1);
- bins = GSL_MIN(boxes / box_per_bin, state->bins_max);
- boxes = box_per_bin * bins;
- state->mode = GSL_VEGAS_MODE_STRATIFIED;
- }
- }
- {
- double tot_boxes = pow ((double) boxes, (double) dim);
- state->calls_per_box = GSL_MAX (calls / tot_boxes, 2);
- calls = state->calls_per_box * tot_boxes;
- }
- /* total volume of x-space/(avg num of calls/bin) */
- state->jac = state->vol * pow ((double) bins, (double) dim) / calls;
- state->boxes = boxes;
- /* If the number of bins changes from the previous invocation, bins
- are expanded or contracted accordingly, while preserving bin
- density */
- if (bins != state->bins)
- {
- resize_grid (state, bins);
- if (state->verbose > 1)
- {
- print_grid (state, dim);
- }
- }
- if (state->verbose >= 0)
- {
- print_head (state,
- dim, calls, state->it_num, state->bins, state->boxes);
- }
- }
- state->it_start = state->it_num;
- cum_int = 0.0;
- cum_sig = 0.0;
- state->chisq = 0.0;
- for (it = 0; it < state->iterations; it++)
- {
- double intgrl = 0.0, intgrl_sq = 0.0;
- double sig = 0.0;
- double wgt;
- size_t calls_per_box = state->calls_per_box;
- double jacbin = state->jac;
- double *x = state->x;
- coord *bin = state->bin;
- state->it_num = state->it_start + it;
- reset_grid_values (state);
- init_box_coord (state, state->box);
-
- do
- {
- double m = 0, q = 0;
- double f_sq_sum = 0.0;
- for (k = 0; k < calls_per_box; k++)
- {
- double fval, bin_vol;
- random_point (x, bin, &bin_vol, state->box, xl, xu, state, r);
- fval = jacbin * bin_vol * GSL_MONTE_FN_EVAL (f, x);
- /* recurrence for mean and variance */
- {
- double d = fval - m;
- m += d / (k + 1.0);
- q += d * d * (k / (k + 1.0));
- }
- if (state->mode != GSL_VEGAS_MODE_STRATIFIED)
- {
- double f_sq = fval * fval;
- accumulate_distribution (state, bin, f_sq);
- }
- }
- intgrl += m * calls_per_box;
- f_sq_sum = q * calls_per_box ;
- sig += f_sq_sum ;
- if (state->mode == GSL_VEGAS_MODE_STRATIFIED)
- {
- accumulate_distribution (state, bin, f_sq_sum);
- }
- }
- while (change_box_coord (state, state->box));
- /* Compute final results for this iteration */
- sig = sig / (calls_per_box - 1.0) ;
- if (sig > 0)
- {
- wgt = 1.0 / sig;
- }
- else if (state->sum_wgts > 0)
- {
- wgt = state->sum_wgts / state->samples;
- }
- else
- {
- wgt = 0.0;
- }
-
- intgrl_sq = intgrl * intgrl;
- state->result = intgrl;
- state->sigma = sqrt(sig);
- if (wgt > 0.0)
- {
- state->samples++ ;
- state->sum_wgts += wgt;
- state->wtd_int_sum += intgrl * wgt;
- state->chi_sum += intgrl_sq * wgt;
- cum_int = state->wtd_int_sum / state->sum_wgts;
- cum_sig = sqrt (1 / state->sum_wgts);
- if (state->samples > 1)
- {
- state->chisq = (state->chi_sum - state->wtd_int_sum * cum_int) /
- (state->samples - 1.0);
- }
- }
- else
- {
- cum_int += (intgrl - cum_int) / (it + 1.0);
- cum_sig = 0.0;
- }
- if (state->verbose >= 0)
- {
- print_res (state,
- state->it_num, intgrl, sqrt (sig), cum_int, cum_sig,
- state->chisq);
- if (it + 1 == state->iterations && state->verbose > 0)
- {
- print_grid (state, dim);
- }
- }
- if (state->verbose > 1)
- {
- print_dist (state, dim);
- }
- refine_grid (state);
- if (state->verbose > 1)
- {
- print_grid (state, dim);
- }
- }
