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- Log ----------------------------------------------------------------- commit a44327a2532c5afc3cbcb1194aa1f15b6dccd2cb Author: Raymond Toy toy.raymond@gmail.com Date: Thu Mar 24 23:01:51 2011 -0400

Update required accuracy for elliptic-pi.n0.q.

diff --git a/rt-tests.lisp b/rt-tests.lisp index ecb401a..c0b59fb 100644 --- a/rt-tests.lisp +++ b/rt-tests.lisp @@ -1037,7 +1037,7 @@ for epi = (elliptic-pi n phi 0) for true = (/ (atan (* (tan phi) (sqrt (- 1 n)))) (sqrt (- 1 n))) - for result = (check-accuracy 208 epi true) + for result = (check-accuracy 204 epi true) unless (eq nil result) append (list (list (list k n phi) result))) nil)

commit 7fa2a4c33e7542f3538bfcfd15cba833120b0daf Author: Raymond Toy toy.raymond@gmail.com Date: Thu Mar 24 23:01:08 2011 -0400

Add DOMAIN-ERROR condition; fix bug in FRESNEL-S-SERIES.

qd-methods.lisp: o Define DOMAIN-ERROR condition to allow signaling errors for incorrect domains.

qd-gamma.lisp: o Signal domain error in CF-INCOMPLETE-GAMMA-TAIL if necessary. o Fix bug in FRESNEL-S-SERIES. We were comparing a real against a complex.

diff --git a/qd-gamma.lisp b/qd-gamma.lisp index 74a9032..78b5a99 100644 --- a/qd-gamma.lisp +++ b/qd-gamma.lisp @@ -235,6 +235,11 @@ ;; ;; See http://functions.wolfram.com/06.06.10.0003.01 (defun cf-incomplete-gamma-tail (a z) + (when (and (zerop (imagpart z)) (minusp (realpart z))) + (error 'domain-error + :function-name 'cf-incomplete-gamma-tail + :format-arguments (list 'z z) + :format-control "Argument ~S should not be on the negative real axis: ~S")) (/ (* (expt z a) (exp (- z))) (let ((z-a (- z a))) @@ -391,7 +396,7 @@ (sum 0) (term pi/2)) (loop for k2 from 0 by 2 - until (< (abs term) (* eps sum)) + until (< (abs term) (* eps (abs sum))) do (incf sum (/ term (+ 3 k2 k2))) (setf term (/ (* term factor) diff --git a/qd-methods.lisp b/qd-methods.lisp index 29b296d..e4df131 100644 --- a/qd-methods.lisp +++ b/qd-methods.lisp @@ -1123,3 +1123,14 @@ the same precision as the argument. The argument can be complex.")) (defmethod float-pi ((z qd-complex)) +pi+)

+ +(define-condition domain-error (simple-error) + ((function-name :accessor condition-function-name + :initarg :function-name)) + (:report (lambda (condition stream) + (format stream "Domain Error for function ~S:~&" + (condition-function-name condition)) + (pprint-logical-block (stream nil :per-line-prefix " ") + (apply #'format stream + (simple-condition-format-control condition) + (simple-condition-format-arguments condition)))))) \ No newline at end of file

commit bed5763d1250d8b3b3793861f6e17449c49dd15b Author: Raymond Toy toy.raymond@gmail.com Date: Thu Mar 24 14:48:55 2011 -0400

Fix incomplete-gamma-tail for negative reals, use series for Fresnel S for small arg and update tests.

qd-gamma.lisp: o INCOMPLETE-GAMMA-TAIL was hanging for arguments on the negative real axis. Use INCOMPLETE-GAMMA in this case too. o Add the series expansion for Fresnel S and use it for evaluating it for small arguments. We were losing accuracy with the existing algorithm.

rt-tests.lisp: o Update thresholds for elliptic-pi-n0.d, elliptic-pi.n2.q, theta3.1.d. o Fix typo in test name. gamma-incomplete-tail.1.q should have been 2.q. o Add tests for gamma-incomplete-tail for arguments on the negative real axis. o Add tests for Fresnel S.

