diff --git a/C/clencurt.h b/C/clencurt.h new file mode 100644 # file too large to diff: C/clencurt.h diff --git a/C/converged.h b/C/converged.h new file mode 100644 --- /dev/null +++ b/C/converged.h @@ -0,0 +1,73 @@ +/* Body of convergence test, shared between hcubature.c and + pcubature.c. We use an #include file because the two routines use + somewhat different data structures, and define macros ERR(j) and + VAL(j) to get the error and value estimates, respectively, for + integrand j. */ +{ + unsigned j; +# define SQR(x) ((x) * (x)) + switch (norm) { + case ERROR_INDIVIDUAL: + for (j = 0; j < fdim; ++j) + if (ERR(j) > reqAbsError && ERR(j) > fabs(VAL(j))*reqRelError) + return 0; + return 1; + + case ERROR_PAIRED: + for (j = 0; j+1 < fdim; j += 2) { + double maxerr, serr, err, maxval, sval, val; + /* scale to avoid overflow/underflow */ + maxerr = ERR(j) > ERR(j+1) ? ERR(j) : ERR(j+1); + maxval = VAL(j) > VAL(j+1) ? VAL(j) : VAL(j+1); + serr = maxerr > 0 ? 1/maxerr : 1; + sval = maxval > 0 ? 1/maxval : 1; + err = sqrt(SQR(ERR(j)*serr) + SQR(ERR(j+1)*serr)) * maxerr; + val = sqrt(SQR(VAL(j)*sval) + SQR(VAL(j+1)*sval)) * maxval; + if (err > reqAbsError && err > val*reqRelError) + return 0; + } + if (j < fdim) /* fdim is odd, do last dimension individually */ + if (ERR(j) > reqAbsError && ERR(j) > fabs(VAL(j))*reqRelError) + return 0; + return 1; + + case ERROR_L1: { + double err = 0, val = 0; + for (j = 0; j < fdim; ++j) { + err += ERR(j); + val += fabs(VAL(j)); + } + return err <= reqAbsError || err <= val*reqRelError; + } + + case ERROR_LINF: { + double err = 0, val = 0; + for (j = 0; j < fdim; ++j) { + double absval = fabs(VAL(j)); + if (ERR(j) > err) err = ERR(j); + if (absval > val) val = absval; + } + return err <= reqAbsError || err <= val*reqRelError; + } + + case ERROR_L2: { + double maxerr = 0, maxval = 0, serr, sval, err = 0, val = 0; + /* scale values by 1/max to avoid overflow/underflow */ + for (j = 0; j < fdim; ++j) { + double absval = fabs(VAL(j)); + if (ERR(j) > maxerr) maxerr = ERR(j); + if (absval > maxval) maxval = absval; + } + serr = maxerr > 0 ? 1/maxerr : 1; + sval = maxval > 0 ? 1/maxval : 1; + for (j = 0; j < fdim; ++j) { + err += SQR(ERR(j) * serr); + val += SQR(fabs(VAL(j)) * sval); + } + err = sqrt(err) * maxerr; + val = sqrt(val) * maxval; + return err <= reqAbsError || err <= val*reqRelError; + } + } + return 1; /* unreachable */ +} diff --git a/C/cubature.h b/C/cubature.h new file mode 100644 --- /dev/null +++ b/C/cubature.h @@ -0,0 +1,123 @@ +/* Adaptive multidimensional integration of a vector of integrands. + * + * Copyright (c) 2005-2013 Steven G. Johnson + * + * Portions (see comments) based on HIntLib (also distributed under + * the GNU GPL, v2 or later), copyright (c) 2002-2005 Rudolf Schuerer. + * (http://www.cosy.sbg.ac.at/~rschuer/hintlib/) + * + * Portions (see comments) based on GNU GSL (also distributed under + * the GNU GPL, v2 or later), copyright (c) 1996-2000 Brian Gough. + * (http://www.gnu.org/software/gsl/) + * + * 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 2 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., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + * + */ + +#ifndef CUBATURE_H +#define CUBATURE_H + +#include /* for size_t */ + +#ifdef __cplusplus +extern "C" +{ +#endif /* __cplusplus */ + +/* USAGE: Call hcubature or pcubature with your function as described + in the README file. */ + +/* a vector integrand - evaluates the function at the given point x + (an array of length ndim) and returns the result in fval (an array + of length fdim). The void* parameter is there in case you have + to pass any additional data through to your function (it corresponds + to the fdata parameter you pass to cubature). Return 0 on + success or nonzero to terminate the integration. */ +typedef int (*integrand) (unsigned ndim, const double *x, void *, + unsigned fdim, double *fval); + +/* a vector integrand of a vector of npt points: x[i*ndim + j] is the + j-th coordinate of the i-th point, and the k-th function evaluation + for the i-th point is returned in fval[i*fdim + k]. Return 0 on success + or nonzero to terminate the integration. */ +typedef int (*integrand_v) (unsigned ndim, size_t npt, + const double *x, void *, + unsigned fdim, double *fval); + +/* Different ways of measuring the absolute and relative error when + we have multiple integrands, given a vector e of error estimates + in the individual components of a vector v of integrands. These + are all equivalent when there is only a single integrand. */ +typedef enum { + ERROR_INDIVIDUAL = 0, /* individual relerr criteria in each component */ + ERROR_PAIRED, /* paired L2 norms of errors in each component, + mainly for integrating vectors of complex numbers */ + ERROR_L2, /* abserr is L_2 norm |e|, and relerr is |e|/|v| */ + ERROR_L1, /* abserr is L_1 norm |e|, and relerr is |e|/|v| */ + ERROR_LINF /* abserr is L_\infty norm |e|, and relerr is |e|/|v| */ +} error_norm; + +/* Integrate the function f from xmin[dim] to xmax[dim], with at most + maxEval function evaluations (0 for no limit), until the given + absolute or relative error is achieved. val returns the integral, + and err returns the estimate for the absolute error in val; both + of these are arrays of length fdim, the dimension of the vector + integrand f(x). The return value of the function is 0 on success + and non-zero if there was an error. */ + +/* adapative integration by partitioning the integration domain ("h-adaptive") + and using the same fixed-degree quadrature in each subdomain, recursively, + until convergence is achieved. */ +int hcubature(unsigned fdim, integrand f, void *fdata, + unsigned dim, const double *xmin, const double *xmax, + size_t maxEval, double reqAbsError, double reqRelError, + error_norm norm, + double *val, double *err); + +/* as hcubature, but vectorized integrand */ +int hcubature_v(unsigned fdim, integrand_v f, void *fdata, + unsigned dim, const double *xmin, const double *xmax, + size_t maxEval, double reqAbsError, double reqRelError, + error_norm norm, + double *val, double *err); + +/* adaptive integration by increasing the degree of (tensor-product + Clenshaw-Curtis) quadrature rules ("p-adaptive"), rather than + subdividing the domain ("h-adaptive"). Possibly better for + smooth integrands in low dimensions. */ +int pcubature_v_buf(unsigned fdim, integrand_v f, void *fdata, + unsigned dim, const double *xmin, const double *xmax, + size_t maxEval, + double reqAbsError, double reqRelError, + error_norm norm, + unsigned *m, + double **buf, size_t *nbuf, size_t max_nbuf, + double *val, double *err); +int pcubature_v(unsigned fdim, integrand_v f, void *fdata, + unsigned dim, const double *xmin, const double *xmax, + size_t maxEval, double reqAbsError, double reqRelError, + error_norm norm, + double *val, double *err); +int pcubature(unsigned fdim, integrand f, void *fdata, + unsigned dim, const double *xmin, const double *xmax, + size_t maxEval, double reqAbsError, double reqRelError, + error_norm norm, + double *val, double *err); + +#ifdef __cplusplus +} /* extern "C" */ +#endif /* __cplusplus */ + +#endif /* CUBATURE_H */ diff --git a/C/hcubature.c b/C/hcubature.c new file mode 100644 --- /dev/null +++ b/C/hcubature.c @@ -0,0 +1,1099 @@ +/* Adaptive multidimensional integration of a vector of integrands. + * + * Copyright (c) 2005-2013 Steven G. Johnson + * + * Portions (see comments) based on HIntLib (also distributed under + * the GNU GPL, v2 or later), copyright (c) 2002-2005 Rudolf Schuerer. + * (http://www.cosy.sbg.ac.at/~rschuer/hintlib/) + * + * Portions (see comments) based on GNU GSL (also distributed under + * the GNU GPL, v2 or later), copyright (c) 1996-2000 Brian Gough. + * (http://www.gnu.org/software/gsl/) + * + * 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 2 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., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + * + */ + +#include +#include +#include +#include +#include +#include + +/* Adaptive multidimensional integration on hypercubes (or, really, + hyper-rectangles) using cubature rules. + + A cubature rule takes a function and a hypercube and evaluates + the function at a small number of points, returning an estimate + of the integral as well as an estimate of the error, and also + a suggested dimension of the hypercube to subdivide. + + Given such a rule, the adaptive integration is simple: + + 1) Evaluate the cubature rule on the hypercube(s). + Stop if converged. + + 2) Pick the hypercube with the largest estimated error, + and divide it in two along the suggested dimension. + + 3) Goto (1). + + The basic algorithm is based on the adaptive cubature described in + + A. C. Genz and A. A. Malik, "An adaptive algorithm for numeric + integration over an N-dimensional rectangular region," + J. Comput. Appl. Math. 6 (4), 295-302 (1980). + + and subsequently extended to integrating a vector of integrands in + + J. Berntsen, T. O. Espelid, and A. Genz, "An adaptive algorithm + for the approximate calculation of multiple integrals," + ACM Trans. Math. Soft. 17 (4), 437-451 (1991). + + Note, however, that we do not use any of code from the above authors + (in part because their code is Fortran 77, but mostly because it is + under the restrictive ACM copyright license). I did make use of some + GPL code from Rudolf Schuerer's HIntLib and from the GNU Scientific + Library as listed in the copyright notice above, on the other hand. + + I am also grateful to Dmitry Turbiner , who + implemented an initial prototype of the "vectorized" functionality + for evaluating multiple points in a single call (as opposed to + multiple functions in a single call). (Although Dmitry implemented + a working version, I ended up re-implementing this feature from + scratch as part of a larger code-cleanup, and in order to have + a single code path for the vectorized and non-vectorized APIs. I + subsequently implemented the algorithm by Gladwell to extract + even more parallelism by evalutating many hypercubes at once.) + + TODO: + + * Putting these routines into the GNU GSL library would be nice. + + * A Python interface would be nice. (Also a Matlab interface, + a GNU Octave interface, ...) + + * For high-dimensional integrals, it would be nice to implement + a sparse-grid cubature scheme using Clenshaw-Curtis quadrature. + Currently, for dimensions > 7 or so, quasi Monte Carlo methods win. + + * Berntsen et. al also describe a "two-level" error estimation scheme + that they claim makes the algorithm more robust. It might be + nice to implement this, at least as an option (although I seem + to remember trying it once and it made the number of evaluations + substantially worse for my test integrands). + +*/ + +/* USAGE: Call cubature with your function as described in cubature.h. + + To compile a test program, compile cubature.c with + -DTEST_INTEGRATOR as described at the end. */ + +#include "cubature.h" + +/* error return codes */ +#define SUCCESS 0 +#define FAILURE 1 + +/***************************************************************************/ +/* Basic datatypes */ + +typedef struct { + double val, err; +} esterr; + +static double errMax(unsigned fdim, const esterr *ee) +{ + double errmax = 0; + unsigned k; + for (k = 0; k < fdim; ++k) + if (ee[k].err > errmax) errmax = ee[k].err; + return errmax; +} + +typedef struct { + unsigned dim; + double *data; /* length 2*dim = center followed by half-widths */ + double vol; /* cache volume = product of widths */ +} hypercube; + +static double compute_vol(const hypercube *h) +{ + unsigned i; + double vol = 1; + for (i = 0; i < h->dim; ++i) + vol *= 2 * h->data[i + h->dim]; + return vol; +} + +static hypercube make_hypercube(unsigned dim, const double *center, const double *halfwidth) +{ + unsigned i; + hypercube h; + h.dim = dim; + h.data = (double *) malloc(sizeof(double) * dim * 2); + h.vol = 0; + if (h.data) { + for (i = 0; i < dim; ++i) { + h.data[i] = center[i]; + h.data[i + dim] = halfwidth[i]; + } + h.vol = compute_vol(&h); + } + return h; +} + +static hypercube make_hypercube_range(unsigned dim, const double *xmin, const double *xmax) +{ + hypercube h = make_hypercube(dim, xmin, xmax); + unsigned i; + if (h.data) { + for (i = 0; i < dim; ++i) { + h.data[i] = 0.5 * (xmin[i] + xmax[i]); + h.data[i + dim] = 0.5 * (xmax[i] - xmin[i]); + } + h.vol = compute_vol(&h); + } + return h; +} + +static void destroy_hypercube(hypercube *h) +{ + free(h->data); + h->dim = 0; +} + +typedef struct { + hypercube h; + unsigned splitDim; + unsigned fdim; /* dimensionality of vector integrand */ + esterr *ee; /* array of length fdim */ + double errmax; /* max ee[k].err */ +} region; + +static region make_region(const hypercube *h, unsigned fdim) +{ + region R; + R.h = make_hypercube(h->dim, h->data, h->data + h->dim); + R.splitDim = 0; + R.fdim = fdim; + R.ee = R.h.data ? (esterr *) malloc(sizeof(esterr) * fdim) : NULL; + R.errmax = HUGE_VAL; + return R; +} + +static void destroy_region(region *R) +{ + destroy_hypercube(&R->h); + free(R->ee); + R->ee = 0; +} + +static int cut_region(region *R, region *R2) +{ + unsigned d = R->splitDim, dim = R->h.dim; + *R2 = *R; + R->h.data[d + dim] *= 0.5; + R->h.vol *= 0.5; + R2->h = make_hypercube(dim, R->h.data, R->h.data + dim); + if (!R2->h.data) return FAILURE; + R->h.data[d] -= R->h.data[d + dim]; + R2->h.data[d] += R->h.data[d + dim]; + R2->ee = (esterr *) malloc(sizeof(esterr) * R2->fdim); + return R2->ee == NULL; +} + +struct rule_s; /* forward declaration */ + +typedef int (*evalError_func)(struct rule_s *r, + unsigned fdim, integrand_v f, void *fdata, + unsigned nR, region *R); +typedef void (*destroy_func)(struct rule_s *r); + + +typedef struct rule_s { + unsigned dim, fdim; /* the dimensionality & number of functions */ + unsigned num_points; /* number of evaluation points */ + unsigned num_regions; /* max number of regions evaluated at once */ + double *pts; /* points to eval: num_regions * num_points * dim */ + double *vals; /* num_regions * num_points * fdim */ + evalError_func evalError; + destroy_func destroy; +} rule; + +static void destroy_rule(rule *r) +{ + if (r) { + if (r->destroy) r->destroy(r); + free(r->pts); + free(r); + } +} + +static int alloc_rule_pts(rule *r, unsigned num_regions) +{ + if (num_regions > r->num_regions) { + free(r->pts); + r->pts = r->vals = NULL; + r->num_regions = 0; + num_regions *= 2; /* allocate extra so that + repeatedly calling alloc_rule_pts with + growing num_regions only needs + a logarithmic number of allocations */ + r->pts = (double *) malloc(sizeof(double) * + (num_regions + * r->num_points * (r->dim + r->fdim))); + if (r->fdim + r->dim > 0 && !r->pts) return FAILURE; + r->vals = r->pts + num_regions * r->num_points * r->dim; + r->num_regions = num_regions; + } + return SUCCESS; +} + +static rule *make_rule(size_t sz, /* >= sizeof(rule) */ + unsigned dim, unsigned fdim, unsigned num_points, + evalError_func evalError, destroy_func destroy) +{ + rule *r; + + if (sz < sizeof(rule)) return NULL; + r = (rule *) malloc(sz); + if (!r) return NULL; + r->pts = r->vals = NULL; + r->num_regions = 0; + r->dim = dim; r->fdim = fdim; r->num_points = num_points; + r->evalError = evalError; + r->destroy = destroy; + return r; +} + +/* note: all regions must have same fdim */ +static int eval_regions(unsigned nR, region *R, + integrand_v f, void *fdata, rule *r) +{ + unsigned iR; + if (nR == 0) return SUCCESS; /* nothing to evaluate */ + if (r->evalError(r, R->fdim, f, fdata, nR, R)) return FAILURE; + for (iR = 0; iR < nR; ++iR) + R[iR].errmax = errMax(R->fdim, R[iR].ee); + return SUCCESS; +} + +/***************************************************************************/ +/* Functions to loop over points in a hypercube. */ + +/* Based on orbitrule.cpp in HIntLib-0.0.10 */ + +/* ls0 returns the least-significant 0 bit of n (e.g. it returns + 0 if the LSB is 0, it returns 1 if the 2 LSBs are 01, etcetera). */ + +static unsigned ls0(unsigned n) +{ +#if defined(__GNUC__) && \ + ((__GNUC__ == 3 && __GNUC_MINOR__ >= 4) || __GNUC__ > 3) + return __builtin_ctz(~n); /* gcc builtin for version >= 3.4 */ +#else + const unsigned bits[256] = { + 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, + 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, + 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, + 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 6, + 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, + 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, + 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, + 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 7, + 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, + 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, + 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, + 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 6, + 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, + 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, + 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, + 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 8, + }; + unsigned bit = 0; + while ((n & 0xff) == 0xff) { + n >>= 8; + bit += 8; + } + return bit + bits[n & 0xff]; +#endif +} + +/** + * Evaluate the integration points for all 2^n points (+/-r,...+/-r) + * + * A Gray-code ordering is used to minimize the number of coordinate updates + * in p, although this doesn't matter as much now that we are saving all pts. + */ +static void evalR_Rfs(double *pts, unsigned dim, double *p, const double *c, const double *r) +{ + unsigned i; + unsigned signs = 0; /* 0/1 bit = +/- for corresponding element of r[] */ + + /* We start with the point where r is ADDed in every coordinate + (this implies signs=0). */ + for (i = 0; i < dim; ++i) + p[i] = c[i] + r[i]; + + /* Loop through the points in Gray-code ordering */ + for (i = 0;; ++i) { + unsigned mask, d; + + memcpy(pts, p, sizeof(double) * dim); pts += dim; + + d = ls0(i); /* which coordinate to flip */ + if (d >= dim) + break; + + /* flip the d-th bit and add/subtract r[d] */ + mask = 1U << d; + signs ^= mask; + p[d] = (signs & mask) ? c[d] - r[d] : c[d] + r[d]; + } +} + +static void evalRR0_0fs(double *pts, unsigned dim, double *p, const double *c, const double *r) +{ + unsigned i, j; + + for (i = 0; i < dim - 1; ++i) { + p[i] = c[i] - r[i]; + for (j = i + 1; j < dim; ++j) { + p[j] = c[j] - r[j]; + memcpy(pts, p, sizeof(double) * dim); pts += dim; + p[i] = c[i] + r[i]; + memcpy(pts, p, sizeof(double) * dim); pts += dim; + p[j] = c[j] + r[j]; + memcpy(pts, p, sizeof(double) * dim); pts += dim; + p[i] = c[i] - r[i]; + memcpy(pts, p, sizeof(double) * dim); pts += dim; + + p[j] = c[j]; /* Done with j -> Restore p[j] */ + } + p[i] = c[i]; /* Done with i -> Restore p[i] */ + } +} + +static void evalR0_0fs4d(double *pts, unsigned dim, double *p, const double *c, + const double *r1, const double *r2) +{ + unsigned i; + + memcpy(pts, p, sizeof(double) * dim); pts += dim; + + for (i = 0; i < dim; i++) { + p[i] = c[i] - r1[i]; + memcpy(pts, p, sizeof(double) * dim); pts += dim; + + p[i] = c[i] + r1[i]; + memcpy(pts, p, sizeof(double) * dim); pts += dim; + + p[i] = c[i] - r2[i]; + memcpy(pts, p, sizeof(double) * dim); pts += dim; + + p[i] = c[i] + r2[i]; + memcpy(pts, p, sizeof(double) * dim); pts += dim; + + p[i] = c[i]; + } +} + +#define num0_0(dim) (1U) +#define numR0_0fs(dim) (2 * (dim)) +#define numRR0_0fs(dim) (2 * (dim) * (dim-1)) +#define numR_Rfs(dim) (1U << (dim)) + +/***************************************************************************/ +/* Based on rule75genzmalik.cpp in HIntLib-0.0.10: An embedded + cubature rule of degree 7 (embedded rule degree 5) due to A. C. Genz + and A. A. Malik. See: + + A. C. Genz and A. A. Malik, "An imbedded [sic] family of fully + symmetric numerical integration rules," SIAM + J. Numer. Anal. 20 (3), 580-588 (1983). +*/ + +typedef struct { + rule parent; + + /* temporary arrays of length dim */ + double *widthLambda, *widthLambda2, *p; + + /* dimension-dependent constants */ + double weight1, weight3, weight5; + double weightE1, weightE3; + double df_scale; +} rule75genzmalik; + +#define real(x) ((double)(x)) +#define to_int(n) ((int)(n)) + +static int isqr(int x) +{ + return x * x; +} + +static void destroy_rule75genzmalik(rule *r_) +{ + rule75genzmalik *r = (rule75genzmalik *) r_; + free(r->p); +} + +static int rule75genzmalik_evalError(rule *r_, unsigned fdim, integrand_v f, void *fdata, unsigned nR, region *R) +{ + /* lambda2 = sqrt(9/70), lambda4 = sqrt(9/10), lambda5 = sqrt(9/19) */ + const double lambda2 = 0.3585685828003180919906451539079374954541; + const double lambda4 = 0.9486832980505137995996680633298155601160; + const double lambda5 = 0.6882472016116852977216287342936235251269; + const double weight2 = 980. / 6561.; + const double weight4 = 200. / 19683.; + const double weightE2 = 245. / 486.; + const double weightE4 = 25. / 729.; + const double ratio = (lambda2 * lambda2) / (lambda4 * lambda4); + + rule75genzmalik *r = (rule75genzmalik *) r_; + unsigned i, j, iR, dim = r_->dim; + size_t npts = 0; + double *diff, *pts, *vals; + + if (alloc_rule_pts(r_, nR)) return FAILURE; + pts = r_->pts; vals = r_->vals; + + for (iR = 0; iR < nR; ++iR) { + const double *center = R[iR].h.data; + const double *halfwidth = R[iR].h.data + dim; + + for (i = 0; i < dim; ++i) + r->p[i] = center[i]; + + for (i = 0; i < dim; ++i) + r->widthLambda2[i] = halfwidth[i] * lambda2; + for (i = 0; i < dim; ++i) + r->widthLambda[i] = halfwidth[i] * lambda4; + + /* Evaluate points in the center, in (lambda2,0,...,0) and + (lambda3=lambda4, 0,...,0). */ + evalR0_0fs4d(pts + npts*dim, dim, r->p, center, + r->widthLambda2, r->widthLambda); + npts += num0_0(dim) + 2 * numR0_0fs(dim); + + /* Calculate points for (lambda4, lambda4, 0, ...,0) */ + evalRR0_0fs(pts + npts*dim, dim, r->p, center, r->widthLambda); + npts += numRR0_0fs(dim); + + /* Calculate points for (lambda5, lambda5, ..., lambda5) */ + for (i = 0; i < dim; ++i) + r->widthLambda[i] = halfwidth[i] * lambda5; + evalR_Rfs(pts + npts*dim, dim, r->p, center, r->widthLambda); + npts += numR_Rfs(dim); + } + + /* Evaluate the integrand function(s) at all the points */ + if (f(dim, npts, pts, fdata, fdim, vals)) + return FAILURE; + + /* we are done with the points, and so we can re-use the pts + array to store the maximum difference diff[i] in each dimension + for each hypercube */ + diff = pts; + for (i = 0; i < dim * nR; ++i) diff[i] = 0; + + for (j = 0; j < fdim; ++j) { + const double *v = vals + j; +# define VALS(i) v[fdim*(i)] + for (iR = 0; iR < nR; ++iR) { + double result, res5th; + double val0, sum2=0, sum3=0, sum4=0, sum5=0; + unsigned k, k0 = 0; + /* accumulate j-th function values into j-th integrals + NOTE: this relies on the ordering of the eval functions + above, as well as on the internal structure of + the evalR0_0fs4d function */ + + val0 = VALS(0); /* central point */ + k0 += 1; + + for (k = 0; k < dim; ++k) { + double v0 = VALS(k0 + 4*k); + double v1 = VALS((k0 + 4*k) + 1); + double v2 = VALS((k0 + 4*k) + 2); + double v3 = VALS((k0 + 4*k) + 3); + + sum2 += v0 + v1; + sum3 += v2 + v3; + + diff[iR * dim + k] += + fabs(v0 + v1 - 2*val0 - ratio * (v2 + v3 - 2*val0)); + } + k0 += 4*k; + + for (k = 0; k < numRR0_0fs(dim); ++k) + sum4 += VALS(k0 + k); + k0 += k; + + for (k = 0; k < numR_Rfs(dim); ++k) + sum5 += VALS(k0 + k); + + /* Calculate fifth and seventh order results */ + result = R[iR].h.vol * (r->weight1 * val0 + weight2 * sum2 + r->weight3 * sum3 + weight4 * sum4 + r->weight5 * sum5); + res5th = R[iR].h.vol * (r->weightE1 * val0 + weightE2 * sum2 + r->weightE3 * sum3 + weightE4 * sum4); + + R[iR].ee[j].val = result; + R[iR].ee[j].err = fabs(res5th - result); + + v += r_->num_points * fdim; + } +# undef VALS + } + + + /* figure out dimension to split: */ + for (iR = 0; iR < nR; ++iR) { + double maxdiff = 0, df = 0; + unsigned dimDiffMax = 0; + + for (j = 0; j < fdim; ++j) + df += R[iR].ee[j].err; + df /= R[iR].h.vol * r->df_scale; + + for (i = 0; i < dim; ++i) { + double delta = diff[iR*dim + i] - maxdiff; + if (delta > df) { + maxdiff = diff[iR*dim + i]; + dimDiffMax = i; + } + else if (fabs(delta) <= df && R[iR].h.data[dim + i] > R[iR].h.data[dim + dimDiffMax]) + dimDiffMax = i; + } + R[iR].splitDim = dimDiffMax; + } + return SUCCESS; +} + +static rule *make_rule75genzmalik(unsigned dim, unsigned fdim) +{ + rule75genzmalik *r; + + if (dim < 2) return NULL; /* this rule does not support 1d integrals */ + + /* Because of the use of a bit-field in evalR_Rfs, we are limited + to be < 32 dimensions (or however many bits are in unsigned). + This is not a practical limitation...long before you reach + 32 dimensions, the Genz-Malik cubature becomes excruciatingly + slow and is superseded by other methods (e.g. Monte-Carlo). */ + if (dim >= sizeof(unsigned) * 8) return NULL; + + r = (rule75genzmalik *) make_rule(sizeof(rule75genzmalik), + dim, fdim, + num0_0(dim) + 2 * numR0_0fs(dim) + + numRR0_0fs(dim) + numR_Rfs(dim), + rule75genzmalik_evalError, + destroy_rule75genzmalik); + if (!r) return NULL; + + r->weight1 = (real(12824 - 9120 * to_int(dim) + 400 * isqr(to_int(dim))) + / real(19683)); + r->weight3 = real(1820 - 400 * to_int(dim)) / real(19683); + r->weight5 = real(6859) / real(19683) / real(1U << dim); + r->weightE1 = (real(729 - 950 * to_int(dim) + 50 * isqr(to_int(dim))) + / real(729)); + r->weightE3 = real(265 - 100 * to_int(dim)) / real(1458); + + r->df_scale = pow(10, dim); /* 10^dim */ + + r->p = (double *) malloc(sizeof(double) * dim * 3); + if (!r->p) { destroy_rule((rule *) r); return NULL; } + r->widthLambda = r->p + dim; + r->widthLambda2 = r->p + 2 * dim; + + return (rule *) r; +} + +/***************************************************************************/ +/* 1d 15-point Gaussian quadrature rule, based on qk15.c and qk.c in + GNU GSL (which in turn is based on QUADPACK). */ + +static int rule15gauss_evalError(rule *r, + unsigned fdim, integrand_v f, void *fdata, + unsigned nR, region *R) +{ + /* Gauss quadrature weights and kronrod quadrature abscissae and + weights as evaluated with 80 decimal digit arithmetic by + L. W. Fullerton, Bell Labs, Nov. 1981. */ + const unsigned n = 8; + const double xgk[8] = { /* abscissae of the 15-point kronrod rule */ + 0.991455371120812639206854697526329, + 0.949107912342758524526189684047851, + 0.864864423359769072789712788640926, + 0.741531185599394439863864773280788, + 0.586087235467691130294144838258730, + 0.405845151377397166906606412076961, + 0.207784955007898467600689403773245, + 0.000000000000000000000000000000000 + /* xgk[1], xgk[3], ... abscissae of the 7-point gauss rule. + xgk[0], xgk[2], ... to optimally extend the 7-point gauss rule */ + }; + static const double wg[4] = { /* weights of the 7-point gauss rule */ + 0.129484966168869693270611432679082, + 0.279705391489276667901467771423780, + 0.381830050505118944950369775488975, + 0.417959183673469387755102040816327 + }; + static const double wgk[8] = { /* weights of the 15-point kronrod rule */ + 0.022935322010529224963732008058970, + 0.063092092629978553290700663189204, + 0.104790010322250183839876322541518, + 0.140653259715525918745189590510238, + 0.169004726639267902826583426598550, + 0.190350578064785409913256402421014, + 0.204432940075298892414161999234649, + 0.209482141084727828012999174891714 + }; + unsigned j, k, iR; + size_t npts = 0; + double *pts, *vals; + + if (alloc_rule_pts(r, nR)) return FAILURE; + pts = r->pts; vals = r->vals; + + for (iR = 0; iR < nR; ++iR) { + const double center = R[iR].h.data[0]; + const double halfwidth = R[iR].h.data[1]; + + pts[npts++] = center; + + for (j = 0; j < (n - 1) / 2; ++j) { + int j2 = 2*j + 1; + double w = halfwidth * xgk[j2]; + pts[npts++] = center - w; + pts[npts++] = center + w; + } + for (j = 0; j < n/2; ++j) { + int j2 = 2*j; + double w = halfwidth * xgk[j2]; + pts[npts++] = center - w; + pts[npts++] = center + w; + } + + R[iR].splitDim = 0; /* no choice but to divide 0th dimension */ + } + + if (f(1, npts, pts, fdata, fdim, vals)) + return FAILURE; + + for (k = 0; k < fdim; ++k) { + const double *vk = vals + k; + for (iR = 0; iR < nR; ++iR) { + const double halfwidth = R[iR].h.data[1]; + double result_gauss = vk[0] * wg[n/2 - 1]; + double result_kronrod = vk[0] * wgk[n - 1]; + double result_abs = fabs(result_kronrod); + double result_asc, mean, err; + + /* accumulate integrals */ + npts = 1; + for (j = 0; j < (n - 1) / 2; ++j) { + int j2 = 2*j + 1; + double v = vk[fdim*npts] + vk[fdim*npts+fdim]; + result_gauss += wg[j] * v; + result_kronrod += wgk[j2] * v; + result_abs += wgk[j2] * (fabs(vk[fdim*npts]) + + fabs(vk[fdim*npts+fdim])); + npts += 2; + } + for (j = 0; j < n/2; ++j) { + int j2 = 2*j; + result_kronrod += wgk[j2] * (vk[fdim*npts] + + vk[fdim*npts+fdim]); + result_abs += wgk[j2] * (fabs(vk[fdim*npts]) + + fabs(vk[fdim*npts+fdim])); + npts += 2; + } + + /* integration result */ + R[iR].ee[k].val = result_kronrod * halfwidth; + + /* error estimate + (from GSL, probably dates back to QUADPACK + ... not completely clear to me why we don't just use + fabs(result_kronrod - result_gauss) * halfwidth */ + mean = result_kronrod * 0.5; + result_asc = wgk[n - 1] * fabs(vk[0] - mean); + npts = 1; + for (j = 0; j < (n - 1) / 2; ++j) { + int j2 = 2*j + 1; + result_asc += wgk[j2] * (fabs(vk[fdim*npts]-mean) + + fabs(vk[fdim*npts+fdim]-mean)); + npts += 2; + } + for (j = 0; j < n/2; ++j) { + int j2 = 2*j; + result_asc += wgk[j2] * (fabs(vk[fdim*npts]-mean) + + fabs(vk[fdim*npts+fdim]-mean)); + npts += 2; + } + err = fabs(result_kronrod - result_gauss) * halfwidth; + result_abs *= halfwidth; + result_asc *= halfwidth; + if (result_asc != 0 && err != 0) { + double scale = pow((200 * err / result_asc), 1.5); + err = (scale < 1) ? result_asc * scale : result_asc; + } + if (result_abs > DBL_MIN / (50 * DBL_EPSILON)) { + double min_err = 50 * DBL_EPSILON * result_abs; + if (min_err > err) err = min_err; + } + R[iR].ee[k].err = err; + + /* increment vk to point to next batch of results */ + vk += 15*fdim; + } + } + return SUCCESS; +} + +static rule *make_rule15gauss(unsigned dim, unsigned fdim) +{ + if (dim != 1) return NULL; /* this rule is only for 1d integrals */ + + return make_rule(sizeof(rule), dim, fdim, 15, + rule15gauss_evalError, 0); +} + +/***************************************************************************/ +/* binary heap implementation (ala _Introduction to Algorithms_ by + Cormen, Leiserson, and Rivest), for use as a priority queue of + regions to integrate. */ + +typedef region heap_item; +#define KEY(hi) ((hi).errmax) + +typedef struct { + size_t n, nalloc; + heap_item *items; + unsigned fdim; + esterr *ee; /* array of length fdim of the total integrand & error */ +} heap; + +static void heap_resize(heap *h, size_t nalloc) +{ + h->nalloc = nalloc; + if (nalloc) + h->items = (heap_item *) realloc(h->items, sizeof(heap_item)*nalloc); + else { + /* BSD realloc does not free for a zero-sized reallocation */ + free(h->items); + h->items = NULL; + } +} + +static heap heap_alloc(size_t nalloc, unsigned fdim) +{ + heap h; + unsigned i; + h.n = 0; + h.nalloc = 0; + h.items = 0; + h.fdim = fdim; + h.ee = (esterr *) malloc(sizeof(esterr) * fdim); + if (h.ee) { + for (i = 0; i < fdim; ++i) h.ee[i].val = h.ee[i].err = 0; + heap_resize(&h, nalloc); + } + return h; +} + +/* note that heap_free does not deallocate anything referenced by the items */ +static void heap_free(heap *h) +{ + h->n = 0; + heap_resize(h, 0); + h->fdim = 0; + free(h->ee); +} + +static int heap_push(heap *h, heap_item hi) +{ + int insert; + unsigned i, fdim = h->fdim; + + for (i = 0; i < fdim; ++i) { + h->ee[i].val += hi.ee[i].val; + h->ee[i].err += hi.ee[i].err; + } + insert = h->n; + if (++(h->n) > h->nalloc) { + heap_resize(h, h->n * 2); + if (!h->items) return FAILURE; + } + + while (insert) { + int parent = (insert - 1) / 2; + if (KEY(hi) <= KEY(h->items[parent])) + break; + h->items[insert] = h->items[parent]; + insert = parent; + } + h->items[insert] = hi; + return SUCCESS; +} + +static int heap_push_many(heap *h, size_t ni, heap_item *hi) +{ + size_t i; + for (i = 0; i < ni; ++i) + if (heap_push(h, hi[i])) return FAILURE; + return SUCCESS; +} + +static heap_item heap_pop(heap *h) +{ + heap_item ret; + int i, n, child; + + if (!(h->n)) { + fprintf(stderr, "attempted to pop an empty heap\n"); + exit(EXIT_FAILURE); + } + + ret = h->items[0]; + h->items[i = 0] = h->items[n = --(h->n)]; + while ((child = i * 2 + 1) < n) { + int largest; + heap_item swap; + + if (KEY(h->items[child]) <= KEY(h->items[i])) + largest = i; + else + largest = child; + if (++child < n && KEY(h->items[largest]) < KEY(h->items[child])) + largest = child; + if (largest == i) + break; + swap = h->items[i]; + h->items[i] = h->items[largest]; + h->items[i = largest] = swap; + } + + { + unsigned i, fdim = h->fdim; + for (i = 0; i < fdim; ++i) { + h->ee[i].val -= ret.ee[i].val; + h->ee[i].err -= ret.ee[i].err; + } + } + return ret; +} + +/***************************************************************************/ + +static int converged(unsigned fdim, const esterr *ee, + double reqAbsError, double reqRelError, error_norm norm) +#define ERR(j) ee[j].err +#define VAL(j) ee[j].val +#include "converged.h" + +/***************************************************************************/ + +/* adaptive integration, analogous to adaptintegrator.cpp in HIntLib */ + +static int rulecubature(rule *r, unsigned fdim, + integrand_v f, void *fdata, + const hypercube *h, + size_t maxEval, + double reqAbsError, double reqRelError, + error_norm norm, + double *val, double *err, int parallel) +{ + size_t numEval = 0; + heap regions; + unsigned i, j; + region *R = NULL; /* array of regions to evaluate */ + size_t nR_alloc = 0; + esterr *ee = NULL; + + if (fdim <= 1) norm = ERROR_INDIVIDUAL; /* norm is irrelevant */ + if (norm < 0 || norm > ERROR_LINF) return FAILURE; /* invalid norm */ + + regions = heap_alloc(1, fdim); + if (!regions.ee || !regions.items) goto bad; + + ee = (esterr *) malloc(sizeof(esterr) * fdim); + if (!ee) goto bad; + + nR_alloc = 2; + R = (region *) malloc(sizeof(region) * nR_alloc); + if (!R) goto bad; + R[0] = make_region(h, fdim); + if (!R[0].ee + || eval_regions(1, R, f, fdata, r) + || heap_push(®ions, R[0])) + goto bad; + numEval += r->num_points; + + while (numEval < maxEval || !maxEval) { + if (converged(fdim, regions.ee, reqAbsError, reqRelError, norm)) + break; + + if (parallel) { /* maximize potential parallelism */ + /* adapted from I. Gladwell, "Vectorization of one + dimensional quadrature codes," pp. 230--238 in + _Numerical Integration. Recent Developments, + Software and Applications_, G. Fairweather and + P. M. Keast, eds., NATO ASI Series C203, Dordrecht + (1987), as described in J. M. Bull and + T. L. Freeman, "Parallel Globally Adaptive + Algorithms for Multi-dimensional Integration," + http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.6638 + (1994). + + Basically, this evaluates in one shot all regions + that *must* be evaluated in order to reduce the + error to the requested bound: the minimum set of + largest-error regions whose errors push the total + error over the bound. + + [Note: Bull and Freeman claim that the Gladwell + approach is intrinsically inefficent because it + "requires sorting", and propose an alternative + algorithm that "only" requires three passes over the + entire set of regions. Apparently, they didn't + realize that one could use a heap data structure, in + which case the time to pop K biggest-error regions + out of N is only O(K log N), much better than the + O(N) cost of the Bull and Freeman algorithm if K << + N, and it is also much simpler.] */ + size_t nR = 0; + for (j = 0; j < fdim; ++j) ee[j] = regions.ee[j]; + do { + if (nR + 2 > nR_alloc) { + nR_alloc = (nR + 2) * 2; + R = (region *) realloc(R, nR_alloc * sizeof(region)); + if (!R) goto bad; + } + R[nR] = heap_pop(®ions); + for (j = 0; j < fdim; ++j) ee[j].err -= R[nR].ee[j].err; + if (cut_region(R+nR, R+nR+1)) goto bad; + numEval += r->num_points * 2; + nR += 2; + if (converged(fdim, ee, reqAbsError, reqRelError, norm)) + break; /* other regions have small errs */ + } while (regions.n > 0 && (numEval < maxEval || !maxEval)); + if (eval_regions(nR, R, f, fdata, r) + || heap_push_many(®ions, nR, R)) + goto bad; + } + else { /* minimize number of function evaluations */ + R[0] = heap_pop(®ions); /* get worst region */ + if (cut_region(R, R+1) + || eval_regions(2, R, f, fdata, r) + || heap_push_many(®ions, 2, R)) + goto bad; + numEval += r->num_points * 2; + } + } + + /* re-sum integral and errors */ + for (j = 0; j < fdim; ++j) val[j] = err[j] = 0; + for (i = 0; i < regions.n; ++i) { + for (j = 0; j < fdim; ++j) { + val[j] += regions.items[i].ee[j].val; + err[j] += regions.items[i].ee[j].err; + } + destroy_region(®ions.items[i]); + } + + /* printf("regions.nalloc = %d\n", regions.nalloc); */ + free(ee); + heap_free(®ions); + free(R); + return SUCCESS; + +bad: + free(ee); + heap_free(®ions); + free(R); + return FAILURE; +} + +static int cubature(unsigned fdim, integrand_v f, void *fdata, + unsigned dim, const double *xmin, const double *xmax, + size_t maxEval, double reqAbsError, double reqRelError, + error_norm norm, + double *val, double *err, int parallel) +{ + rule *r; + hypercube h; + int status; + unsigned i; + + if (fdim == 0) /* nothing to do */ return SUCCESS; + if (dim == 0) { /* trivial integration */ + if (f(0, 1, xmin, fdata, fdim, val)) return FAILURE; + for (i = 0; i < fdim; ++i) err[i] = 0; + return SUCCESS; + } + r = dim == 1 ? make_rule15gauss(dim, fdim) + : make_rule75genzmalik(dim, fdim); + if (!r) { + for (i = 0; i < fdim; ++i) { + val[i] = 0; + err[i] = HUGE_VAL; + } + return FAILURE; + } + h = make_hypercube_range(dim, xmin, xmax); + status = !h.data ? FAILURE + : rulecubature(r, fdim, f, fdata, &h, + maxEval, reqAbsError, reqRelError, norm, + val, err, parallel); + destroy_hypercube(&h); + destroy_rule(r); + return status; +} + +int hcubature_v(unsigned fdim, integrand_v f, void *fdata, + unsigned dim, const double *xmin, const double *xmax, + size_t maxEval, double reqAbsError, double reqRelError, + error_norm norm, + double *val, double *err) +{ + return cubature(fdim, f, fdata, dim, xmin, xmax, + maxEval, reqAbsError, reqRelError, norm, val, err, 1); +} + +#include "vwrapper.h" + +int hcubature(unsigned fdim, integrand f, void *fdata, + unsigned dim, const double *xmin, const double *xmax, + size_t maxEval, double reqAbsError, double reqRelError, + error_norm norm, + double *val, double *err) +{ + int ret; + fv_data d; + + if (fdim == 0) return SUCCESS; /* nothing to do */ + + d.f = f; d.fdata = fdata; + ret = cubature(fdim, fv, &d, dim, xmin, xmax, + maxEval, reqAbsError, reqRelError, norm, val, err, 0); + return ret; +} + +/***************************************************************************/ diff --git a/C/mintegration.c b/C/mintegration.c new file mode 100644 --- /dev/null +++ b/C/mintegration.c @@ -0,0 +1,28 @@ +#include "cubature.h" + +double mintegration( + char version, + int f(unsigned, const double*, void*, unsigned, double*), + unsigned dim, + double* xmin, + double* xmax, + double relError, + double* errorEstimate +) +{ + double value; + switch(version){ + case 'h': + hcubature( + 1, f, NULL, dim, xmin, xmax, 0, 0, + relError, ERROR_INDIVIDUAL, &value, errorEstimate + ); + break; + case 'p': + pcubature( + 1, f, NULL, dim, xmin, xmax, 0, 0, + relError, ERROR_INDIVIDUAL, &value, errorEstimate + ); + } + return value; +} diff --git a/C/pcubature.c b/C/pcubature.c new file mode 100644 --- /dev/null +++ b/C/pcubature.c @@ -0,0 +1,424 @@ +/* Adaptive multidimensional integration of a vector of integrands. + * + * Copyright (c) 2005-2013 Steven G. Johnson + * + * 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 2 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., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + * + */ + +/* p-adaptive cubature (adaptive by increasing the degree of the + cubature rule rather than subdividing the domain), using products + of Clenshaw-Curtis rules. This algorithm may be superior to + Genz-Malik for smooth integrands lacking strongly-localized + features, in moderate dimensions. */ + +#include +#include +#include + +#include "cubature.h" + +/* error return codes */ +#define SUCCESS 0 +#define FAILURE 1 + +/* pre-generated Clenshaw-Curtis rules and weights */ +#include "clencurt.h" + +/* no point in supporting very high dimensional integrals here */ +#define MAXDIM (20U) + +/***************************************************************************/ +/* For adaptive cubature, thanks to the nesting of the C-C rules, we + can re-use the values from coarser grids for finer grids, and the + coarser grids are also used for error estimation. + + A grid is determined by an m[dim] array, where m[i] denotes + 2^(m[i]+1)+1 points in the i-th dimension. +*/ + +/* cache of the values for the m[dim] grid. If mi < dim, then we only + store the values corresponding to the difference between the m grid + and the grid with m[mi] -> m[mi]-1. (m[mi]-1 == -1 corresponds to + the trivial grid of one point in the center.) */ +typedef struct cacheval_s { + unsigned m[MAXDIM]; + unsigned mi; + double *val; +} cacheval; + +/* array of ncache cachevals c[i] */ +typedef struct valcache_s { + size_t ncache; + cacheval *c; +} valcache; + +static void free_cachevals(valcache *v) +{ + if (!v) return; + if (v->c) { + size_t i; + for (i = 0; i < v->ncache; ++i) + free(v->c[i].val); + free(v->c); + v->c = NULL; + } + v->ncache = 0; +} + +/***************************************************************************/ + +/* recursive loop over all cubature points for the given (m,mi) cache entry: + add each point to the buffer buf, evaluating all at once whenever the + buffer is full or when we are done */ +static int compute_cacheval(const unsigned *m, unsigned mi, + double *val, size_t *vali, + unsigned fdim, integrand_v f, void *fdata, + unsigned dim, unsigned id, double *p, + const double *xmin, const double *xmax, + double *buf, size_t nbuf, size_t *ibuf) +{ + if (id == dim) { /* add point to buffer of points */ + memcpy(buf + (*ibuf)++ * dim, p, sizeof(double) * dim); + if (*ibuf == nbuf) { /* flush buffer */ + if (f(dim, nbuf, buf, fdata, fdim, val + *vali)) + return FAILURE; + *vali += *ibuf * fdim; + *ibuf = 0; + } + } + else { + double c = (xmin[id] + xmax[id]) * 0.5; + double r = (xmax[id] - xmin[id]) * 0.5; + const double *x = clencurt_x + + ((id == mi) ? (m[id] ? (1 << (m[id] - 1)) : 0) : 0); + unsigned i, nx = (id == mi ? (m[id] ? (1 << (m[id] - 1)) : 1) + : (1 << (m[id]))); + if (id != mi) { + p[id] = c; + if (compute_cacheval(m, mi, val, vali, fdim, f, fdata, + dim, id + 1, p, + xmin, xmax, buf, nbuf, ibuf)) + return FAILURE; + } + for (i = 0; i < nx; ++i) { + p[id] = c + r * x[i]; + if (compute_cacheval(m, mi, val, vali, fdim, f, fdata, + dim, id + 1, p, + xmin, xmax, buf, nbuf, ibuf)) + return FAILURE; + p[id] = c - r * x[i]; + if (compute_cacheval(m, mi, val, vali, fdim, f, fdata, + dim, id + 1, p, + xmin, xmax, buf, nbuf, ibuf)) + return FAILURE; + } + } + return SUCCESS; +} + +static size_t num_cacheval(const unsigned *m, unsigned mi, unsigned dim) +{ + unsigned i; + size_t nval = 1; + for (i = 0; i < dim; ++i) { + if (i == mi) + nval *= m[i] == 0 ? 