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- Log ----------------------------------------------------------------- commit c31b1cdd112f26334b7762014d3afb781917ebda Author: Raymond Toy toy.raymond@gmail.com Date: Mon Mar 28 22:37:32 2011 -0400

More accurate incomplete-gamma function, add debugging to lentz, and some random clean ups.

o Add *DEBUG-CF-EVAL* to enable debugging prints in LENTZ. o Modify LENTZ to terminate with an error if *MAX-CF-ITERATIONS* is reached. o Modify LENTZ to return the function value, the number of iterations, and the number of times a zero value had to be replaced. o Adjust cf-incomplete-gamma and cf-incomplete-gamma-tail not to signal overflow prematurely when calculating z^a*exp(-z). o Fix doc bug in reference for continued fraction for (original) cf-incomplete-gamma. o Add new version of cf-incomplete-gamma using a different continued fraction. This appears to converge faster and to be more accurate than the original, especially for points near the negative real axis.

diff --git a/qd-gamma.lisp b/qd-gamma.lisp index 78b5a99..e020a91 100644 --- a/qd-gamma.lisp +++ b/qd-gamma.lisp @@ -177,32 +177,62 @@ ;; b0 + ---- ---- ---- ;; b1 + b2 + b3 + ;; + +(defvar *debug-cf-eval* + nil + "When true, enable some debugging prints when evaluating a + continued fraction.") + +;; Max number of iterations allowed when evaluating the continued +;; fraction. When this is reached, we assume that the continued +;; fraction did not converge. +(defvar *max-cf-iterations* + 10000 + "Max number of iterations allowed when evaluating the continued + fraction. When this is reached, we assume that the continued + fraction did not converge.") + (defun lentz (bf af) - (flet ((value-or-tiny (v) - (if (zerop v) - (etypecase v - ((or double-float cl:complex) - least-positive-normalized-double-float) - ((or qd-real qd-complex) - (make-qd least-positive-normalized-double-float))) - v))) - (let* ((f (value-or-tiny (funcall bf 0))) - (c f) - (d 0) - (eps (epsilon f))) - (loop - for j from 1 - for an = (funcall af j) - for bn = (funcall bf j) - do (progn - (setf d (value-or-tiny (+ bn (* an d)))) - (setf c (value-or-tiny (+ bn (/ an c)))) - (setf d (/ d)) - (let ((delta (* c d))) - (setf f (* f delta)) - (when (<= (abs (- delta 1)) eps) - (return))))) - f))) + (let ((tiny-value-count 0)) + (flet ((value-or-tiny (v) + (if (zerop v) + (progn + (incf tiny-value-count) + (etypecase v + ((or double-float cl:complex) + least-positive-normalized-double-float) + ((or qd-real qd-complex) + (make-qd least-positive-normalized-double-float)))) + v))) + (let* ((f (value-or-tiny (funcall bf 0))) + (c f) + (d 0) + (eps (epsilon f))) + (loop + for j from 1 upto *max-cf-iterations* + for an = (funcall af j) + for bn = (funcall bf j) + do (progn + (setf d (value-or-tiny (+ bn (* an d)))) + (setf c (value-or-tiny (+ bn (/ an c)))) + (when *debug-cf-eval* + (format t "~&j = ~d~%" j) + (format t " an = ~s~%" an) + (format t " bn = ~s~%" bn) + (format t " c = ~s~%" c) + (format t " d = ~s~%" d)) + (let ((delta (/ c d))) + (setf d (/ d)) + (setf f (* f delta)) + (when *debug-cf-eval* + (format t " dl= ~S~%" delta) + (format t " f = ~S~%" f)) + (when (<= (abs (- delta 1)) eps) + (return-from lentz (values f j tiny-value-count))))) + finally + (error 'simple-error + :format-control "~<Continued fraction failed to converge after ~D iterations.~% Delta = ~S~>" + :format-arguments (list *max-cf-iterations* (/ c d))))))))

