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- Log ----------------------------------------------------------------- commit c388f81713d7b2a483000d3cee1af030ed2c1cac Author: Raymond Toy toy.raymond@gmail.com Date: Mon Dec 5 19:58:44 2011 -0800

Better exp-integral-e computation and fix for incomplete-gamma-tail.

For exp-integral-e, use the series for small z and the incomplete-gamma-tail for near the negative real axis. Otherwise, use the continued fraction.

In incomplete-gamma-tail, we were using the continued fraction instead of the incomplete-gamma function for the region just below the negative real axis. We should use the cf except in that region.

diff --git a/qd-gamma.lisp b/qd-gamma.lisp index f13b9a8..48f8345 100644 --- a/qd-gamma.lisp +++ b/qd-gamma.lisp @@ -396,7 +396,7 @@ ;; Use the incomplete gamma function to evaluate in this ;; region. (Arbitrarily selected the region to be a sector. ;; But what is the correct size of this sector?) - (if (<= (phase z) 3.1) + (if (<= (abs (phase z)) 3.1) (cf-incomplete-gamma-tail a z) (- (gamma a) (cf-incomplete-gamma a z)))))))

@@ -460,15 +460,7 @@ (/ (incomplete-gamma 1/2 (* z z)) (sqrt (float-pi z))))))

-(defun exp-integral-e (v z) - "Exponential integral E: - - E(v,z) = integrate(exp(-t)/t^v, t, 1, inf)" - ;; E(v,z) = z^(v-1) * integrate(t^(-v)*exp(-t), t, z, inf); - ;; - ;; for |arg(z)| < pi. - ;; - ;; +(defun cf-exp-integral-e (v z) ;; We use the continued fraction ;; ;; E(v,z) = exp(-z)/cf(z) @@ -485,7 +477,45 @@ (+ z+v (* 2 k))) #'(lambda (k) (* (- k) - (1- (+ k v)))))))) + (+ k v -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)))) + +(defun exp-integral-e (v z) + "Exponential integral E: + + E(v,z) = integrate(exp(-t)/t^v, t, 1, inf)" + ;; E(v,z) = z^(v-1) * integrate(t^(-v)*exp(-t), t, z, inf); + ;; + ;; for |arg(z)| < pi. + ;; + ;; + (cond ((< (abs z) 1) + ;; Use series for small z + (s-exp-integral-e v z)) + ((>= (abs (phase z)) 3.1) + ;; The continued fraction doesn't converge on the negative + ;; real axis, and converges very slowly near the negative + ;; real axis, so use the incomplete-gamma-tail function in + ;; this region. "Closeness" to the negative real axis is + ;; teken to mean that z is in a sector near the axis. + ;; + ;; E(v,z) = z^(v-1)*incomplete_gamma_tail(1-v,z) + (* (expt z (- v 1)) + (incomplete-gamma-tail (- 1 v) z))) + (t + ;; Use continued fraction for everything else. + (cf-exp-integral-e v z))))

;; Series for Fresnel S ;;

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Summary of changes: qd-gamma.lisp | 52 +++++++++++++++++++++++++++++++++++++++++----------- 1 files changed, 41 insertions(+), 11 deletions(-)

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