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- Log ----------------------------------------------------------------- commit 390a7483f5658fe802d5d239070cdaa573adf4a5 Author: Raymond Toy toy.raymond@gmail.com Date: Sat Mar 12 13:35:18 2011 -0500

Add prelimary support for integrals of the 3rd kind.

qd-elliptic.lisp: o Clean up for unused variable in ELLIPTIC-K o Add Carlson's Rj functions o Implement elliptic-pi using Carlson's method.

rt-tests.lisp: o Add many tests for elliptic-pi. Some tests pass, and some fail. The failing tests are not enabled because I don't know if the failure is because the test itself is wrong or if the integral is wrong.

diff --git a/qd-elliptic.lisp b/qd-elliptic.lisp index 342290e..1b80f13 100644 --- a/qd-elliptic.lisp +++ b/qd-elliptic.lisp @@ -408,8 +408,7 @@ (cond ((= m 0) (/ (float +pi+ m) 2)) (t - (let ((precision (float-contagion m))) - (carlson-rf 0 (- 1 m) 1))))) + (carlson-rf 0 (- 1 m) 1))))

;; Elliptic integral of the first kind. This is computed using ;; Carlson's Rf function: @@ -532,3 +531,186 @@ E(m) = integrate(sqrt(1-m*sin(x)^2), x, 0, %pi/2)" (- (carlson-rf 0 y 1) (* (/ m 3) (carlson-rd 0 y 1))))))) + +;; Carlson's Rc function. +;; +;; Some interesting identities: +;; +;; log(x) = (x-1)*rc(((1+x)/2)^2, x), x > 0 +;; asin(x) = x * rc(1-x^2, 1), |x|<= 1 +;; acos(x) = sqrt(1-x^2)*rc(x^2,1), 0 <= x <=1 +;; atan(x) = x * rc(1,1+x^2) +;; asinh(x) = x * rc(1+x^2,1) +;; acosh(x) = sqrt(x^2-1) * rc(x^2,1), x >= 1 +;; atanh(x) = x * rc(1,1-x^2), |x|<=1 +;; + +(defun carlson-rc (x y) + "Compute Carlson's Rc function: + + Rc(x,y) = integrate(1/2*(t+x)^(-1/2)*(t+y)^(-1), t, 0, inf)" + (let* ((precision (float-contagion x y)) + (yn (apply-contagion y precision)) + (x (apply-contagion x precision)) + xn z w a an pwr4 n epslon lambda sn s) + (cond ((and (zerop (imagpart yn)) + (minusp (realpart yn))) + (setf xn (- x y)) + (setf yn (- yn)) + (setf z yn) + (setf w (sqrt (/ x xn)))) + (t + (setf xn x) + (setf z yn) + (setf w 1))) + (setf a (/ (+ xn yn yn) 3)) + (setf epslon (/ (abs (- a xn)) (errtol x y))) + (setf an a) + (setf pwr4 1) + (setf n 0) + (loop while (> (* epslon pwr4) (abs an)) + do + (setf pwr4 (/ pwr4 4)) + (setf lambda (+ (* 2 (sqrt xn) (sqrt yn)) yn)) + (setf an (/ (+ an lambda) 4)) + (setf xn (/ (+ xn lambda) 4)) + (setf yn (/ (+ yn lambda) 4)) + (incf n)) + ;; c2=3/10,c3=1/7,c4=3/8,c5=9/22,c6=159/208,c7=9/8 + (setf sn (/ (* pwr4 (- z a)) an)) + (setf s (* sn sn (+ 3/10 + (* sn (+ 1/7 + (* sn (+ 3/8 + (* sn (+ 9/22 + (* sn (+ 159/208 + (* sn 9/8)))))))))))) + (/ (* w (+ 1 s)) + (sqrt an)))) + +(defun carlson-rj1 (x y z p) + (let* ((xn x) + (yn y) + (zn z) + (pn p) + (en (* (- pn xn) + (- pn yn) + (- pn zn))) + (sigma 0) + (power4 1) + (k 0) + (a (/ (+ xn yn zn pn pn) 5)) + (epslon (/ (max (abs (- a xn)) + (abs (- a yn)) + (abs (- a zn)) + (abs (- a pn))) + (errtol x y z p))) + (an a) + xnroot ynroot znroot pnroot lam dn) + (loop while (> (* power4 epslon) (abs an)) + do + (setf xnroot (sqrt xn)) + (setf ynroot (sqrt yn)) + (setf znroot (sqrt zn)) + (setf pnroot (sqrt pn)) + (setf lam (+ (* xnroot ynroot) + (* xnroot znroot) + (* ynroot znroot))) + (setf dn (* (+ pnroot xnroot) + (+ pnroot ynroot) + (+ pnroot znroot))) + (setf sigma (+ sigma + (/ (* power4 + (carlson-rc 1 (+ 1 (/ en (* dn dn))))) + dn))) + (setf power4 (* power4 1/4)) + (setf en (/ en 64)) + (setf xn (* (+ xn lam) 1/4)) + (setf yn (* (+ yn lam) 1/4)) + (setf zn (* (+ zn lam) 1/4)) + (setf pn (* (+ pn lam) 1/4)) + (setf an (* (+ an lam) 