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- Log ----------------------------------------------------------------- commit c239a8514ff5f1fc96d33a4c581cbaec834f6641 Author: Raymond Toy toy.raymond@gmail.com Date: Thu Mar 10 22:23:50 2011 -0500

Implement elliptic integrals of the first kind.

o Add FLOAT-CONTAGION to determine the max precision of the given arguments so we can do appopriate contagion in the routines. o Add some docstrings and other documentation of the algorithms. o Add implmentation of ELLIPTIC-K and ELLIPTIC-F for the complete and incomplete elliptic integrals of the first kind, respectively.

diff --git a/qd-elliptic.lisp b/qd-elliptic.lisp index 2fadecd..63c3687 100644 --- a/qd-elliptic.lisp +++ b/qd-elliptic.lisp @@ -29,6 +29,57 @@

(declaim (inline descending-transform ascending-transform))

+;; Determine which of x and y has the higher precision and return the +;; value of the higher precision number. If both x and y are +;; rationals, just return 1f0, for a single-float value. +(defun float-contagion-2 (x y) + (etypecase x + (cl:rational + (etypecase y + (cl:rational + 1f0) + (cl:float + y) + (qd-real + y))) + (single-float + (etypecase y + ((or cl:rational single-float) + x) + ((or double-float qd-real) + y))) + (double-float + (etypecase y + ((or cl:rational single-float double-float) + x) + (qd-real + y))) + (qd-real + x))) + +;; Return a floating point (or complex) type of the highest precision +;; among all of the given arguments. +(defun float-contagion (&rest args) + ;; It would be easy if we could just add the args together and let + ;; normal contagion do the work, but we could easily introduce + ;; overflows or other errors that way. So look at each argument and + ;; determine the precision and choose the highest precision. + (let ((complexp (some #'complexp args)) + (max-type + (etypecase (reduce #'float-contagion-2 (mapcar #'realpart (if (cdr args) + args + (list (car args) 0)))) + (single-float 'single-float) + (double-float 'double-float) + (qd-real 'qd-real)))) + (if complexp + (if (eq max-type 'qd-real) + 'qd-complex + `(cl:complex ,max-type)) + max-type))) + +;;; Jacobian elliptic functions + (defun ascending-transform (u m) ;; A&S 16.14.1 ;; @@ -153,12 +204,14 @@ 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" ;; Use the Gauss transformation from ;; http://functions.wolfram.com/09.29.16.0013.01: ;; @@ -192,6 +245,7 @@ (+ 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) @@ -202,17 +256,32 @@ (/ (- (* d d) root-mu1) d)))))  +;;; 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(2^(-fpprec)). Not sure this is - ;; quite right, but it makes the routines more accurate as fpprec - ;; increases. + ;; 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) - double-float-epsilon + single-float-epsilon (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* ((xn x) (yn y) (zn z) @@ -252,8 +321,90 @@ (* -3/44 ee2 ee3)))) (/ s (sqrt an)))))

+;; 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)))) + (let ((precision (float-contagion m))) + (carlson-rf (coerce 0 precision) (- 1 m) (coerce 1 precision)))))) + +;; 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) +(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 (coerce x precision)) + (m (coerce 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 (cl: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 (cl: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))) + 1.0)))) + ((< 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)))))))

commit 46359140b7aa3c0c4af6b3aabc7bcbb4cf248301 Author: Raymond Toy toy.raymond@gmail.com Date: Thu Mar 10 22:21:26 2011 -0500

Add documentation for EPSILON and add FLOAT-PI.

FLOAT-PI returns a value of pi that matches the precision of the argument.

diff --git a/qd-methods.lisp b/qd-methods.lisp index 6f41857..966d85f 100644 --- a/qd-methods.lisp +++ b/qd-methods.lisp @@ -1063,7 +1063,13 @@ underlying floating-point format" (frob two-arg-- cl:- sub-qd sub-d-qd sub-qd-d) (frob two-arg-* cl:* mul-qd mul-d-qd mul-qd-d) (frob two-arg-/ cl:/ div-qd nil nil)) - + +(defgeneric epsilon (m) + (:documentation +"Return an epsilon value of the same precision as the argument. It is +the smallest number x such that 1+x /= x. The argument can be +complex")) + (defmethod epsilon ((m cl:float)) (etypecase m (single-float single-float-epsilon) @@ -1082,3 +1088,23 @@ underlying floating-point format"

(defmethod epsilon ((m qd-complex)) (epsilon (realpart m))) + +(defgeneric float-pi (x) + (:documentation +"Return a floating-point value of the mathematical constant pi that is +the same precision as the argument. The argument can be complex.")) + +(defmethod float-pi ((x cl:rational)) + (float pi 1f0)) + +(defmethod float-pi ((x cl:float)) + (float pi x)) + +(defmethod float-pi ((x qd-real)) + +pi+) + +(defmethod float-pi ((z cl:complex)) + (float pi (realpart z))) + +(defmethod float-pi ((z qd-complex)) + +pi+) \ No newline at end of file

commit 85f3e2b22f917ff35d080180ab1e2dda4476e881 Author: Raymond Toy toy.raymond@gmail.com Date: Thu Mar 10 22:20:27 2011 -0500

Move inline function NEG-QD-T before first use.

diff --git a/qd.lisp b/qd.lisp index af5a779..fb3e4e1 100644 --- a/qd.lisp +++ b/qd.lisp @@ -476,11 +476,6 @@ If TARGET is given, TARGET is destructively modified to contain the result." (%store-qd-d target (+ a0 b0) 0d0 0d0 0d0) (%store-qd-d target s0 s1 s2 s3)))))))))))

-(defun neg-qd (a &optional (target #+oct-array (%make-qd-d 0d0 0d0 0d0 0d0))) - "Return the negative of the %QUAD-DOUBLE number A. -If TARGET is given, TARGET is destructively modified to contain the result." - (neg-qd-t a target)) - (defun neg-qd-t (a target) (declare (type %quad-double a #+oct-array target) #+(and cmu (not oct-array)) (ignore target)) @@ -489,6 +484,13 @@ If TARGET is given, TARGET is destructively modified to contain the result." (declare (double-float a0 a1 a2 a3)) (%store-qd-d target (cl:- a0) (cl:- a1) (cl:- a2) (cl:- a3))))

+(defun neg-qd (a &optional (target #+oct-array (%make-qd-d 0d0 0d0 0d0 0d0))) + "Return the negative of the %QUAD-DOUBLE number A. +If TARGET is given, TARGET is destructively modified to contain the result." + (neg-qd-t a target)) + + + (defun sub-qd (a b &optional (target #+oct-array (%make-qd-d 0d0 0d0 0d0 0d0))) "Return the difference between the %QUAD-DOUBLE numbers A and B. If TARGET is given, TARGET is destructively modified to contain the result."

-----------------------------------------------------------------------

Summary of changes: qd-elliptic.lisp | 161 ++++++++++++++++++++++++++++++++++++++++++++++++++++-- qd-methods.lisp | 28 +++++++++- qd.lisp | 12 ++-- 3 files changed, 190 insertions(+), 11 deletions(-)

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