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- Log ----------------------------------------------------------------- commit 535d2509c2974c113e6c1bfa27ece5d5b8f99bd1 Author: Raymond Toy toy.raymond@gmail.com Date: Sun Mar 6 21:30:15 2011 -0500

Add support for Jacobi elliptic functions

qd-elliptic.lisp: o Implementation for Jacobi sn, cn, and dn functions.

qd-methods.lisp: o Add EPSILON method to return the floating-point epsilon value for the given float.

oct.asd: o Build qd-elliptic.lisp.

diff --git a/oct.asd b/oct.asd index 947b047..153706c 100644 --- a/oct.asd +++ b/oct.asd @@ -58,6 +58,8 @@ :depends-on ("qd-methods")) (:file "qd-complex" :depends-on ("qd-methods")) + (:file "qd-elliptic" + :depends-on ("qd-methods")) ))

(defmethod perform ((op test-op) (c (eql (find-system :oct)))) diff --git a/qd-elliptic.lisp b/qd-elliptic.lisp new file mode 100644 index 0000000..9c56fcd --- /dev/null +++ b/qd-elliptic.lisp @@ -0,0 +1,180 @@ +;;;; -*- 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) + +(declaim (inline descending-transform ascending-transform)) + +(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). + (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. + (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 + (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) + (let ((s (elliptic-sn-descending u m))) + (if (and (realp u) (realp m)) + (realpart s) + s))) + +(defun jacobi-dn (u m) + ;; 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) + ;; Use the ascending Landen transformation, A&S 16.14.3. + (multiple-value-bind (v mu root-mu1) + (ascending-transform u m) + (let ((d (dn v mu))) + (* (/ (+ 1 root-mu1) mu) + (/ (- (* d d) root-mu1) + d))))) \ No newline at end of file diff --git a/qd-methods.lisp b/qd-methods.lisp index b6a83cf..3af7bd0 100644 --- a/qd-methods.lisp +++ b/qd-methods.lisp @@ -1089,3 +1089,23 @@ underlying floating-point format" ;; and make a real qd-real float, instead of the hackish ;; %qd-real. (set-dispatch-macro-character ## #\Q #'qd-class-reader) + + +(defmethod epsilon ((m cl:float)) + (etypecase m + (single-float single-float-epsilon) + (double-float double-float-epsilon))) + +(defmethod epsilon ((m cl:complex)) + (epsilon (realpart m))) + +(defmethod epsilon ((m qd-real)) + ;; What is the epsilon value for a quad-double? This is complicated + ;; by the fact that things like (+ #q1 #q1q-100) is representable as + ;; a quad-double. For most purposes we want epsilon to be close to + ;; the 212 bits of precision (4*53 bits) that we normally have with + ;; a quad-double. + (scale-float #q1 -212)) + +(defmethod epsilon ((m qd-complex)) + (epsilon (realpart m)))

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Summary of changes: oct.asd | 2 + qd-elliptic.lisp | 180 ++++++++++++++++++++++++++++++++++++++++++++++++++++++ qd-methods.lisp | 20 ++++++ 3 files changed, 202 insertions(+), 0 deletions(-) create mode 100644 qd-elliptic.lisp

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