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;;;; -*- Mode: lisp -*-
;;;;
;;;; Copyright (c) 2011 Raymond Toy
;;;; Permission is hereby granted, free of charge, to any person
;;;; obtaining a copy of this software and associated documentation
;;;; files (the "Software"), to deal in the Software without
;;;; restriction, including without limitation the rights to use,
;;;; copy, modify, merge, publish, distribute, sublicense, and/or sell
;;;; copies of the Software, and to permit persons to whom the
;;;; Software is furnished to do so, subject to the following
;;;; conditions:
;;;;
;;;; The above copyright notice and this permission notice shall be
;;;; included in all copies or substantial portions of the Software.
;;;;
;;;; THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
;;;; EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
;;;; OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
;;;; NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
;;;; HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
;;;; WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
;;;; FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
;;;; OTHER DEALINGS IN THE SOFTWARE.
(in-package #:oct)
(eval-when (:compile-toplevel :load-toplevel :execute)
(setf *readtable* *oct-readtable*))
(declaim (inline descending-transform ascending-transform))
;;; Jacobian elliptic functions
(defun ascending-transform (u m)
;; A&S 16.14.1
;;
;; Take care in computing this transform. For the case where
;; m is complex, we should compute sqrt(mu1) first as
;; (1-sqrt(m))/(1+sqrt(m)), and then square this to get mu1.
;; If not, we may choose the wrong branch when computing
;; sqrt(mu1).
;;
;; mu = 4*sqrt(m)/(1+sqrt(m))^2
;; sqrt(mu1) = (1-sqrt(m))/(1+sqrt(m))
;; v = u/(1+sqrt(mu1))
;;
;; Return v, mu, sqrt(mu1)
(let* ((root-m (sqrt m))
(mu (/ (* 4 root-m)
(expt (1+ root-m) 2)))
(root-mu1 (/ (- 1 root-m) (+ 1 root-m)))
(v (/ u (1+ root-mu1))))
(values v mu root-mu1)))
(defun descending-transform (u m)
;; Note: Don't calculate mu first, as given in 16.12.1. We
;; should calculate sqrt(mu) = (1-sqrt(m1)/(1+sqrt(m1)), and
;; then compute mu = sqrt(mu)^2. If we calculate mu first,
;; sqrt(mu) loses information when m or m1 is complex.
;;
;; sqrt(mu) = (1-sqrt(m1))/(1+sqrt(m1))
;; v = u/(1+sqrt(mu))
;;
;; where m1 = 1-m
;;
;; Return v, mu, sqrt(mu)
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(let* ((root-m1 (sqrt (- 1 m)))
(root-mu (/ (- 1 root-m1) (+ 1 root-m1)))
(mu (* root-mu root-mu))
(v (/ u (1+ root-mu))))
(values v mu root-mu)))
;; Could use the descending transform, but some of my tests show
;; that it has problems with roundoff errors.
;; WARNING: This doesn't work very well for u > 1000 or so. For
;; example (elliptic-dn-ascending 1000b0 .5b0) -> 3.228b324, but dn <= 1.
#+nil
(defun elliptic-dn-ascending (u m)
(cond ((zerop m)
;; A&S 16.6.3
1.0)
((< (abs (- 1 m)) (* 4 (epsilon u)))
;; A&S 16.6.3
(/ (cosh u)))
(t
(multiple-value-bind (v mu root-mu1)
(ascending-transform u m)
;; A&S 16.14.4
(let* ((new-dn (elliptic-dn-ascending v mu)))
(* (/ (- 1 root-mu1) mu)
(/ (+ root-mu1 (* new-dn new-dn))
new-dn)))))))
;; Don't use the descending version because it requires cn, dn, and
;; sn.
;;
;; WARNING: This doesn't work very well for large u.
;; (elliptic-cn-ascending 1000b0 .5b0) -> 4.565b324. But |cn| <= 1.
#+nil
(defun elliptic-cn-ascending (u m)
(cond ((zerop m)
;; A&S 16.6.2
(cos u))
((< (abs (- 1 m)) (* 4 (epsilon u)))
;; A&S 16.6.2
(/ (cl:cosh u)))
(t
(multiple-value-bind (v mu root-mu1)
(ascending-transform u m)
;; A&S 16.14.3
(let* ((new-dn (elliptic-dn-ascending v mu)))
(* (/ (+ 1 root-mu1) mu)
(/ (- (* new-dn new-dn) root-mu1)
new-dn)))))))
;;
;; This appears to work quite well for both real and complex values
;; of u.
