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- Log ----------------------------------------------------------------- commit 94d15953c5116999ac15377515b6c7e5ae430ea7 Author: Raymond Toy toy.raymond@gmail.com Date: Thu Apr 25 20:11:27 2013 -0700
Add continued fractions for erf, erfc, w and Dawson's integral.
qd-gamma.lisp:: Add continued fractions
rt-tests.lisp:: Add a test for erfc
diff --git a/qd-gamma.lisp b/qd-gamma.lisp index 40a45eb..11e9822 100644 --- a/qd-gamma.lisp +++ b/qd-gamma.lisp @@ -275,6 +275,107 @@ #'(lambda (n) (* 4 n z2)))))))
+;; From Cuyt +;; +;; sqrt(%pi)*z*exp(z^2)*erf(z) = K +;; +;; where K is the continued fraction with terms F[m]*z^2/(1 + G[m]*z^2) +;; with F[1] = 2 and F[m] = 4*(m-1)/(2*m-3)/(2*m-1) and G[m] = -2/(2*m-1) +;; +(defun cf-erf (z) + (let ((z2 (* z z))) + (* (/ (exp (- z2)) + (sqrt (float-pi z)) + z) + (lentz #'(lambda (n) + (if (zerop n) + (float 0 (realpart z)) + (1+ (/ (* -2 z2) + (+ n n -1))))) + #'(lambda (n) + (if (= n 1) + (* 2 z2) + (/ (* z2 4 (- n 1)) + (* (+ n n -3) (+ n n -1))))))))) + +;; From the above, we also have Dawson's integral: +;; +;; exp(-z^-2)*integrate(exp(t^2), t, 0, z) = %i*sqrt(%pi)/2*exp(-z^2)*erf(-%i*z); +;; +;; -2*z*exp(-z^2)*integrate(exp(t^2), t, 0, z) = K +;; +;; with K = -F[m)*z^2/(1 - G[m]*z^2), where F[m] and G[m] are as above. +;; +;; Also erf(-%i*z) = dawson(z) * 2*exp(-z^2)/(*%i*sqrt(%pi)) +(defun cf-dawson (z) + (let ((z2 (* z z))) + (/ (lentz #'(lambda (n) + (if (zerop n) + (float 0 (realpart z)) + (- 1 (/ (* -2 z2) + (+ n n -1))))) + #'(lambda (n) + (if (= n 1) + (* -2 z2) + (/ (* z2 -4 (- n 1)) + (* (+ n n -3) (+ n n -1)))))) + (* -2 z)))) + +;; erfc(z) = z/sqrt(%pi)*exp(-z^2)*K +;; +;; where K is the continued fraction with a[1] = 1, a[m] = (m-1)/2, +;; for m >= 2 and b[0] = 0, b[2*m+1] = z^2, b[2*m] = 1. +;; +;; This is valid only if Re(z) > 0. + +(defun cf-erfc (z) + (let ((z2 (* z z)) + (zero (float 0 (realpart z))) + (one (float 1 (realpart z)))) + (* (exp (- z2)) + z + (/ (sqrt (float-pi (realpart z)))) + (lentz #'(lambda (n) + (if (zerop n) + zero + (if (evenp n) + one + z2))) + #'(lambda (n) + (if (= n 1) + one + (/ (- n 1) 2))))))) + +;; w(z) = exp(-z^2)*erfc(-%i*z) +;; +;; = -%i*z/sqrt(%pi)*K +;; +;; where K is the continued fraction with a[n] the same as for erfc +;; and b[0] = 0, b[2*m+1] = -z^2, b[2*m] = 1. +;; +;; This is valid only if Im(z) > 0. We can use the following +;; identities: +;; +;; w(-z) = 2*exp(-z^2) - w(z) +;; w(conj(z)) = conj(w(-z)) + +(defun cf-w (z) + (let ((z2 (* z z)) + (zero (float 0 (realpart z))) + (one (float 1 (realpart z)))) + (* #c(0 -1) + z + (/ (sqrt (float-pi (realpart z)))) + (lentz #'(lambda (n) + (if (zerop n) + zero + (if (evenp n) + one + (- z2)))) + #'(lambda (n) + (if (= n 1) + one + (/ (- n 1) 2))))))) ;; Tail of the incomplete gamma function: ;; integrate(x^(a-1)*exp(-x), x, z, inf) ;; diff --git a/rt-tests.lisp b/rt-tests.lisp index af1d00c..caf568f 100644 --- a/rt-tests.lisp +++ b/rt-tests.lisp @@ -1729,3 +1729,8 @@ when result append (list (list (list k x) result))) nil) + +(rt:deftest erfc + (check-accuracy 210 (erfc #q-4) + #q1.9999999845827420997199811478403265131159514278547464108088316570950057869589732) + nil)
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Summary of changes: qd-gamma.lisp | 101 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ rt-tests.lisp | 5 +++ 2 files changed, 106 insertions(+)
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