2 : (1 << (m[i])); + else + nval *= (1 << (m[i] + 1)) + 1; + } + return nval; +} + +static int add_cacheval(valcache *vc, + const unsigned *m, unsigned mi, + unsigned fdim, integrand_v f, void *fdata, + unsigned dim, const double *xmin, const double *xmax, + double *buf, size_t nbuf) +{ + size_t ic = vc->ncache; + size_t nval, vali = 0, ibuf = 0; + double p[MAXDIM]; + + vc->c = (cacheval *) realloc(vc->c, sizeof(cacheval) * ++(vc->ncache)); + if (!vc->c) return -1; + + vc->c[ic].mi = mi; + memcpy(vc->c[ic].m, m, sizeof(unsigned) * dim); + nval = fdim * num_cacheval(m, mi, dim); + vc->c[ic].val = (double *) malloc(sizeof(double) * nval); + if (!vc->c[ic].val) return FAILURE; + + if (compute_cacheval(m, mi, vc->c[ic].val, &vali, + fdim, f, fdata, + dim, 0, p, xmin, xmax, + buf, nbuf, &ibuf)) + return FAILURE; + + if (ibuf > 0) /* flush remaining buffer */ + return f(dim, ibuf, buf, fdata, fdim, vc->c[ic].val + vali); + + return SUCCESS; +} + +/***************************************************************************/ + +/* recursive loop to evaluate the integral contribution from the cache + entry c, accumulating in val, for the given m[] except with m[md] + -> m[md] - 1 if md < dim, using the cached values (cm,cmi,cval). id is the + current loop dimension (from 0 to dim-1). */ +static unsigned eval(const unsigned *cm, unsigned cmi, double *cval, + const unsigned *m, unsigned md, + unsigned fdim, unsigned dim, unsigned id, + double weight, double *val) +{ + size_t voff = 0; /* amount caller should offset cval array afterwards */ + if (id == dim) { + unsigned i; + for (i = 0; i < fdim; ++i) val[i] += cval[i] * weight; + voff = fdim; + } + else if (m[id] == 0 && id == md) /* using trivial rule for this dim */ { + voff = eval(cm, cmi, cval, m, md, fdim, dim, id+1, weight*2, val); + voff += fdim * (1 << cm[id]) * 2 + * num_cacheval(cm + id+1, cmi - (id+1), dim - (id+1)); + } + else { + unsigned i; + unsigned mid = m[id] - (id == md); /* order of C-C rule */ + const double *w = clencurt_w + mid + (1 << mid) - 1 + + (id == cmi ? (cm[id] ? 1 + (1 << (cm[id]-1)) : 1) : 0); + unsigned cnx = (id == cmi ? (cm[id] ? (1 << (cm[id]-1)) : 1) + : (1 << (cm[id]))); + unsigned nx = cm[id] <= mid ? cnx : (1 << mid); + + if (id != cmi) { + voff = eval(cm, cmi, cval, m, md, fdim, dim, id + 1, + weight * w[0], val); + ++w; + } + for (i = 0; i < nx; ++i) { + voff += eval(cm, cmi, cval + voff, m, md, fdim, dim, id + 1, + weight * w[i], val); + voff += eval(cm, cmi, cval + voff, m, md, fdim, dim, id + 1, + weight * w[i], val); + } + + voff += (cnx - nx) * fdim * 2 + * num_cacheval(cm + id+1, cmi - (id+1), dim - (id+1)); + } + return voff; +} + +/* loop over all cache entries that contribute to the integral, + (with m[md] decremented by 1) */ +static void evals(valcache vc, const unsigned *m, unsigned md, + unsigned fdim, unsigned dim, + double V, double *val) +{ + size_t i; + + memset(val, 0, sizeof(double) * fdim); + for (i = 0; i < vc.ncache; ++i) { + if (vc.c[i].mi >= dim || + vc.c[i].m[vc.c[i].mi] + (vc.c[i].mi == md) <= m[vc.c[i].mi]) + eval(vc.c[i].m, vc.c[i].mi, vc.c[i].val, + m, md, fdim, dim, 0, V, val); + } +} + +/* evaluate the integrals for the given m[] using the cached values in vc, + storing the integrals in val[], the error estimate in err[], and the + dimension to subdivide next (the largest error contribution) in *mi */ +static void eval_integral(valcache vc, const unsigned *m, + unsigned fdim, unsigned dim, double V, + unsigned *mi, double *val, double *err, double *val1) +{ + double maxerr = 0; + unsigned i, j; + + evals(vc, m, dim, fdim, dim, V, val); + + /* error estimates along each dimension by comparing val with + lower-order rule in that dimension; overall (conservative) + error estimate from maximum error of lower-order rules. */ + memset(err, 0, sizeof(double) * fdim); + *mi = 0; + for (i = 0; i < dim; ++i) { + double emax = 0; + evals(vc, m, i, fdim, dim, V, val1); + for (j = 0; j < fdim; ++j) { + double e = fabs(val[j] - val1[j]); + if (e > emax) emax = e; + if (e > err[j]) err[j] = e; + } + if (emax > maxerr) { + maxerr = emax; + *mi = i; + } + } + /* printf("eval: %g +/- %g (dim %u)\n", val[0], err[0], *mi); */ +} + +/***************************************************************************/ + +static int converged(unsigned fdim, const double *vals, const double *errs, + double reqAbsError, double reqRelError, error_norm norm) +#define ERR(j) errs[j] +#define VAL(j) vals[j] +#include "converged.h" + +/***************************************************************************/ +/* Vectorized version with user-supplied buffer to store points and values. + The buffer *buf should be of length *nbuf * dim on entry (these parameters + are changed upon return to the final buffer and length that was used). + The buffer length will be kept <= max(max_nbuf, 1) * dim. + + Also allows the caller to specify an array m[dim] of starting degrees + for the rule, which upon return will hold the final degrees. The + number of points in each dimension i is 2^(m[i]+1) + 1. */ + +int pcubature_v_buf(unsigned fdim, integrand_v f, void *fdata, + unsigned dim, const double *xmin, const double *xmax, + size_t maxEval, + double reqAbsError, double reqRelError, + error_norm norm, + unsigned *m, + double **buf, size_t *nbuf, size_t max_nbuf, + double *val, double *err) +{ + int ret = FAILURE; + double V = 1; + size_t numEval = 0, new_nbuf; + unsigned i; + valcache vc = {0, NULL}; + double *val1 = NULL; + + if (fdim <= 1) norm = ERROR_INDIVIDUAL; /* norm is irrelevant */ + if (norm < 0 || norm > ERROR_LINF) return FAILURE; /* invalid norm */ + + if (fdim == 0) return SUCCESS; /* nothing to do */ + if (dim > MAXDIM) return FAILURE; /* unsupported */ + if (dim == 0) { /* trivial case */ + if (f(0, 1, xmin, fdata, fdim, val)) return FAILURE; + for (i = 0; i < fdim; ++i) err[i] = 0; + return SUCCESS; + } + + for (i = 0; i < fdim; ++i) { + val[i] = 0; + err[i] = HUGE_VAL; + } + + for (i = 0; i < dim; ++i) + V *= (xmax[i] - xmin[i]) * 0.5; /* scale factor for C-C volume */ + + new_nbuf = num_cacheval(m, dim, dim); + + if (max_nbuf < 1) max_nbuf = 1; + if (new_nbuf > max_nbuf) new_nbuf = max_nbuf; + if (*nbuf < new_nbuf) { + free(*buf); + *buf = (double *) malloc(sizeof(double) + * (*nbuf = new_nbuf) * dim); + if (!*buf) goto done; + } + + /* start by evaluating the m=0 cubature rule */ + if (add_cacheval(&vc, m, dim, fdim, f, fdata, dim, xmin, xmax, + *buf, *nbuf) != SUCCESS) + goto done; + + val1 = (double *) malloc(sizeof(double) * fdim); + + while (1) { + unsigned mi; + + eval_integral(vc, m, fdim, dim, V, &mi, val, err, val1); + if (converged(fdim, val, err, reqAbsError, reqRelError, norm) + || (numEval > maxEval && maxEval)) { + ret = SUCCESS; + goto done; + } + m[mi] += 1; + if (m[mi] > clencurt_M) goto done; /* FAILURE */ + + new_nbuf = num_cacheval(m, mi, dim); + if (new_nbuf > *nbuf && *nbuf < max_nbuf) { + *nbuf = new_nbuf; + if (*nbuf > max_nbuf) *nbuf = max_nbuf; + free(*buf); + *buf = (double *) malloc(sizeof(double) * *nbuf * dim); + if (!