1/4)) + (incf k)) + (let* ((xndev (/ (* (- a x) power4) an)) + (yndev (/ (* (- a y) power4) an)) + (zndev (/ (* (- a z) power4) an)) + (pndev (* -0.5 (+ xndev yndev zndev))) + (ee2 (+ (* xndev yndev) + (* xndev zndev) + (* yndev zndev) + (* -3 pndev pndev))) + (ee3 (+ (* xndev yndev zndev) + (* 2 ee2 pndev) + (* 4 pndev pndev pndev))) + (ee4 (* (+ (* 2 xndev yndev zndev) + (* ee2 pndev) + (* 3 pndev pndev pndev)) + pndev)) + (ee5 (* xndev yndev zndev pndev pndev)) + (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)))))) + (+ (* 6 sigma) + (/ (* power4 s) + (sqrt (* an an an))))))) + +(defun carlson-rj (x y z p) + "Compute Carlson's Rj function: + + Rj(x,y,z,p) = integrate(3/2*(t+x)^(-1/2)*(t+y)^(-1/2)*(t+z)^(-1/2)*(t+p)^(-1), t, 0, inf)" + (let* ((precision (float-contagion x y z p)) + (xn (apply-contagion x precision)) + (yn (apply-contagion y precision)) + (zn (apply-contagion z precision)) + (p (apply-contagion p precision)) + (qn (- p))) + (cond ((and (and (zerop (imagpart xn)) (>= (realpart xn) 0)) + (and (zerop (imagpart yn)) (>= (realpart yn) 0)) + (and (zerop (imagpart zn)) (>= (realpart zn) 0)) + (and (zerop (imagpart qn)) (> (realpart qn) 0))) + (destructuring-bind (xn yn zn) + (sort (list xn yn zn) #'<) + (let* ((pn (+ yn (* (- zn yn) (/ (- yn xn) (+ yn qn))))) + (s (- (* (- pn yn) (carlson-rj1 xn yn zn pn)) + (* 3 (carlson-rf xn yn zn))))) + (setf s (+ s (* 3 (sqrt (/ (* xn yn zn) + (+ (* xn zn) (* pn qn)))) + (carlson-rc (+ (* xn zn) (* pn qn)) (* pn qn))))) + (/ s (+ yn qn))))) + (t + (carlson-rj1 x y z p))))) + +;; Elliptic integral of the third kind: +;; +;; (A&S 17.2.14) +;; +;; PI(n; phi|m) = integrate(1/sqrt(1-m*sin(x)^2)/(1-n*sin(x)^2), x, 0, phi) +;; +(defun elliptic-pi (n phi m) + "Compute elliptic integral of the third kind: + + PI(n; phi|m) = integrate(1/sqrt(1-m*sin(x)^2)/(1-n*sin(x)^2), x, 0, phi)" + ;; Note: Carlson's DRJ has n defined as the negative of the n given + ;; in A&S. + (let* ((precision (float-contagion n phi m)) + (n (apply-contagion n precision)) + (phi (apply-contagion phi precision)) + (m (apply-contagion m precision)) + (nn (- n)) + (sin-phi (sin phi)) + (cos-phi (cos phi)) + (k (sqrt m)) + (k2sin (* (- 1 (* k sin-phi)) + (+ 1 (* k sin-phi))))) + (- (* sin-phi (carlson-rf (expt cos-phi 2) k2sin 1)) + (* (/ nn 3) (expt sin-phi 3) + (carlson-rj (expt cos-phi 2) k2sin 1 + (+ 1 (* nn (expt sin-phi 2)))))))) diff --git a/rt-tests.lisp b/rt-tests.lisp index 7635611..80d5c57 100644 --- a/rt-tests.lisp +++ b/rt-tests.lisp @@ -928,4 +928,143 @@ (let ((rf (carlson-rf 0 2 #q1q0)) (true #q1.311028777146059905232419794945559706841377475715811581408410851900395q0)) (check-accuracy 212 rf true)) - nil) \ No newline at end of file + nil) + +;; Elliptic integral of the third kind + +;; elliptic-pi(0,phi,m) = elliptic-f(phi, m) +(rt:deftest oct.elliptic-pi.1d + (loop for k from 0 to 100 + for phi = (random (/ pi 2)) + for m = (random 1d0) + for epi = (elliptic-pi 0 phi m) + for ef = (elliptic-f phi m) + for result = (check-accuracy 53 epi ef) + unless (eq nil result) + append (list (list phi m) result)) + nil) + +(rt:deftest oct.elliptic-pi.1q + (loop for k from 0 below 100 + for phi = (random (/ +pi+ 2)) + for m = (random #q1) + for epi = (elliptic-pi 0 phi m) + for ef = (elliptic-f phi m) + for result = (check-accuracy 53 epi ef) + unless (eq nil result) + append (list (list phi m) result)) + nil) + +;; DLMF 19.6.3 +;; +;; PI(n; pi/2 | 0) = pi/(2*sqrt(1-n)) +(rt:deftest oct.elliptic-pi.19.6.3.d + (loop for k from 0 below 100 + for n = (random 1d0) + for epi = (elliptic-pi n (/ pi 2) 0) + for true = (/ pi (* 2 (sqrt (- 1 n)))) + for result = (check-accuracy 49 epi true) + unless (eq nil result) + append (list (list (list k n) result))) + nil) + +(rt:deftest oct.elliptic-pi.19.6.3.q + (loop for k from 0 below 100 + for n = (random #q1) + for epi = (elliptic-pi n (/ (float-pi n) 2) 0) + for true = (/ (float-pi n) (* 2 (sqrt (- 1 n)))) + for result = (check-accuracy 210 epi true) + unless (eq nil result) + append (list (list (list k n) result))) + nil) + +#+nil +(rt:deftest oct.elliptic-pi.19.6.2.d + (loop for k from 0 below 100 + for n = (random 1d0) + for epi = (elliptic-pi (- n) (/ (float-pi n) 2) n) + for true = (+ (/ (float-pi n) 4 (sqrt (+ 1 (sqrt n)))) + (/ (elliptic-k n) 2)) + for result = (check-accuracy 53 epi true) + when result + append (list (list (list k n) result))) + nil) + + +#|| +;; elliptic-pi(n, phi, 0) = +;; atanh(sqrt(1-n)*tan(phi))/sqrt(1-n) n < 1 +;; atanh(sqrt(n-1)*tan(phi))/sqrt(n-1) n > 1 +;; tan(phi) n = 1 +(rt:deftest oct.elliptic-pi.n0.d + (loop for k from 0 below 100 + for phi = (random (/ pi 2)) + for n = (random 1d0) + for epi = (elliptic-pi n phi 0) + for true = (/ (atanh (* (tan phi) (sqrt (- 1 n)))) + (sqrt (- 1 n))) + for result = (check-accuracy 53 epi true) + unless (eq nil result) + append (list (list (list k n phi) result))) + nil) + +(rt:deftest oct.elliptic-pi.n1.d + (loop for k from 0 below 100 + for phi = (random (/ pi 2)) + for epi = (elliptic-pi 0 phi 0) + for true = (tan phi) + for result = (check-accuracy 53 epi true) + unless (eq nil result) + append (list (list (list k phi) result))) + nil) + +(rt:deftest oct.elliptic-pi.n2.d + (loop for k from 0 below 100 + for phi = (random (/ pi 2)) + for n = (+ 1d0 (random 100d0)) + for epi = (elliptic-pi n phi 0) + for true = (/ (atanh (* (tan phi) (sqrt (- n 1)))) + (sqrt (- n 1))) + for result = (check-accuracy 52 epi true) + ;; Not sure if this formula holds when atanh gives a complex + ;; result. Wolfram doesn't say + when (and (not (complexp true)) result) + append (list (list (list k n phi) result))) + nil) + +(rt:deftest oct.elliptic-pi.n0.q + (loop for k from 0 below 100 + for phi = (random (/ +pi+ 2)) + for n = (random #q1) + for epi = (elliptic-pi n phi 0) + for true = (/ (atanh (* (tan phi) (sqrt (- 1 n)))) + (sqrt (- 1 n))) + for result = (check-accuracy 212 epi true) + unless (eq nil result) + append (list (list (list k n phi) result))) + nil) + +(rt:deftest oct.elliptic-pi.n1.q + (loop for k from 0 below 100 + for phi = (random (/ +pi+ 2)) + for epi = (elliptic-pi 0 phi 0) + for true = (tan phi) + for result = (check-accuracy 212 epi true) + unless (eq nil result) + append (list (list (list k phi) result))) + nil) + +(rt:deftest oct.elliptic-pi.n2.q + (loop for k from 0 below 100 + for phi = (random (/ +pi+ 2)) + for n = (+ #q1 (random #q1)) + for epi = (elliptic-pi n phi 0) + for true = (/ (atanh (* (tan phi) (sqrt (- n 1)))) + (sqrt (- n 1))) + for result = (check-accuracy 209 epi true) + ;; Not sure if this formula holds when atanh gives a complex + ;; result. Wolfram doesn't say + when (and (not (complexp true)) result) + append (list (list (list k n phi) result))) + nil) +||#

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Summary of changes: qd-elliptic.lisp | 186 +++++++++++++++++++++++++++++++++++++++++++++++++++++- rt-tests.lisp | 141 ++++++++++++++++++++++++++++++++++++++++- 2 files changed, 324 insertions(+), 3 deletions(-)

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