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- Log ----------------------------------------------------------------- commit 5bebc8d76a83b85cdba6d803d026aae741b6e7c8 Author: Raymond Toy toy.raymond@gmail.com Date: Thu Mar 10 23:05:08 2011 -0500

Test carlson-rf and carlson-rd.

diff --git a/rt-tests.lisp b/rt-tests.lisp index 82ca0b7..85f9b95 100644 --- a/rt-tests.lisp +++ b/rt-tests.lisp @@ -833,3 +833,34 @@ (val (jacobi-dn ek #q.5))) (check-accuracy 212 val true)) nil) + +(rt:deftest oct.carlson-rf.1d + ;; Rf(0,2,1) = integrate(1/sqrt(1-s^4), s, 0 ,1) + ;; = 1/4*beta(1/2,1/2) + ;; = sqrt(%pi)/4*gamma(1/4)/gamma(3/4) + (let ((rf (carlson-rf 0d0 2d0 1d0)) + (true 1.31102877714605990523241979494d0)) + (check-accuracy 53 rf true)) + nil) + +(rt:deftest oct.carlson-rf.1q + ;; Rf(0,2,1) = integrate(1/sqrt(1-s^4), s, 0 ,1) + (let ((rf (carlson-rf #q0 #q2 #q1)) + (true #q1.311028777146059905232419794945559706841377475715811581408410851900395q0)) + (check-accuracy 212 rf true)) + nil) + +(rt:deftest oct.carlson-rd.1d + ;; Rd(0,2,1) = 3*integrate(s^2/sqrt(1-s^4), s, 0 ,1) + ;; = 3*beta(3/4,1/2)/4 + ;; = 3*sqrt(%pi)*gamma(3/4)/gamma(1/4) + (let ((rd (carlson-rd 0d0 2d0 1d0)) + (true 1.7972103521033883d0)) + (check-accuracy 51 rd true)) + nil) + +(rt:deftest oct.carlson-rd.1q + (let ((rd (carlson-rd #q0 #q2 #q1)) + (true #q1.797210352103388311159883738420485817340818994823477337395512429419599q0)) + (check-accuracy 212 rd true)) + nil)

commit 5f6c628014268b683e61baae5470a56b078d1c16 Author: Raymond Toy toy.raymond@gmail.com Date: Thu Mar 10 23:04:28 2011 -0500

Add elliptic integrals of the second kind.

diff --git a/qd-elliptic.lisp b/qd-elliptic.lisp index 63c3687..6c8070a 100644 --- a/qd-elliptic.lisp +++ b/qd-elliptic.lisp @@ -321,6 +321,73 @@ (* -3/44 ee2 ee3)))) (/ s (sqrt an)))))