*buf) goto done; /* FAILURE */ + } + + if (add_cacheval(&vc, m, mi, fdim, f, fdata, + dim, xmin, xmax, *buf, *nbuf) != SUCCESS) + goto done; /* FAILURE */ + numEval += new_nbuf; + } + +done: + free(val1); + free_cachevals(&vc); + return ret; +} + +/***************************************************************************/ + +#define DEFAULT_MAX_NBUF (1U << 20) + +int pcubature_v(unsigned fdim, integrand_v f, void *fdata, + unsigned dim, const double *xmin, const double *xmax, + size_t maxEval, double reqAbsError, double reqRelError, + error_norm norm, + double *val, double *err) +{ + int ret; + size_t nbuf = 0; + unsigned m[MAXDIM]; + double *buf = NULL; + memset(m, 0, sizeof(unsigned) * dim); + ret = pcubature_v_buf(fdim, f, fdata, dim, xmin, xmax, + maxEval, reqAbsError, reqRelError, norm, + m, &buf, &nbuf, DEFAULT_MAX_NBUF, val, err); + free(buf); + return ret; +} + +#include "vwrapper.h" + +int pcubature(unsigned fdim, integrand f, void *fdata, + unsigned dim, const double *xmin, const double *xmax, + size_t maxEval, double reqAbsError, double reqRelError, + error_norm norm, + double *val, double *err) +{ + int ret; + size_t nbuf = 0; + unsigned m[MAXDIM]; + double *buf = NULL; + fv_data d; + + d.f = f; d.fdata = fdata; + memset(m, 0, sizeof(unsigned) * dim); + ret = pcubature_v_buf( + fdim, fv, &d, dim, xmin, xmax, + maxEval, reqAbsError, reqRelError, norm, + m, &buf, &nbuf, 16 /* max_nbuf > 0 to amortize function overhead */, + val, err); + free(buf); + return ret; +} diff --git a/C/vwrapper.h b/C/vwrapper.h new file mode 100644 --- /dev/null +++ b/C/vwrapper.h @@ -0,0 +1,16 @@ +/* vectorized wrapper around non-vectorized integrands */ +typedef struct fv_data_s { integrand f; void *fdata; } fv_data; +static int fv(unsigned ndim, size_t npt, + const double *x, void *d_, + unsigned fdim, double *fval) +{ + fv_data *d = (fv_data *) d_; + integrand f = d->f; + void *fdata = d->fdata; + unsigned i; + /* printf("npt = %u\n", npt); */ + for (i = 0; i < npt; ++i) + if (f(ndim, x + i*ndim, fdata, fdim, fval + i*fdim)) + return FAILURE; + return SUCCESS; +} diff --git a/CHANGELOG.md b/CHANGELOG.md new file mode 100644 --- /dev/null +++ b/CHANGELOG.md @@ -0,0 +1,5 @@ +# Changelog for `adaptive-cubature` + +## 0.1.0.0 - 2023-09-15 + +First release. diff --git a/LICENSE b/LICENSE new file mode 100644 --- /dev/null +++ b/LICENSE @@ -0,0 +1,674 @@ + GNU GENERAL PUBLIC LICENSE + Version 3, 29 June 2007 + + Copyright (C) 2007 Free Software Foundation, Inc. + Everyone is permitted to copy and distribute verbatim copies + of this license document, but changing it is not allowed. + + Preamble + + The GNU General Public License is a free, copyleft license for +software and other kinds of works. + + The licenses for most software and other practical works are designed +to take away your freedom to share and change the works. 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But first, please read +. diff --git a/README.md b/README.md new file mode 100644 --- /dev/null +++ b/README.md @@ -0,0 +1,31 @@ +# adaptive-cubature + +*Adaptive integration of a multivariate function on an axis-aligned hyperrectangle.* + +___ + +This package is powered by the C library [cubature](https://github.com/stevengj/cubature). +Follow the link for details. + +### Usage + +```haskell +cubature :: Char -- ^ cubature version, 'h' or 'p' + -> ([Double] -> Double) -- ^ integrand + -> Int -- ^ dimension (number of variables) + -> [Double] -- ^ lower limits of integration + -> [Double] -- ^ upper limits of integration + -> Double -- ^ desired relative error + -> IO Result -- ^ output: integral value and error estimate +``` + +### Example + +```haskell +fExample :: [Double] -> Double +fExample x = exp (-0.5 * (sum $ zipWith (*) x x)) + +example :: IO Result -- should give 2pi ≈ 6.283185307179586 +example = cubature 'h' fExample 2 [-6, -6] [6, 6] 1e-10 +-- Result {_integral = 6.283185282383672, _error = 6.280185128024888e-10} +``` diff --git a/Setup.hs b/Setup.hs new file mode 100644 --- /dev/null +++ b/Setup.hs @@ -0,0 +1,2 @@ +import Distribution.Simple +main = defaultMain diff --git a/adaptive-cubature.cabal b/adaptive-cubature.cabal new file mode 100644 --- /dev/null +++ b/adaptive-cubature.cabal @@ -0,0 +1,43 @@ +cabal-version: 2.2 +name: adaptive-cubature +version: 0.1.0.0 +synopsis: Multidimensional integration +description: Adaptive integration of a multivariate function on a hyperrectangle. +homepage: https://github.com/stla/adaptive-cubature#readme +license: GPL-3.0-only +license-file: LICENSE +author: Stéphane Laurent +maintainer: laurent_step@outlook.fr +copyright: 2023 Stéphane Laurent +category: Numerical +build-type: Simple +extra-source-files: README.md + CHANGELOG.md + +library + hs-source-dirs: src + exposed-modules: Numerical.Cubature + build-depends: base >= 4.7 && < 5 + other-extensions: ForeignFunctionInterface + include-dirs: C + C-sources: C/mintegration.c + , C/hcubature.c + , C/pcubature.c + install-includes: C/cubature.h + , C/converged.h + , C/clencurt.h + , C/vwrapper.h + default-language: Haskell2010 + ghc-options: -Wall + -Wcompat + -Widentities + -Wincomplete-record-updates + -Wincomplete-uni-patterns + -Wmissing-export-lists + -Wmissing-home-modules + -Wpartial-fields + -Wredundant-constraints + +source-repository head + type: git + location: https://github.com/stla/adaptive-cubature diff --git a/src/Numerical/Cubature.hs b/src/Numerical/Cubature.hs new file mode 100644 --- /dev/null +++ b/src/Numerical/Cubature.hs @@ -0,0 +1,63 @@ +{-# LANGUAGE ForeignFunctionInterface #-} +module Numerical.Cubature + (cubature) + where +import Foreign.C.Types (CUInt(..)) +import Foreign.Marshal.Alloc (free, mallocBytes) +import Foreign.Marshal.Array (peekArray, pokeArray) +import Foreign.Ptr (FunPtr, Ptr, freeHaskellFunPtr) +import Foreign.Storable (poke, peek, sizeOf) + +type Integrand = CUInt -> Ptr Double -> Ptr () -> CUInt -> Ptr Double -> IO Int + +data Result = Result + { _integral :: Double, _error :: Double } + deriving (Show) + +foreign import ccall safe "wrapper" integrandPtr + :: Integrand -> IO (FunPtr Integrand) + +foreign import ccall safe "mintegration" c_cubature + :: Char + -> FunPtr Integrand + -> Int + -> Ptr Double + -> Ptr Double + -> Double + -> Ptr Double + -> IO Double + +fun2integrand :: ([Double] -> Double) -> Int -> Integrand +fun2integrand f n _ x _ _ fval = do + list <- peekArray n x + poke fval (f list) + return 0 + +-- | Multivariate integration on an axis-aligned box. +cubature :: Char -- ^ cubature version, 'h' or 'p' + -> ([Double] -> Double) -- ^ integrand + -> Int -- ^ dimension (number of variables) + -> [Double] -- ^ lower limits of integration + -> [Double] -- ^ upper limits of integration + -> Double -- ^ desired relative error + -> IO Result -- ^ output: integral value and error estimate +cubature version f n xmin xmax relError = do + fPtr <- integrandPtr (fun2integrand f n) + xminPtr <- mallocBytes (n * sizeOf (0.0 :: Double)) + pokeArray xminPtr xmin + xmaxPtr <- mallocBytes (n * sizeOf (0.0 :: Double)) + pokeArray xmaxPtr xmax + errorPtr <- mallocBytes (sizeOf (0.0 :: Double)) + result <- c_cubature version fPtr n xminPtr xmaxPtr relError errorPtr + errorEstimate <- peek errorPtr + free errorPtr + free xmaxPtr + free xminPtr + freeHaskellFunPtr fPtr + return Result { _integral = result, _error = errorEstimate } + +-- fExample :: [Double] -> Double +-- fExample list = exp (-0.5 * (sum $ zipWith (*) list list)) + +-- example :: IO Result +-- example = cubature 'h' fExample 2 [-6,-6] [6,6] 1e-10