Commit 390a7483 by Raymond Toy

### Add prelimary support for integrals of the 3rd kind.

qd-elliptic.lisp:
o Clean up for unused variable in ELLIPTIC-K
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.
parent 147fa2c7
 ... ... @@ -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))))))))
 ... ... @@ -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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