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- Log ----------------------------------------------------------------- commit 6ab5226ca3e32b443d87934ec138ff0efc8aaecc Author: Raymond Toy toy.raymond@gmail.com Date: Wed Apr 11 20:32:23 2012 -0700

First cut at Bessel Y. Not working yet.

diff --git a/qd-bessel.lisp b/qd-bessel.lisp index e7ca034..588b5af 100644 --- a/qd-bessel.lisp +++ b/qd-bessel.lisp @@ -376,6 +376,7 @@ (format t " term = ~S~%" term) (format t " sum = ~S~%" sum))))))

+;; ;; TODO: ;; o For |z| <= 1 use the series. ;; o Currently accuracy is not good for large z and half-integer @@ -402,6 +403,41 @@ (+ (/ -1 z) (sum-ab big-n v z) (sum-big-ia big-n v z))))))))) + +;; Bessel Y +;; +;; bessel_y(v, z) = 1/(2*%pi*%i)*(exp(-%i*v*%pi/2)*I(%i*v,z) - exp(%i*v*%pi/2)*I(-%i*z, v)) +;; + z/v/%pi*((1-cos(v*%pi)/z) + S(N,z,v)*cos(v*%pi)-S(N,z,-v)) +;; +;; where +;; +;; S(N,z,v) = sum(alpha[n](z)*a[n](0,v) + beta[n](z)*sum(exp(-k*z)*a[n](k,v),k,1,N),n,0,inf) +;; + sum(A[n](v)*I[n](N+1/2,z,v),n,0,inf) +;; +(defun bessel-y (v z) + (flet ((ipart (v z) + (let* ((iz (* #c(0 1) z)) + (c+ (exp (* v (float-pi z) 1/2))) + (c- (exp (* v (float-pi z) -1/2))) + (i+ (exp-arc-i-2 iz v)) + (i- (exp-arc-i-2 (- iz) v))) + (/ (- (* c- i+) (* c+ i-)) + (* #c(0 2) (float-pi z))))) + (s (big-n z v) + (+ (sum-ab big-n v z) + (sum-big-ia big-n v z)))) + (let* ((big-n 100) + (vpi (* v (float-pi z))) + (c (cos vpi))) + (+ (ipart v z) + (* (/ z vpi) + (+ (/ (- 1 c) + z) + (* c + (s big-n z v)) + (- (s big-n z (- v))))))))) + +  (defun paris-series (v z n) (labels ((pochhammer (a k)

commit d795ba718dc53f591c82994811f50250aceec1d7 Author: Raymond Toy toy.raymond@gmail.com Date: Wed Apr 11 20:15:47 2012 -0700

Add TODO list for bessel-j.

diff --git a/qd-bessel.lisp b/qd-bessel.lisp index 3958ff2..e7ca034 100644 --- a/qd-bessel.lisp +++ b/qd-bessel.lisp @@ -375,13 +375,25 @@ (format t " f = ~S~%" f) (format t " term = ~S~%" term) (format t " sum = ~S~%" sum)))))) - + +;; TODO: +;; o For |z| <= 1 use the series. +;; o Currently accuracy is not good for large z and half-integer +;; order. +;; o For real v and z, return a real number instead of complex. +;; o Handle the case of Re(z) < 0. (The formulas are for Re(z) > 0: +;; bessel_j(v,z*exp(m*%pi*%i)) = exp(m*v*%pi*%i)*bessel_j(v, z) +;; o The paper suggests using +;; bessel_i(v,z) = exp(-v*%pi*%i/2)*bessel_j(v, %i*z) +;; when Im(z) >> Re(z) +;; (defun bessel-j (v z) (let ((vv (ftruncate v))) (cond ((= vv v) ;; v is an integer (integer-bessel-j-exp-arc v z)) (t + ;; Need to fine-tune the value of big-n. (let ((big-n 100) (vpi (* v (float-pi (realpart z))))) (+ (integer-bessel-j-exp-arc v z)

commit c0e12ddf6b61f571555d46c6f168e6bebabc80b1 Author: Raymond Toy toy.raymond@gmail.com Date: Wed Apr 11 20:06:38 2012 -0700

Correct the computation of alpha and beta.

diff --git a/qd-bessel.lisp b/qd-bessel.lisp index 0197e7f..3958ff2 100644 --- a/qd-bessel.lisp +++ b/qd-bessel.lisp @@ -213,13 +213,13 @@

(defun alpha (n z) (let ((n (float n (realpart z)))) - (/ (cf-incomplete-gamma (1+ n) (/ z 2)) + (/ (incomplete-gamma (1+ n) (/ z 2)) (expt z (1+ n)))))

(defun beta (n z) (let ((n (float n (realpart z)))) - (/ (- (cf-incomplete-gamma (1+ n) (/ z 2)) - (cf-incomplete-gamma (1+ n) (/ z -2))) + (/ (- (incomplete-gamma (1+ n) (/ z 2)) + (incomplete-gamma (1+ n) (/ z -2))) (expt z (1+ n)))))

;; a[0](k,v) := (k+sqrt(k^2+1))^(-v);

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Summary of changes: qd-bessel.lisp | 56 ++++++++++++++++++++++++++++++++++++++++++++++++++++---- 1 files changed, 52 insertions(+), 4 deletions(-)

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