;; Continued fraction for erf(b): ;; @@ -240,8 +270,13 @@ :function-name 'cf-incomplete-gamma-tail :format-arguments (list 'z z) :format-control "Argument ~S should not be on the negative real axis: ~S")) - (/ (* (expt z a) - (exp (- z))) + (/ (handler-case (* (expt z a) + (exp (- z))) + (arithmetic-error () + ;; z^a*exp(-z) can overflow prematurely. In this case, use + ;; the equivalent exp(a*log(z)-z). We don't use this latter + ;; form because it has more roundoff error than the former. + (exp (- (* a (log z)) z)))) (let ((z-a (- z a))) (lentz #'(lambda (n) (+ n n 1 z-a)) @@ -251,18 +286,23 @@ ;; Incomplete gamma function: ;; integrate(x^(a-1)*exp(-x), x, 0, z) ;; -;; The continued fraction, valid for all z except the negative real -;; axis: +;; The continued fraction, valid for all z: ;; ;; b[n] = n - 1 + z + a ;; a[n] = -z*(a + n) ;; -;; See http://functions.wolfram.com/06.06.10.0005.01. We modified the +;; See http://functions.wolfram.com/06.06.10.0007.01. We modified the ;; continued fraction slightly and discarded the first quotient from ;; the fraction. +#+nil (defun cf-incomplete-gamma (a z) - (/ (* (expt z a) - (exp (- z))) + (/ (handler-case (* (expt z a) + (exp (- z))) + (arithmetic-error () + ;; z^a*exp(-z) can overflow prematurely. In this case, use + ;; the equivalent exp(a*log(z)-z). We don't use this latter + ;; form because it has more roundoff error than the former. + (exp (- (* a (log z)) z)))) (let ((za1 (+ z a 1))) (- a (/ (* a z) (lentz #'(lambda (n) @@ -270,6 +310,35 @@ #'(lambda (n) (- (* z (+ a n))))))))))

+;; Incomplete gamma function: +;; integrate(x^(a-1)*exp(-x), x, 0, z) +;; +;; The continued fraction, valid for all z: +;; +;; b[n] = a + n +;; a[n] = -(a+n/2)*z if n odd +;; n/2*z if n even +;; +;; See http://functions.wolfram.com/06.06.10.0009.01. +;; +;; Some experiments indicate that this converges faster than the above +;; and is actually quite a bit more accurate, expecially near the +;; negative real axis. +(defun cf-incomplete-gamma (a z) + (/ (handler-case (* (expt z a) + (exp (- z))) + (arithmetic-error () + ;; z^a*exp(-z) can overflow prematurely. In this case, use + ;; the equivalent exp(a*log(z)-z). We don't use this latter + ;; form because it has more roundoff error than the former. + (exp (- (* a (log z)) z)))) + (lentz #'(lambda (n) + (+ n a)) + #'(lambda (n) + (if (evenp n) + (* (ash n -1) z) + (- (* (+ a (ash n -1)) z))))))) + ;; Series expansion for incomplete gamma. Intended for |a|<1 and ;; |z|<1. The series is ;; @@ -283,8 +352,6 @@ when (< (abs term) (* (abs sum) eps)) return (* sum (expt z a)))))

- - ;; Tail of the incomplete gamma function. (defun incomplete-gamma-tail (a z) "Tail of the incomplete gamma function defined by:

commit 88cff63cfb4996e2e90e499afd34e1fe16ecc179 Author: Raymond Toy toy.raymond@gmail.com Date: Mon Mar 28 09:08:37 2011 -0400

Fix typo in #q() number in fresnel-s.2q test.

diff --git a/rt-tests.lisp b/rt-tests.lisp index c0b59fb..66aab0c 100644 --- a/rt-tests.lisp +++ b/rt-tests.lisp @@ -1279,7 +1279,7 @@ nil)

(rt:deftest fresnel-s.2q - (let* ((z #q(1q-3 1q-3)) + (let* ((z #q(#q1q-3 #q1q-3)) (s (fresnel-s z)) (true (fresnel-s-series z))) (check-accuracy 212 s true))

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Summary of changes: qd-gamma.lisp | 135 ++++++++++++++++++++++++++++++++++++++++++-------------- rt-tests.lisp | 2 +- 2 files changed, 102 insertions(+), 35 deletions(-)

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