- /* By setting stage to 1 further calls will generate independent
- estimates based on the same grid, although it may be rebinned. */
- state->stage = 1;
- *result = cum_int;
- *abserr = cum_sig;
- return GSL_SUCCESS;
- }
- gsl_monte_vegas_state *
- gsl_monte_vegas_alloc (size_t dim)
- {
- gsl_monte_vegas_state *s =
- (gsl_monte_vegas_state *) malloc (sizeof (gsl_monte_vegas_state));
- if (s == 0)
- {
- GSL_ERROR_VAL ("failed to allocate space for vegas state struct",
- GSL_ENOMEM, 0);
- }
- s->delx = (double *) malloc (dim * sizeof (double));
- if (s->delx == 0)
- {
- free (s);
- GSL_ERROR_VAL ("failed to allocate space for delx", GSL_ENOMEM, 0);
- }
- s->d = (double *) malloc (BINS_MAX * dim * sizeof (double));
- if (s->d == 0)
- {
- free (s->delx);
- free (s);
- GSL_ERROR_VAL ("failed to allocate space for d", GSL_ENOMEM, 0);
- }
- s->xi = (double *) malloc ((BINS_MAX + 1) * dim * sizeof (double));
- if (s->xi == 0)
- {
- free (s->d);
- free (s->delx);
- free (s);
- GSL_ERROR_VAL ("failed to allocate space for xi", GSL_ENOMEM, 0);
- }
- s->xin = (double *) malloc ((BINS_MAX + 1) * sizeof (double));
- if (s->xin == 0)
- {
- free (s->xi);
- free (s->d);
- free (s->delx);
- free (s);
- GSL_ERROR_VAL ("failed to allocate space for xin", GSL_ENOMEM, 0);
- }
- s->weight = (double *) malloc (BINS_MAX * sizeof (double));
- if (s->weight == 0)
- {
- free (s->xin);
- free (s->xi);
- free (s->d);
- free (s->delx);
- free (s);
- GSL_ERROR_VAL ("failed to allocate space for xin", GSL_ENOMEM, 0);
- }
- s->box = (coord *) malloc (dim * sizeof (coord));
- if (s->box == 0)
- {
- free (s->weight);
- free (s->xin);
- free (s->xi);
- free (s->d);
- free (s->delx);
- free (s);
- GSL_ERROR_VAL ("failed to allocate space for box", GSL_ENOMEM, 0);
- }
- s->bin = (coord *) malloc (dim * sizeof (coord));
- if (s->bin == 0)
- {
- free (s->box);
- free (s->weight);
- free (s->xin);
- free (s->xi);
- free (s->d);
- free (s->delx);
- free (s);
- GSL_ERROR_VAL ("failed to allocate space for bin", GSL_ENOMEM, 0);
- }
- s->x = (double *) malloc (dim * sizeof (double));
- if (s->x == 0)
- {
- free (s->bin);
- free (s->box);
- free (s->weight);
- free (s->xin);
- free (s->xi);
- free (s->d);
- free (s->delx);
- free (s);
- GSL_ERROR_VAL ("failed to allocate space for x", GSL_ENOMEM, 0);
- }
- s->dim = dim;
- s->bins_max = BINS_MAX;
- gsl_monte_vegas_init (s);
- return s;
- }
- /* Set some default values and whatever */
- int
- gsl_monte_vegas_init (gsl_monte_vegas_state * state)
- {
- state->stage = 0;
- state->alpha = 1.5;
- state->verbose = -1;
- state->iterations = 5;
- state->mode = GSL_VEGAS_MODE_IMPORTANCE;
- state->chisq = 0;
- state->bins = state->bins_max;
- state->ostream = stdout;
- return GSL_SUCCESS;
- }
- void
- gsl_monte_vegas_free (gsl_monte_vegas_state * s)
- {
- free (s->x);
- free (s->delx);
- free (s->d);
- free (s->xi);
- free (s->xin);
- free (s->weight);
- free (s->box);
- free (s->bin);
- free (s);
- }
- static void
- init_box_coord (gsl_monte_vegas_state * s, coord box[])
- {
- size_t i;
- size_t dim = s->dim;
- for (i = 0; i < dim; i++)
- {
- box[i] = 0;
- }
- }
- /* change_box_coord steps through the box coord like
- {0,0}, {0, 1}, {0, 2}, {0, 3}, {1, 0}, {1, 1}, {1, 2}, ...
