source: qd-bessel.lisp @ 1d404a

Last change on this file since 1d404a was 1d404a, checked in by Raymond Toy <toy.raymond@…>, 3 years ago

Add some comments, rename bessel-j-exp-arc to integer-bessel-j-exp-arc.

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File size: 6.6 KB
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1;;;; -*- Mode: lisp -*-
2;;;;
3;;;; Copyright (c) 2011 Raymond Toy
4;;;; Permission is hereby granted, free of charge, to any person
5;;;; obtaining a copy of this software and associated documentation
6;;;; files (the "Software"), to deal in the Software without
7;;;; restriction, including without limitation the rights to use,
8;;;; copy, modify, merge, publish, distribute, sublicense, and/or sell
9;;;; copies of the Software, and to permit persons to whom the
10;;;; Software is furnished to do so, subject to the following
11;;;; conditions:
12;;;;
13;;;; The above copyright notice and this permission notice shall be
14;;;; included in all copies or substantial portions of the Software.
15;;;;
16;;;; THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
17;;;; EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
18;;;; OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
19;;;; NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
20;;;; HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
21;;;; WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
22;;;; FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
23;;;; OTHER DEALINGS IN THE SOFTWARE.
24
25(in-package #:oct)
26
27;;; References:
28;;;
29;;; [1] Borwein, Borwein, Crandall, "Effective Laguerre Asymptotics",
30;;; http://people.reed.edu/~crandall/papers/Laguerre-f.pdf
31;;;
32;;; [2] Borwein, Borwein, Chan, "The Evaluation of Bessel Functions
33;;; via Exp-Arc Integrals", http://web.cs.dal.ca/~jborwein/bessel.pdf
34;;;
35
36(defvar *debug-exparc* nil)
37
38;; B[k](p) = 1/2^(k+3/2)*integrate(exp(-p*u)*u^(k-1/2),u,0,1)
39;;         = 1/2^(k+3/2)/p^(k+1/2)*integrate(t^(k-1/2)*exp(-t),t,0,p)
40;;         = 1/2^(k+3/2)/p^(k+1/2) * g(k+1/2, p)
41;;
42;; where g(a,z) is the lower incomplete gamma function.
43;;
44;; There is the continued fraction expansion for g(a,z) (see
45;; cf-incomplete-gamma in qd-gamma.lisp):
46;;
47;;  g(a,z) = z^a*exp(-z)/ CF
48;;
49;; So
50;;
51;;  B[k](p) = 1/2^(k+3/2)/p^(k+1/2)*p^(k+1/2)*exp(-p)/CF
52;;          = exp(-p)/2^(k+3/2)/CF
53;;
54;;
55;; Note also that [2] gives a recurrence relationship for B[k](p) in
56;; eq (2.6), but there is an error there.  The correct relationship is
57;;
58;;  B[k](p) = -exp(-p)/(p*sqrt(2)*2^(k+1)) + (k-1/2)*B[k-1](p)/(2*p)
59;;
60;; The paper is missing the division by p in the term containing
61;; B[k-1](p).  This is easily derived from the recurrence relationship
62;; for the (lower) incomplete gamma function.
63;;
64;; Note too that as k increases, the recurrence appears to be unstable
65;; and B[k](p) begins to increase even though it is strictly bounded.
66;; (This is also easy to see from the integral.)  Hence, we do not use
67;; the recursion.  However, it might be stable for use with
68;; double-float precision; this has not been tested.
69;;
70(defun bk (k p)
71  (/ (exp (- p))
72     (* (sqrt (float 2 (realpart p))) (ash 1 (+ k 1)))
73     (let ((a (float (+ k 1/2) (realpart p))))
74       (lentz #'(lambda (n)
75                  (+ n a))
76              #'(lambda (n)
77                  (if (evenp n)
78                      (* (ash n -1) p)
79                      (- (* (+ a (ash n -1)) p))))))))
80
81;; exp-arc I function, as given in the Laguerre paper
82;;
83;; I(p, q) = 4*exp(p) * sum(g[k](-2*%i*q)/(2*k)!*B[k](p), k, 0, inf)
84;;
85;; where g[k](p) = product(p^2+(2*j-1)^2, j, 1, k) and B[k](p) as above.
86;;
87;; For computation, note that g[k](p) = g[k-1](p) * (p^2 + (2*k-1)^2)
88;; and (2*k)! = (2*k-2)! * (2*k-1) * (2*k).  Then, let
89;;
90;;  R[k](p) = g[k](p)/(2*k)!
91;;
92;; Then
93;;
94;;  R[k](p) = g[k](p)/(2*k)!
95;;          = g[k-1](p)/(2*k-2)! * (p^2 + (2*k-1)^2)/((2*k-1)*(2*k)
96;;          = R[k-1](p) * (p^2 + (2*k-1)^2)/((2*k-1)*(2*k)
