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- Log ----------------------------------------------------------------- commit 0d5870201359817c679921a2d740fdd1697469b2 Author: Raymond Toy toy.raymond@gmail.com Date: Wed Mar 21 18:45:44 2012 -0700

Implement psi and fix exp-integral-e for integral values of v. Needs some more work.

diff --git a/qd-gamma.lisp b/qd-gamma.lisp index 48f8345..02b5547 100644 --- a/qd-gamma.lisp +++ b/qd-gamma.lisp @@ -479,17 +479,44 @@ (* (- k) (+ k v -1)))))))

+ +;; For v not an integer: +;; +;; E(v,z) = gamma(1-v)*z^(v-1) - sum((-1)^k*z^k/(k-v+1)/k!, k, 0, inf) +;; +;; For v an integer: +;; +;; E(v,z) = (-z)^(v-1)/(v-1)!*(psi(v)-log(z)) +;; - sum((-1)^k*z^k/(k-v+1)/k!, k, 0, inf, k != n-1) +;; (defun s-exp-integral-e (v z) ;; E(v,z) = gamma(1-v)*z^(v-1) - sum((-1)^k*z^k/(k-v+1)/k!, k, 0, inf) (let ((-z (- z)) (-v (- v)) (eps (epsilon z))) - (loop for k from 0 - for term = 1 then (* term (/ -z k)) - for sum = (/ (- 1 v)) then (+ sum (/ term (+ k 1 -v))) - when (< (abs term) (* (abs sum) eps)) - return (- (* (gamma (- 1 v)) (expt z (- v 1))) - sum)))) + (if (and (realp v) + (= v (ftruncate v))) + ;; v is an integer + (let ((n (truncate v))) + (- (* (/ (expt -z (- v 1)) + (gamma v)) + (- (psi v) (log z))) + (loop for k from 0 below n + for term = 1 then (* term (/ -z k)) + for sum = (/ (- 1 v)) then (+ sum (/ term (+ k 1 -v))) + when (< (abs term) (* (abs sum) eps)) + return sum) + (loop for k from n + for term = 1 then (* term (/ -z k)) + for sum = 0 then (+ sum (/ term (+ k 1 -v))) + when (< (abs term) (* (abs sum) eps)) + return sum))) + (loop for k from 0 + for term = 1 then (* term (/ -z k)) + for sum = (/ (- 1 v)) then (+ sum (/ term (+ k 1 -v))) + when (< (abs term) (* (abs sum) eps)) + return (- (* (gamma (- 1 v)) (expt z (- v 1))) + sum)))))

(defun exp-integral-e (v z) "Exponential integral E: @@ -679,3 +706,109 @@ (if (and (realp z) (plusp z)) (realpart (ci z)) (ci z)))) + +(defconstant bern-values + (make-array 55 + :initial-contents + '(1 + -1/2 + 1/6 + 0 + -1/30 + 0 + 1/42 + 0 + -1/30 + 0 + 5/66 + 0 + -691/2730 + 0 + 7/6 + 0 + -3617/510 + 0 + 43867/798 + 0 + -174611/330 + 0 + 854513/138 + 0 + -236364091/2730 + 0 + 8553103/6 + 0 + -23749461029/870 + 0 + 8615841276005/14322 + 0 + -7709321041217/510 + 0 + 2577687858367/6 + 0 + -26315271553053477373/1919190 + 0 + 2929993913841559/6 + 0 + -261082718496449122051/13530 + 0 + 1520097643918070802691/1806 + 0 + -27833269579301024235023/690 + 0 + 596451111593912163277961/282 + 0 + -5609403368997817686249127547/46410 + 0 + 495057205241079648212477525/66 + 0 + -801165718135489957347924991853/1590 + 0 + 29149963634884862421418123812691/798 + ))) + +(defun bern (k) + (aref bern-values k)) + +(defun psi (z) + "Digamma function defined by + + - %gamma + sum(1/k-1/(k+z-1), k, 1, inf) + + where %gamma is Euler's constant" + + ;; A&S 6.3.7: Reflection formula + ;; + ;; psi(1-z) = psi(z) + %pi*cot(%pi*z) + ;; + ;; A&S 6.3.6: Recurrence formula + ;; + ;; psi(n+z) = 1/(z+n-1)+1/(z+n-2)+...+1/(z+2)+1/(1+z)+psi(1+z) + ;; + ;; A&S 6.3.8: Asymptotic formula + ;; + ;; psi(z) ~ log(z) - sum(bern(2*n)/(2*n*z^(2*n)), n, 1, inf) + ;; + ;; So use reflection formula if Re(z) < 0. For z > 0, use the recurrence + ;; formula to increase the argument and then apply the asymptotic formula. + + (cond ((minusp (realpart z)) + (- (psi (- 1 z)) + (* +pi+ (/ (tan (* +pi+ z)))))) + (t + (let* ((k (* 2 (1+ (floor (* .41 (- (log (epsilon z) 10))))))) + (m 0) + (y (expt (+ z k) 2)) + (x 0)) + (loop for i from 1 upto (floor k 2) do + (progn + (incf m (+ (/ (+ z i i -1)) + (/ (+ z i i -2)))) + (setf x (/ (+ x (/ (bern (+ k 2 (* -2 i))) + (- k i i -2))) + y)))) + (- (log (+ z k)) + (/ (* 2 (+ z k))) + x + m))))) +

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Summary of changes: qd-gamma.lisp | 145 ++++++++++++++++++++++++++++++++++++++++++++++++++++++-- 1 files changed, 139 insertions(+), 6 deletions(-)

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