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- Log ----------------------------------------------------------------- commit 64c424b7f41413424bbab93d4f5dd4f8461b784d Author: Raymond Toy toy.raymond@gmail.com Date: Mon Apr 16 20:45:12 2012 -0700

Add iterative versions for some functions.

diff --git a/qd-bessel.lisp b/qd-bessel.lisp index 14743af..37a4c8e 100644 --- a/qd-bessel.lisp +++ b/qd-bessel.lisp @@ -78,6 +78,21 @@ (* (ash n -1) p) (- (* (+ a (ash n -1)) p))))))))

+;; Use the recursion +(defun bk-iter (k p old-bk) + (with-floating-point-contagion (p old-bk) + (if (zerop k) + (* (sqrt (/ (float-pi p) 8)) + (let ((rp (sqrt p))) + (/ (erf rp) + rp))) + (- (* (- k 1/2) + (/ old-bk (* 2 p))) + (/ (exp (- p)) + p + (ash 1 (+ k 1)) + (sqrt (float 2 (realpart p)))))))) + ;; exp-arc I function, as given in the Laguerre paper ;; ;; I(p, q) = 4*exp(p) * sum(g[k](-2*%i*q)/(2*k)!*B[k](p), k, 0, inf) @@ -178,6 +193,35 @@ (format t " term = ~S~%" term) (format t " sum - ~S~%" sum)))))

+(defun exp-arc-i-3 (p q) + (let* ((v (* #c(0 -2) q)) + (v2 (expt v 2)) + (eps (epsilon (realpart p)))) + (do* ((k 0 (1+ k)) + (bk (bk 0 p) + (bk-iter k p bk)) + ;; Compute g[k](p)/(2*k)!, not r[2*k+1](p)/(2*k)! + (ratio 1 + (* ratio (/ (+ v2 (expt (1- (* 2 k)) 2)) + (* 2 k (1- (* 2 k)))))) + (term (* ratio bk) + (* ratio bk)) + (sum term (+ sum term))) + ((< (abs term) (* (abs sum) eps)) + (when *debug-exparc* + (format t "Final k= ~D~%" k) + (format t " bk = ~S~%" bk) + (format t " ratio = ~S~%" ratio) + (format t " term = ~S~%" term) + (format t " sum - ~S~%" sum)) + (* sum 4 (exp p))) + (when *debug-exparc* + (format t "k = ~D~%" k) + (format t " bk = ~S~%" bk) + (format t " ratio = ~S~%" ratio) + (format t " term = ~S~%" term) + (format t " sum - ~S~%" sum))))) +

;; Not really just for Bessel J for integer orders, but in that case, ;; this is all that's needed to compute Bessel J. For other values, @@ -212,6 +256,7 @@ ;; ;; alpha[n](z) = - exp(-z/2)/2^n/z + n/z*alpha[n-1](z) ;; beta[n]z) = ((-1)^n*exp(z/2)-exp(-z/2))/2^n/z + n/z*beta[n-1](z) +;; = (-1)^n/(2^n)*2*sinh(z/2)/z + n/z*beta[n-1](z) ;; ;; We also note that ;; @@ -223,12 +268,34 @@ (/ (incomplete-gamma (1+ n) (/ z 2)) (expt z (1+ n)))))

+(defun alpha-iter (n z alpha-old) + (if (zerop n) + ;; (1- exp(-z/2))/z. + (/ (- 1 (exp (* z -1/2))) + z) + (- (* (/ n z) alpha-old) + (/ (exp (- (* z 1/2))) + z + (ash 1 n))))) + (defun beta (n z) (let ((n (float n (realpart z)))) (/ (- (incomplete-gamma (1+ n) (/ z 2)) (incomplete-gamma (1+ n) (/ z -2))) (expt z (1+ n)))))

+(defun beta-iter (n z old-beta) + (if (zerop n) + ;; integrate(exp(-z*s),s,-1/2,1/2) + ;; = (exp(z/2)-exp(-z/2)/z + ;; = 2*sinh(z/2)/z + ;; = sinh(z/2)/(z/2) + (* 2 (/ (sinh (* 1/2 z)) z)) + (+ (* n (/ old-beta z)) + (* (/ (sinh (* 1/2 z)) (* 1/2 z)) + (scale-float (float (if (evenp n) 1 -1) (realpart z)) (- n)))))) + + ;; a[0](k,v) := (k+sqrt(k^2+1))^(-v); ;; a[1](k,v) := -v*a[0](k,v)/sqrt(k^2+1); ;; a[n](k,v) := 1/(k^2+1)/(n-1)/n*((v^2-(n-2)^2)*a[n-2](k,v)-k*(n-1)*(2*n-3)*a[n-1](k,v)); @@ -305,6 +372,26 @@ (format t " term = ~S~%" term) (format t " sum = ~S~%" sum)))))

+(defun sum-ab-2 (big-n v z) + (let ((eps (epsilon (realpart z)))) + (an-clrhash) + (do* ((n 0 (+ 1 n)) + (alphan (alpha-iter 0 z 0) + (alpha-iter n z alphan)) + (betan (beta-iter 0 z 0) + (beta-iter n z betan)) + (term (+ (* alphan (an n 0 v)) + (* betan (sum-an big-n n v z))) + (+ (* alphan (an n 0 v)) + (* betan (sum-an big-n n v z)))) + (sum term (+ sum term))) + ((<= (abs term) (* eps (abs sum))) + sum) + (when nil + (format t "n = ~D~%" n) + (format t " term = ~S~%" term) + (format t " sum = ~S~%" sum))))) + ;; Convert to iteration instead of this quick-and-dirty memoization? (let ((hash (make-hash-table :test 'equal))) (defun %big-a-clrhash ()

commit 8c5195a87137fd61952a175a5dae676c70b480ef Author: Raymond Toy toy.raymond@gmail.com Date: Mon Apr 16 19:30:12 2012 -0700

Make big-n a defvar so we can change it easily.

diff --git a/qd-bessel.lisp b/qd-bessel.lisp index 48a0dac..14743af 100644 --- a/qd-bessel.lisp +++ b/qd-bessel.lisp @@ -410,6 +410,7 @@ ;; bessel_i(v,z) = exp(-v*%pi*%i/2)*bessel_j(v, %i*z) ;; when Im(z) >> Re(z) ;; +(defvar *big-n* 100) (defun bessel-j (v z) (let ((vv (ftruncate v))) ;; Clear the caches for now. @@ -420,7 +421,7 @@ (integer-bessel-j-exp-arc v z)) (t ;; Need to fine-tune the value of big-n. - (let ((big-n 10) + (let ((big-n *big-n*) (vpi (* v (float-pi (realpart z))))) (+ (integer-bessel-j-exp-arc v z) (if (= vv v)

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Summary of changes: qd-bessel.lisp | 90 +++++++++++++++++++++++++++++++++++++++++++++++++++++++- 1 files changed, 89 insertions(+), 1 deletions(-)

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