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- Log ----------------------------------------------------------------- commit c6f0a5ef00138c7f504432e0d0f485c23328274a Author: Raymond Toy toy.raymond@gmail.com Date: Mon Mar 14 20:47:19 2011 -0400

Add docstrings; simplify elliptic-c, elliptic-ec.

diff --git a/qd-elliptic.lisp b/qd-elliptic.lisp index 01a9e21..385f00f 100644 --- a/qd-elliptic.lisp +++ b/qd-elliptic.lisp @@ -413,7 +413,13 @@ ;; Elliptic integral of the first kind. This is computed using ;; Carlson's Rf function: ;; -;; F(phi, m) = sin(phi) * Rf(cos(phi)^2, 1 - m*sin(phi)^2, 1) +;; F(phi, m) = sin(phi) * Rf(cos(phi)^2, 1 - m*sin(phi)^2, 1) +;; +;; Also, DLMF 19.25.5 says +;; +;; F(phi, k) = Rf(c-1, c-k^2, c) +;; +;; where c = csc(phi)^2. (defun elliptic-f (x m) "Incomplete Elliptic integral of the first kind:

@@ -485,6 +491,30 @@ ;; ;; E(phi, m) = integrate(sqrt(1-m*sin(x)^2), x, 0, phi) ;; +;; Carlson says +;; +;; E(phi, k) = sin(phi) * Rf(cos(phi)^2, 1-k^2*sin(phi)^2, 1) +;; - (k^2/3)*sin(phi)^3 * Rd(cos(phi)^2, 1 - k^2*sin(phi)^2, 1) +;; +;; But note that DLMF 19.25.9 says +;; +;; E(phi, k) = Rf(c-1, c - k^2, c) - k^2/3*Rd(c-1, c-k^2, c) +;; +;; where c = csc(phi)^2. Also DLMF 19.25.10 has +;; +;; E(phi, k) = k1^2*Rf(c-1,c-k^2,c) + k^/3*k1^2*Rd(c-1, c, c-k^2) +;; + k^2*sqrt((c-1)/(c*(c-k^2))) +;; +;; where k1^2 = 1-k^2 and c > k^2. Finally, DLMF 19.25.11 says +;; +;; E(phi, k) = -k1^2/3*Rd(c-k^2,c,c-1) + sqrt((c-k^2)/(c*(c-1))) +;; +;; for phi /= pi/2. +;; +;; One possible advantage is that all terms on the rhs are positive +;; for 19.25.9 if k^2 <= 0; 19.25.10 if 0 <= k^2 <= 1; 19.25.11 if 1 +;; <= k^2 <= c. It might be beneficial to use this idea so that we +;; don't have subtractive cancellation. (defun elliptic-e (phi m) "Incomplete elliptic integral of the second kind:

@@ -499,21 +529,26 @@ E(phi, m) = integrate(sqrt(1-m*sin(x)^2), x, 0, phi)" ;; A&S 17.4.25 (sin phi)) (t - (let* ((sin-phi (sin phi)) - (cos-phi (cos phi)) - (k (sqrt m)) - (y (* (- 1 (* k sin-phi)) - (+ 1 (* k sin-phi))))) - (- (* sin-phi - (carlson-rf (* cos-phi cos-phi) y 1)) - (* (/ m 3) - (expt sin-phi 3) - (carlson-rd (* cos-phi cos-phi) y 1)))))))) + ;; For quad-doubles, it's significantly faster to compute + ;; cis(phi) than to compute sin and cos separately. + (multiple-value-bind (cos-phi sin-phi) + (let ((cis (cis phi))) + (values (realpart cis) (imagpart cis))) + (let ((y (- 1 (* m sin-phi sin-phi)))) + (- (* sin-phi + (carlson-rf (* cos-phi cos-phi) y 1)) + (* (/ m 3) + (expt sin-phi 3) + (carlson-rd (* cos-phi cos-phi) y 1)))))))))

;; Complete elliptic integral of second kind. ;; ;; E(phi) = integrate(sqrt(1-m*sin(x)^2), x, 0, %pi/2) ;; +;; Carlson says +;; +;; E(k) = Rf(0, 1-k^2,1) - (k^2/3)*Rd(0,1-k^2,1) +;; (defun elliptic-ec (m) "Complete elliptic integral of the second kind:

@@ -525,12 +560,10 @@ E(m) = integrate(sqrt(1-m*sin(x)^2), x, 0, %pi/2)" ;; A&S 17.4.25 (float 1 m)) (t - (let* ((k (sqrt m)) - (y (* (- 1 k) - (+ 1 k)))) - (- (carlson-rf 0 y 1) + (let* ((m1 (- 1 m))) + (- (carlson-rf 0 m1 1) (* (/ m 3) - (carlson-rd 0 y 1))))))) + (carlson-rd 0 m1 1)))))))

;; Carlson's Rc function. ;;

commit 80be2f3a917f7f429012985cd3a003d77de22ce7 Author: Raymond Toy toy.raymond@gmail.com Date: Mon Mar 14 20:45:28 2011 -0400

Export functions, add docstrings.

qd-package.lisp: o Export the elliptic theta functions and the Carlson elliptic integrals.

qd-theta.lisp: o Add simple docstrings for elliptic theta functions. o Add ELLIPTIC-THETA function.

diff --git a/qd-package.lisp b/qd-package.lisp index 9587038..b102435 100644 --- a/qd-package.lisp +++ b/qd-package.lisp @@ -286,7 +286,15 @@ #:elliptic-k #:elliptic-f #:elliptic-e - #:elliptic-ec) + #:elliptic-ec + #:carlson-rd + #:carlson-rf + #:carlson-rj + #:elliptic-theta-1 + #:elliptic-theta-2 + #:elliptic-theta-3 + #:elliptic-theta-4 + #:elliptic-theta) ;; Constants (:export #:+pi+ #:+pi/2+ diff --git a/qd-theta.lisp b/qd-theta.lisp index caae788..fb41b82 100644 --- a/qd-theta.lisp +++ b/qd-theta.lisp @@ -70,6 +70,9 @@ ;; [ 0 0 1 ]

(defun elliptic-theta-1 (z q) + "Elliptic theta function 1 + + theta1(z, q) = 2*q^(1/4)*sum((-1)^n*q^(n*(n+1))*sin((2*n+1)*z), n, 0, inf)" (let* ((precision (float-contagion z q)) (z (apply-contagion z precision)) (q (apply-contagion q precision)) @@ -97,6 +100,9 @@ ;; k = 1 [ ] ;; [ 0 0 1 ] (defun elliptic-theta-3 (z q) + "Elliptic theta function 3 + + theta3(z, q) = 1 + 2 * sum(q^(n^2)*cos(2*n*z), n, 1, inf)" (let* ((precision (float-contagion z q)) (z (apply-contagion z precision)) (q (apply-contagion q precision)) @@ -115,6 +121,9 @@

;; theta[2](z,q) = theta[1](z+%pi/2, q) (defun elliptic-theta-2 (z q) + "Elliptic theta function 2 + + theta2(z, q) = 2*q^(1/4)*sum(q^(n*(n+1))*cos((2*n+1)*z), n, 0, inf)" (let* ((precision (float-contagion z q)) (z (apply-contagion z precision)) (q (apply-contagion q precision))) @@ -122,14 +131,26 @@

;; theta[4](z,q) = theta[3](z+%pi/2,q) (defun elliptic-theta-4 (z q) + "Elliptic theta function 4 + + theta4(z, q) = 1 + 2*sum((-1)^n*q^(n^2)*cos(2*n*z), n, 1, inf)" (let* ((precision (float-contagion z q)) (z (apply-contagion z precision)) (q (apply-contagion q precision))) (elliptic-theta-3 (+ z (/ (float-pi z) 2)) q)))

+(defun elliptic-theta (n z q) + "Elliptic Theta function n where n = 1, 2, 3, or 4." + (ecase n + (1 (elliptic-theta-1 z q)) + (2 (elliptic-theta-2 z q)) + (3 (elliptic-theta-3 z q)) + (4 (elliptic-theta-4 z q)))) + ;; The nome, q, is given by q = exp(-%pi*K'/K) where K and %i*K' are ;; the quarter periods. (defun elliptic-nome (m) + "Compute the elliptic nome, q, from the parameter m" (exp (- (/ (* (float-pi m) (elliptic-k (- 1 m))) (elliptic-k m)))))

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Summary of changes: qd-elliptic.lisp | 65 ++++++++++++++++++++++++++++++++++++++++------------- qd-package.lisp | 10 +++++++- qd-theta.lisp | 21 +++++++++++++++++ 3 files changed, 79 insertions(+), 17 deletions(-)

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