(defun elliptic-sn-descending (u m)
(cond ((= m 1)
;; A&S 16.6.1
(tanh u))
((< (abs m) (epsilon u))
;; A&S 16.6.1
(sin u))
(t
;; A&S 16.12.2
;;
;; sn(u|m) = (1 + sqrt(mu))*sn(v|u)/(1 + sqrt(mu)*sn(v|mu)^2)
(multiple-value-bind (v mu root-mu)
(descending-transform u m)
(let* ((new-sn (elliptic-sn-descending v mu)))
(/ (* (1+ root-mu) new-sn)
(1+ (* root-mu new-sn new-sn))))))))
;; We don't use the ascending transform here because it requires
;; evaluating sn, cn, and dn. The ascending transform only needs
;; sn.
#+nil
(defun elliptic-sn-ascending (u m)
(if (< (abs (- 1 m)) (* 4 flonum-epsilon))
;; A&S 16.6.1
(tanh u)
(multiple-value-bind (v mu root-mu1)
(ascending-transform u m)
;; A&S 16.14.2
(let* ((new-cn (elliptic-cn-ascending v mu))
(new-dn (elliptic-dn-ascending v mu))
(new-sn (elliptic-sn-ascending v mu)))
(/ (* (+ 1 root-mu1) new-sn new-cn)
new-dn)))))
(defun jacobi-sn (u m)
"Compute Jacobian sn for argument u and parameter m"
(let ((s (elliptic-sn-descending u m)))
(if (and (realp u) (realp m))
(realpart s)
s)))
(defun jacobi-dn (u m)
"Compute Jacobi dn for argument u and parameter m"
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;; Use the Gauss transformation from
;; http://functions.wolfram.com/09.29.16.0013.01:
;;
;;
;; dn((1+sqrt(m))*z, 4*sqrt(m)/(1+sqrt(m))^2)
;; = (1-sqrt(m)*sn(z, m)^2)/(1+sqrt(m)*sn(z,m)^2)
;;
;; So
;;
;; dn(y, mu) = (1-sqrt(m)*sn(z, m)^2)/(1+sqrt(m)*sn(z,m)^2)
;;
;; where z = y/(1+sqrt(m)) and mu=4*sqrt(m)/(1+sqrt(m))^2.
;;
;; Solve for m, and we get
;;
;; sqrt(m) = -(mu+2*sqrt(1-mu)-2)/mu or (-mu+2*sqrt(1-mu)+2)/mu.
;;
;; I don't think it matters which sqrt we use, so I (rtoy)
;; arbitrarily choose the first one above.
;;
;; Note that (1-sqrt(1-mu))/(1+sqrt(1-mu)) is the same as
;; -(mu+2*sqrt(1-mu)-2)/mu. Also, the former is more
;; accurate for small mu.
(let* ((root (let ((root-1-m (sqrt (- 1 m))))
(/ (- 1 root-1-m)
(+ 1 root-1-m))))
(z (/ u (+ 1 root)))
(s (elliptic-sn-descending z (* root root)))
(p (* root s s )))
(/ (- 1 p)
(+ 1 p))))
(defun jacobi-cn (u m)
"Compute Jacobi cn for argument u and parameter m"
;; Use the ascending Landen transformation, A&S 16.14.3.
;;
;; cn(u,m) = (1+sqrt(mu1))/mu * (dn(v,mu)^2-sqrt(mu1))/dn(v,mu)
(multiple-value-bind (v mu root-mu1)
(ascending-transform u m)
(* (/ (+ 1 root-mu1) mu)
(/ (- (* d d) root-mu1)
;;; Elliptic Integrals
;;;
;; Translation of Jim FitzSimons' bigfloat implementation of elliptic
;; integrals from http://www.getnet.com/~cherry/elliptbf3.mac.
;;
;; The algorithms are based on B.C. Carlson's "Numerical Computation
;; of Real or Complex Elliptic Integrals". These are updated to the
;; algorithms in Journal of Computational and Applied Mathematics 118
;; (2000) 71-85 "Reduction Theorems for Elliptic Integrands with the
;; Square Root of two quadritic factors"
;;
(defun errtol (&rest args)
;; Compute error tolerance as sqrt(<float-precision>). Not sure
;; this is quite right, but it makes the routines more accurate as
;; precision increases increases.
(sqrt (reduce #'min (mapcar #'(lambda (x)
(if (rationalp x)
(epsilon x)))
args))))
(defun carlson-rf (x y z)
"Compute Carlson's Rf function:
Rf(x, y, z) = 1/2*integrate((t+x)^(-1/2)*(t+y)^(-1/2)*(t+z)^(-1/2), t, 0, inf)"
(let* ((precision (float-contagion x y z))
(xn (apply-contagion x precision))
(yn (apply-contagion y precision))
(zn (apply-contagion z precision))
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(a (/ (+ xn yn zn) 3))
(epslon (/ (max (abs (- a xn))
(abs (- a yn))
(abs (- a zn)))
(errtol x y z)))
(an a)
(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 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)))
(ee2 (- (* xndev yndev) (* 6 zndev zndev)))
(ee3 (* xndev yndev zndev))
(s (+ 1
(* -1/10 ee2)
(* 1/14 ee3)
(* 1/24 ee2 ee2)
(* -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* ((precision (float-contagion x y z))
(xn (apply-contagion x precision))
(yn (apply-contagion y precision))
(zn (apply-contagion z precision))
(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:
;;
;; K(m) = Rf(0, 1 - m, 1)
(defun elliptic-k (m)
"Complete elliptic integral of the first kind K for parameter m
K(m) = integrate(1/sqrt(1-m*sin(x)^2), x, 0, %pi/2).