- */
- static int
- change_box_coord (gsl_monte_vegas_state * s, coord box[])
- {
- int j = s->dim - 1;
- int ng = s->boxes;
- while (j >= 0)
- {
- box[j] = (box[j] + 1) % ng;
- if (box[j] != 0)
- {
- return 1;
- }
- j--;
- }
- return 0;
- }
- static void
- init_grid (gsl_monte_vegas_state * s, double xl[], double xu[], size_t dim)
- {
- size_t j;
- double vol = 1.0;
- s->bins = 1;
- for (j = 0; j < dim; j++)
- {
- double dx = xu[j] - xl[j];
- s->delx[j] = dx;
- vol *= dx;
- COORD (s, 0, j) = 0.0;
- COORD (s, 1, j) = 1.0;
- }
- s->vol = vol;
- }
- static void
- reset_grid_values (gsl_monte_vegas_state * s)
- {
- size_t i, j;
- size_t dim = s->dim;
- size_t bins = s->bins;
- for (i = 0; i < bins; i++)
- {
- for (j = 0; j < dim; j++)
- {
- VALUE (s, i, j) = 0.0;
- }
- }
- }
- static void
- accumulate_distribution (gsl_monte_vegas_state * s, coord bin[], double y)
- {
- size_t j;
- size_t dim = s->dim;
- for (j = 0; j < dim; j++)
- {
- int i = bin[j];
- VALUE (s, i, j) += y;
- }
- }
- static void
- random_point (double x[], coord bin[], double *bin_vol,
- const coord box[], const double xl[], const double xu[],
- gsl_monte_vegas_state * s, gsl_rng * r)
- {
- /* Use the random number generator r to return a random position x
- in a given box. The value of bin gives the bin location of the
- random position (there may be several bins within a given box) */
- double vol = 1.0;
- size_t j;
- size_t dim = s->dim;
- size_t bins = s->bins;
- size_t boxes = s->boxes;
- DISCARD_POINTER(xu); /* prevent warning about unused parameter */
- for (j = 0; j < dim; ++j)
- {
- /* box[j] + ran gives the position in the box units, while z
- is the position in bin units. */
- double z = ((box[j] + gsl_rng_uniform_pos (r)) / boxes) * bins;
- int k = z;
- double y, bin_width;
- bin[j] = k;
- if (k == 0)
- {
- bin_width = COORD (s, 1, j);
- y = z * bin_width;
- }
- else
- {
- bin_width = COORD (s, k + 1, j) - COORD (s, k, j);
- y = COORD (s, k, j) + (z - k) * bin_width;
- }
- x[j] = xl[j] + y * s->delx[j];
- vol *= bin_width;
- }
- *bin_vol = vol;
- }
- static void
- resize_grid (gsl_monte_vegas_state * s, unsigned int bins)
- {
- size_t j, k;
- size_t dim = s->dim;
- /* weight is ratio of bin sizes */
- double pts_per_bin = (double) s->bins / (double) bins;
- for (j = 0; j < dim; j++)
- {
- double xold;
- double xnew = 0;
- double dw = 0;
- int i = 1;
- for (k = 1; k <= s->bins; k++)
- {
- dw += 1.0;
- xold = xnew;
- xnew = COORD (s, k, j);
- for (; dw > pts_per_bin; i++)
- {
- dw -= pts_per_bin;
- NEW_COORD (s, i) = xnew - (xnew - xold) * dw;
- }
- }
- for (k = 1 ; k < bins; k++)
- {
- COORD(s, k, j) = NEW_COORD(s, k);
- }
- COORD (s, bins, j) = 1;
- }
- s->bins = bins;
- }
- static void
- refine_grid (gsl_monte_vegas_state * s)
- {
- size_t i, j, k;
- size_t dim = s->dim;
- size_t bins = s->bins;
- for (j = 0; j < dim; j++)
- {
- double grid_tot_j, tot_weight;
- double * weight = s->weight;
- double oldg = VALUE (s, 0, j);
- double newg = VALUE (s, 1, j);
- VALUE (s, 0, j) = (oldg + newg) / 2;
- grid_tot_j = VALUE (s, 0, j);
- /* This implements gs[i][j] = (gs[i-1][j]+gs[i][j]+gs[i+1][j])/3 */
- for (i = 1; i < bins - 1; i++)
- {
- double rc = oldg + newg;
- oldg = newg;
- newg = VALUE (s, i + 1, j);
- VALUE (s, i, j) = (rc + newg) / 3;
- grid_tot_j += VALUE (s, i, j);
- }
- VALUE (s, bins - 1, j) = (newg + oldg) / 2;