+;; rd(x,y,z) = integrate(3/2*(t+x)^(-1/2)*(t+y)^(-1/2)*(t+z)^(-3/2), t, 0, inf) +;; +;; E(K) = rf(0, 1-K^2, 1) - (K^2/3)*rd(0,1-K^2,1) +;; +;; B = integrate(s^2/sqrt(1-s^4), s, 0 ,1) +;; = beta(3/4,1/2)/4 +;; = sqrt(%pi)*gamma(3/4)/gamma(1/4) +;; = 1/3*rd(0,2,1) +(defun carlson-rd (x y z) + "Compute Carlson's Rd function: + + Rd(x,y,z) = integrate(3/2*(t+x)^(-1/2)*(t+y)^(-1/2)*(t+z)^(-3/2), t, 0, inf)" + (let* ((xn x) + (yn y) + (zn z) + (a (/ (+ xn yn (* 3 zn)) 5)) + (epslon (/ (max (abs (- a xn)) + (abs (- a yn)) + (abs (- a zn))) + (errtol x y z))) + (an a) + (sigma 0) + (power4 1) + (n 0) + xnroot ynroot znroot lam) + (loop while (> (* power4 epslon) (abs an)) + do + (setf xnroot (sqrt xn)) + (setf ynroot (sqrt yn)) + (setf znroot (sqrt zn)) + (setf lam (+ (* xnroot ynroot) + (* xnroot znroot) + (* ynroot znroot))) + (setf sigma (+ sigma (/ power4 + (* znroot (+ zn lam))))) + (setf power4 (* power4 1/4)) + (setf xn (* (+ xn lam) 1/4)) + (setf yn (* (+ yn lam) 1/4)) + (setf zn (* (+ zn lam) 1/4)) + (setf an (* (+ an lam) 1/4)) + (incf n)) + ;; c1=-3/14,c2=1/6,c3=9/88,c4=9/22,c5=-3/22,c6=-9/52,c7=3/26 + (let* ((xndev (/ (* (- a x) power4) an)) + (yndev (/ (* (- a y) power4) an)) + (zndev (- (* (+ xndev yndev) 1/3))) + (ee2 (- (* xndev yndev) (* 6 zndev zndev))) + (ee3 (* (- (* 3 xndev yndev) + (* 8 zndev zndev)) + zndev)) + (ee4 (* 3 (- (* xndev yndev) (* zndev zndev)) zndev zndev)) + (ee5 (* xndev yndev zndev zndev zndev)) + (s (+ 1 + (* -3/14 ee2) + (* 1/6 ee3) + (* 9/88 ee2 ee2) + (* -3/22 ee4) + (* -9/52 ee2 ee3) + (* 3/26 ee5) + (* -1/16 ee2 ee2 ee2) + (* 3/10 ee3 ee3) + (* 3/20 ee2 ee4) + (* 45/272 ee2 ee2 ee3) + (* -9/68 (+ (* ee2 ee5) (* ee3 ee4)))))) + (+ (* 3 sigma) + (/ (* power4 s) + (expt an 3/2)))))) + ;; Complete elliptic integral of the first kind. This can be computed ;; from Carlson's Rf function: ;; @@ -408,3 +475,55 @@ (* (- 1 (* k sin-x)) (+ 1 (* k sin-x))) 1))))))) + +;; Incomplete elliptic integral of the second kind. +;; +;; E(phi, m) = integrate(sqrt(1-m*sin(x)^2), x, 0, phi) +;; +(defun elliptic-e (phi m) + "Incomplete elliptic integral of the second kind: + +E(phi, m) = integrate(sqrt(1-m*sin(x)^2), x, 0, phi)" + (let* ((precision (float-contagion phi m)) + (phi (coerce phi precision)) + (m (coerce m precision))) + (cond ((= m 0) + ;; A&S 17.4.23 + phi) + ((= m 1) + ;; 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 (coerce 1 precision))) + (* (/ m 3) + (expt sin-phi 3) + (carlson-rd (* cos-phi cos-phi) y (coerce 1 precision))))))))) + +;; Complete elliptic integral of second kind. +;; +;; E(phi) = integrate(sqrt(1-m*sin(x)^2), x, 0, %pi/2) +;; +(defun elliptic-ec (m) + "Complete elliptic integral of the second kind: + +E(m) = integrate(sqrt(1-m*sin(x)^2), x, 0, %pi/2)" + (let ((precision (float-contagion m))) + (cond ((= m 0) + ;; A&S 17.4.23 + (/ (float-pi m) 2)) + ((= m 1) + ;; A&S 17.4.25 + (coerce 1 precision)) + (t + (let* ((k (sqrt m)) + (y (* (- 1 k) + (+ 1 k)))) + (- (carlson-rf 0.0 y 1.0) + (* (/ m 3) + (carlson-rd 0.0 y 1.0))))))))

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Summary of changes: qd-elliptic.lisp | 119 ++++++++++++++++++++++++++++++++++++++++++++++++++++++ rt-tests.lisp | 31 ++++++++++++++ 2 files changed, 150 insertions(+), 0 deletions(-)

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