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;;; nsmc --- n-sphere Monte Carlo method
;;; Copyright © 2021 Arun I <arunisaac@systemreboot.net>
;;; Copyright © 2021 Murugesan Venkatapathi <murugesh@iisc.ac.in>
;;;
;;; This file is part of nsmc.
;;;
;;; nsmc 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.
;;;
;;; nsmc 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 nsmc. If not, see <https://www.gnu.org/licenses/>.
(pre-include "math.h")
(pre-include "gsl/gsl_blas.h")
(pre-include "gsl/gsl_randist.h")
(pre-include "gsl/gsl_sf_gamma.h")
(pre-include "gsl/gsl_vector.h")
(pre-include "utils.h")
(define (random-direction-vector r x)
(void (const gsl-rng*) gsl-vector*)
"Generate a random vector distributed uniformly on the unit
sphere. Write the result to X."
(gsl-ran-dir-nd r (: x size) (: x data)))
(define (rotate-from-nth-canonical x orient)
((static void) gsl-vector* (const gsl-vector*))
(let* ((n size-t (: x size))
(xn double (gsl-vector-get x (- n 1)))
(mun double (gsl-vector-get orient (- n 1)))
(orient-sub gsl-vector-const-view
(gsl-vector-const-subvector orient 0 (- n 1)))
(b double (gsl-blas-dnrm2 (address-of (struct-get orient-sub vector))))
(a double (/ (- (dot-product orient x)
(* xn mun))
b))
(s double (sqrt (- 1 (gsl-pow-2 mun)))))
(gsl-blas-daxpy (/ (+ (* xn s)
(* a (- mun 1)))
b)
orient
x)
(gsl-vector-set x
(- n 1)
(+ (gsl-vector-get x (- n 1))
(* xn (- mun 1))
(- (* a s))
(- (/ (* mun (+ (* xn s)
(* a (- mun 1))))
b))))))
(define (beta-inc-unnormalized a b x) ((static double) double double double)
(return (* (gsl-sf-beta-inc a b x)
(gsl-sf-beta a b))))
(define (incomplete-wallis-integral theta m) ((static double) double (unsigned int))
"Return the incomplete Wallis integral \\int_0^\\theta \\sin^m x
dx. THETA should be in [0,pi]."
(cond
((or (< theta 0) (> theta M-PI))
(GSL-ERROR "Incomplete Wallis integral only allows theta in [0,pi]" GSL-EDOM))
((< theta M-PI-2)
(return (/ (beta-inc-unnormalized (* 0.5 (+ m 1)) 0.5 (gsl-pow-2 (sin theta)))
2)))
(else
(return (/ (+ (gsl-sf-beta (* 0.5 (+ m 1)) 0.5)
(beta-inc-unnormalized 0.5 (* 0.5 (+ m 1)) (gsl-pow-2 (cos theta))))
2)))))
(define (planar-angle->solid-angle planar-angle dimension) (double double (unsigned int))
(return (/ (* 2
(pow M-PI (* 0.5 (- dimension 1)))
(incomplete-wallis-integral planar-angle (- dimension 2)))
(gsl-sf-gamma (* 0.5 (- dimension 1))))))
(define (solid-angle->planar-angle solid-angle dimension) (double double (unsigned int))
(define (f planar-angle params) (double double void*)
(return (- (planar-angle->solid-angle planar-angle dimension)
solid-angle)))
(declare gsl-f (struct-variable gsl-function (function (address-of f))))
;; TODO: Equality comparisons to floating point values may be
;; problematic. Fix it.
(cond
((= solid-angle 0) (return 0))
((= solid-angle (surface-area-of-ball dimension)) (return M-PI))
(else (return (bisection (address-of gsl-f) 0 M-PI)))))
;; TODO: There is an edge case when mean is the (n-1)th canonical
;; basis vector. Fix it.
(define (hollow-cone-random-vector r mean theta-min theta-max x)
(void (const gsl-rng*) (const gsl-vector*) double double gsl-vector*)
;; Generate random vector around the nth canonical basis vector.
(let* ((n size-t (: x size)))
(gsl-ran-dir-nd r (- n 1) (: x data))
(gsl-vector-scale x (* (cos theta-max)
(tan (solid-angle->planar-angle
(gsl-ran-flat r
(planar-angle->solid-angle theta-min n)
(planar-angle->solid-angle theta-max n))
n))))
(gsl-vector-set x (- n 1) (cos theta-max)))
(gsl-vector-scale x (/ 1 (gsl-blas-dnrm2 x)))
;; Rotate to arbitrary basis.
(rotate-from-nth-canonical x mean))
(define (cone-random-vector r mean theta-max x)
(void (const gsl-rng*) (const gsl-vector*) double gsl-vector*)
(hollow-cone-random-vector r mean 0 theta-max x))
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