97;;
98;; In the exp-arc paper, the function is defined (equivalently) as
99;;
100;; I(p, q) = 2*%i*exp(p)/q * sum(r[2*k+1](-2*%i*q)/(2*k)!*B[k](p), k, 0, inf)
101;;
102;; where r[2*k+1](p) = p*product(p^2 + (2*j-1)^2, j, 1, k)
103;;
104;; Let's note some properties of I(p, q).
105;;
106;; I(-%i*z, v) = 2*%i*exp(-%i*z)/q * sum(r[2*k+1](-2*%i*v)/(2*k)!*B[k](-%i*z))
107;;
108;; Note thate B[k](-%i*z) = 1/2^(k+3/2)*integrate(exp(%i*z*u)*u^(k-1/2),u,0,1)
109;;                        = conj(B[k](%i*z).
110;;
111;; Hence I(-%i*z, v) = conj(I(%i*z, v)) when both z and v are real.
112(defun exp-arc-i (p q)
113  (let* ((sqrt2 (sqrt (float 2 (realpart p))))
114         (exp/p/sqrt2 (/ (exp (- p)) p sqrt2))
115         (v (* #c(0 -2) q))
116         (v2 (expt v 2))
117         (eps (epsilon (realpart p))))
118    (when *debug-exparc*
119      (format t "sqrt2 = ~S~%" sqrt2)
120      (format t "exp/p/sqrt2 = ~S~%" exp/p/sqrt2))
121    (do* ((k 0 (1+ k))
122          (bk (/ (incomplete-gamma 1/2 p)
123                 2 sqrt2 (sqrt p))
124              (- (/ (* bk (- k 1/2)) 2 p)
125                 (/ exp/p/sqrt2 (ash 1 (+ k 1)))))
126          ;; ratio[k] = r[2*k+1](v)/(2*k)!.
127          ;; r[1] = v and r[2*k+1](v) = r[2*k-1](v)*(v^2 + (2*k-1)^2)
128          ;; ratio[0] = v
129          ;; and ratio[k] = r[2*k-1](v)*(v^2+(2*k-1)^2) / ((2*k-2)! * (2*k-1) * 2*k)
130          ;;              = ratio[k]*(v^2+(2*k-1)^2)/((2*k-1) * 2 * k)
131          (ratio v
132                 (* ratio (/ (+ v2 (expt (1- (* 2 k)) 2))
133                             (* 2 k (1- (* 2 k))))))
134          (term (* ratio bk)
135                (* ratio bk))
136          (sum term (+ sum term)))
137         ((< (abs term) (* (abs sum) eps))
138          (* sum #c(0 2) (/ (exp p) q)))
139      (when *debug-exparc*
140        (format t "k      = ~D~%" k)
141        (format t " bk    = ~S~%" bk)
142        (format t " ratio = ~S~%" ratio)
143        (format t " term  = ~S~%" term)
144        (format t " sum   - ~S~%" sum)))))
145
146(defun exp-arc-i-2 (p q)
147  (let* ((sqrt2 (sqrt (float 2 (realpart p))))
148         (exp/p/sqrt2 (/ (exp (- p)) p sqrt2))
149         (v (* #c(0 -2) q))
150         (v2 (expt v 2))
151         (eps (epsilon (realpart p))))
152    (when *debug-exparc*
153      (format t "sqrt2 = ~S~%" sqrt2)
154      (format t "exp/p/sqrt2 = ~S~%" exp/p/sqrt2))
155    (do* ((k 0 (1+ k))
156          (bk (bk 0 p)
157              (bk k p))
158          (ratio v
159                 (* ratio (/ (+ v2 (expt (1- (* 2 k)) 2))
160                             (* 2 k (1- (* 2 k))))))
161          (term (* ratio bk)
162                (* ratio bk))
163          (sum term (+ sum term)))
164         ((< (abs term) (* (abs sum) eps))
165          (* sum #c(0 2) (/ (exp p) q)))
166      (when *debug-exparc*
167        (format t "k      = ~D~%" k)
168        (format t " bk    = ~S~%" bk)
169        (format t " ratio = ~S~%" ratio)
170        (format t " term  = ~S~%" term)
171        (format t " sum   - ~S~%" sum)))))
172
173
174;; This currently only works for v an integer.
175;;
176(defun integer-bessel-j-exp-arc (v z)
177  (let* ((iz (* #c(0 1) z))
178         (i+ (exp-arc-i-2 iz v))
179         (i- (exp-arc-i-2 (- iz ) v)))
180    (/ (+ (* (cis (* v (float-pi i+) -1/2))
181             i+)
182          (* (cis (* v (float-pi i+) 1/2))
183             i-))
184       (float-pi i+)
185       2)))
186
187(defun paris-series (v z n)
188  (labels ((pochhammer (a k)
189             (/ (gamma (+ a k))
190                (gamma a)))
191           (a (v k)
192             (* (/ (pochhammer (+ 1/2 v) k)
193                   (gamma (float (1+ k) z)))
194                (pochhammer (- 1/2 v) k))))
195    (* (loop for k from 0 below n
196             sum (* (/ (a v k)
197                       (expt (* 2 z) k))
198                    (/ (cf-incomplete-gamma (+ k v 1/2) (* 2 z))
199                       (gamma (+ k v 1/2)))))
200       (/ (exp z)
201          (sqrt (* 2 (float-pi z) z))))))
202
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