- grid_tot_j += VALUE (s, bins - 1, j);
- tot_weight = 0;
- for (i = 0; i < bins; i++)
- {
- weight[i] = 0;
- if (VALUE (s, i, j) > 0)
- {
- oldg = grid_tot_j / VALUE (s, i, j);
- /* damped change */
- weight[i] = pow (((oldg - 1) / oldg / log (oldg)), s->alpha);
- }
- tot_weight += weight[i];
- #ifdef DEBUG
- printf("weight[%d] = %g\n", i, weight[i]);
- #endif
- }
- {
- double pts_per_bin = tot_weight / bins;
- double xold;
- double xnew = 0;
- double dw = 0;
- i = 1;
- for (k = 0; k < bins; k++)
- {
- dw += weight[k];
- xold = xnew;
- xnew = COORD (s, k + 1, j);
- for (; dw > pts_per_bin; i++)
- {
- dw -= pts_per_bin;
- NEW_COORD (s, i) = xnew - (xnew - xold) * dw / weight[k];
- }
- }
- for (k = 1 ; k < bins ; k++)
- {
- COORD(s, k, j) = NEW_COORD(s, k);
- }
- COORD (s, bins, j) = 1;
- }
- }
- }
- static void
- print_lim (gsl_monte_vegas_state * state,
- double xl[], double xu[], unsigned long dim)
- {
- unsigned long j;
- fprintf (state->ostream, "The limits of integration are:\n");
- for (j = 0; j < dim; ++j)
- fprintf (state->ostream, "\nxl[%lu]=%f xu[%lu]=%f", j, xl[j], j, xu[j]);
- fprintf (state->ostream, "\n");
- fflush (state->ostream);
- }
- static void
- print_head (gsl_monte_vegas_state * state,
- unsigned long num_dim, unsigned long calls,
- unsigned int it_num, unsigned int bins, unsigned int boxes)
- {
- fprintf (state->ostream,
- "\nnum_dim=%lu, calls=%lu, it_num=%d, max_it_num=%d ",
- num_dim, calls, it_num, state->iterations);
- fprintf (state->ostream,
- "verb=%d, alph=%.2f,\nmode=%d, bins=%d, boxes=%d\n",
- state->verbose, state->alpha, state->mode, bins, boxes);
- fprintf (state->ostream,
- "\n single.......iteration ");
- fprintf (state->ostream, "accumulated......results \n");
- fprintf (state->ostream,
- "iteration integral sigma integral ");
- fprintf (state->ostream, " sigma chi-sq/it\n\n");
- fflush (state->ostream);
- }
- static void
- print_res (gsl_monte_vegas_state * state,
- unsigned int itr,
- double res, double err,
- double cum_res, double cum_err,
- double chi_sq)
- {
- fprintf (state->ostream,
- "%4d %6.4e %10.2e %6.4e %8.2e %10.2e\n",
- itr, res, err, cum_res, cum_err, chi_sq);
- fflush (state->ostream);
- }
- static void
- print_dist (gsl_monte_vegas_state * state, unsigned long dim)
- {
- unsigned long i, j;
- int p = state->verbose;
- if (p < 1)
- return;
- for (j = 0; j < dim; ++j)
- {
- fprintf (state->ostream, "\n axis %lu \n", j);
- fprintf (state->ostream, " x g\n");
- for (i = 0; i < state->bins; i++)
- {
- fprintf (state->ostream, "weight [%11.2e , %11.2e] = ",
- COORD (state, i, j), COORD(state,i+1,j));
- fprintf (state->ostream, " %11.2e\n", VALUE (state, i, j));
- }
- fprintf (state->ostream, "\n");
- }
- fprintf (state->ostream, "\n");
- fflush (state->ostream);
- }
- static void
- print_grid (gsl_monte_vegas_state * state, unsigned long dim)
- {
- unsigned long i, j;
- int p = state->verbose;
- if (p < 1)
- return;
- for (j = 0; j < dim; ++j)
- {
- fprintf (state->ostream, "\n axis %lu \n", j);
- fprintf (state->ostream, " x \n");
- for (i = 0; i <= state->bins; i++)
- {
- fprintf (state->ostream, "%11.2e", COORD (state, i, j));
- if (i % 5 == 4)
- fprintf (state->ostream, "\n");
- }
- fprintf (state->ostream, "\n");
- }
- fprintf (state->ostream, "\n");
- fflush (state->ostream);
- }
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