Note: K(m) = F(%pi/2, m), where F is the (incomplete) elliptic
integral of the first kind."
(cond ((= m 0)
(/ (float +pi+ m) 2))
(t
(carlson-rf 0 (- 1 m) 1))))
;; 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)
;;
;; 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:
F(x, m) = integrate(1/sqrt(1-m*sin(phi)^2), phi, 0, x)
Note for the complete elliptic integral, you can use elliptic-k"
(let* ((precision (float-contagion x m))
(x (apply-contagion x precision))
(m (apply-contagion m precision)))
(cond ((and (realp m) (realp x))
(cond ((> m 1)
;; A&S 17.4.15
;;
;; F(phi|m) = 1/sqrt(m)*F(theta|1/m)
;;
;; with sin(theta) = sqrt(m)*sin(phi)
(/ (elliptic-f (asin (* (sqrt m) (sin x))) (/ m))
(sqrt m)))
((< m 0)
;; A&S 17.4.17
(let* ((m (- m))
(m+1 (+ 1 m))
(root (sqrt m+1))
(m/m+1 (/ m m+1)))
(- (/ (elliptic-f (/ (float-pi m) 2) m/m+1)
root)
(/ (elliptic-f (- (/ (float-pi x) 2) x) m/m+1)
root))))
((= m 0)
;; A&S 17.4.19
x)
((= m 1)
;; A&S 17.4.21
;;
;; F(phi,1) = log(sec(phi)+tan(phi))
;; = log(tan(pi/4+pi/2))
(log (tan (+ (/ x 2) (/ (float-pi x) 4)))))
((minusp x)
(- (elliptic-f (- x) m)))
((> x (float-pi x))
;; A&S 17.4.3
(multiple-value-bind (s x-rem)
(truncate x (float-pi x))
(+ (* 2 s (elliptic-k m))
(elliptic-f x-rem m))))
((<= x (/ (float-pi x) 2))
(let ((sin-x (sin x))
(cos-x (cos x))
(k (sqrt m)))
(* sin-x
(carlson-rf (* cos-x cos-x)
(* (- 1 (* k sin-x))
(+ 1 (* k sin-x)))
((< x (float-pi x))
(+ (* 2 (elliptic-k m))
(elliptic-f (- x (float pi x)) m)))))
(t
(let ((sin-x (sin x))
(cos-x (cos x))
(k (sqrt m)))
(* sin-x
(carlson-rf (* cos-x cos-x)
(* (- 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)
;;
;; 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:
E(phi, m) = integrate(sqrt(1-m*sin(x)^2), x, 0, phi)"
(let* ((precision (float-contagion phi m))
(phi (apply-contagion phi precision))
(m (apply-contagion m precision)))
(cond ((= m 0)
;; A&S 17.4.23
phi)
((= m 1)
;; A&S 17.4.25
(sin phi))
(t
;; 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:
E(m) = integrate(sqrt(1-m*sin(x)^2), x, 0, %pi/2)"
(cond ((= m 0)
;; A&S 17.4.23
(/ (float-pi m) 2))
((= m 1)
;; A&S 17.4.25
(float 1 m))
(t
(let* ((m1 (- 1 m)))
(- (carlson-rf 0 m1 1)
(carlson-rd 0 m1 1)))))))
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;; 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)
;;
;;
;; Carlson writes
;;
;; P(phi,k,n) = integrate((1+n*sin(t)^2)^(-1)*(1-k^2*sin(t)^2)^(-1/2), t, 0, phi)
;; = sin(phi)*Rf(cos(phi)^2, 1-k^2*sin(phi)^2, 1)
;; - n/3*sin(phi)^3*Rj(cos(phi)^2, 1-k^2*sin(phi)^2, 1, 1+n*sin(phi)^2)
;;
;; Note that this definition as a different sign for the n parameter from A&S!
(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)"
(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))
(m-sin2 (- 1 (* m sin-phi sin-phi))))
(- (* sin-phi (carlson-rf (expt cos-phi 2) m-sin2 1))
(* (/ nn 3) (expt sin-phi 3)
(carlson-rj (expt cos-phi 2) m-sin2 1
(+ 1 (* nn (expt sin-phi 2))))))))