diff --git a/qd-gamma.lisp b/qd-gamma.lisp index c0cacaa..74a9032 100644 --- a/qd-gamma.lisp +++ b/qd-gamma.lisp @@ -288,10 +288,17 @@ (let* ((prec (float-contagion a z)) (a (apply-contagion a prec)) (z (apply-contagion z prec))) - (if (and (realp a) (realp z)) + (if (and (zerop (imagpart a)) + (zerop (imagpart z))) ;; For real values, we split the result to compute either the - ;; tail directly or from gamma(a) - incomplete-gamma - (if (> (abs z) (abs (- a 1))) + ;; tail directly from the continued fraction or from gamma(a) + ;; - incomplete-gamma. The continued fraction doesn't + ;; converge on the negative real axis, so we can't use that + ;; there. And accuracy appears to be better if z is "small". + ;; We take this to mean |z| < |a-1|. Note that |a-1| is the + ;; peak of the integrand. + (if (and (> (abs z) (abs (- a 1))) + (not (minusp (realpart z)))) (cf-incomplete-gamma-tail a z) (- (gamma a) (incomplete-gamma a z))) (cf-incomplete-gamma-tail a z)))) @@ -364,27 +371,90 @@ (* (expt z (- v 1)) (incomplete-gamma-tail (- 1 v) z)))

+;; Series for Fresnel S +;; +;; S(z) = z^3*sum((%pi/2)^(2*k+1)(-z^4)^k/(2*k+1)!/(4*k+3), k, 0, inf) +;; +;; Compute as +;; +;; S(z) = z^3*sum(a(k)/(4*k+3), k, 0, inf) +;; +;; where +;; +;; a(k+1) = -a(k) * (%pi/2)^2 * z^4 / (2*k+2) / (2*k+3) +;; +;; a(0) = %pi/2. +(defun fresnel-s-series (z) + (let* ((pi/2 (* 1/2 (float-pi z))) + (factor (- (* (expt z 4) pi/2 pi/2))) + (eps (epsilon z)) + (sum 0) + (term pi/2)) + (loop for k2 from 0 by 2 + until (< (abs term) (* eps sum)) + do + (incf sum (/ term (+ 3 k2 k2))) + (setf term (/ (* term factor) + (* (+ k2 2) + (+ k2 3))))) + (* sum (expt z 3)))) + (defun fresnel-s (z) "Fresnel S:

S(z) = integrate(sin(%pi*t^2/2), t, 0, z) " - (let ((sqrt-pi (sqrt (float-pi z)))) + (let ((prec (float-contagion z)) + (sqrt-pi (sqrt (float-pi z)))) (flet ((fs (z) ;; Wolfram gives ;; ;; S(z) = (1+%i)/4*(erf(c*z) - %i*erf(conjugate(c)*z)) ;; ;; where c = sqrt(%pi)/2*(1+%i). - (* #c(1/4 1/4) - (- (erf (* #c(1/2 1/2) sqrt-pi z)) - (* #c(0 1) - (erf (* #c(1/2 -1/2) sqrt-pi z))))))) - (if (realp z) - ;; FresnelS is real for a real argument. And it is odd. - (if (minusp z) - (- (realpart (fs (- z)))) - (realpart (fs z))) - (fs z))))) + ;; + ;; But for large z, we should use erfc. Then + ;; S(z) = 1/2 - (1+%i)/4*(erfc(c*z) - %i*erfc(conjugate(c)*z)) + (if (and t (> (abs z) 2)) + (- 1/2 + (* #c(1/4 1/4) + (- (erfc (* #c(1/2 1/2) sqrt-pi z)) + (* #c(0 1) + (erfc (* #c(1/2 -1/2) sqrt-pi z)))))) + (* #c(1/4 1/4) + (- (erf (* #c(1/2 1/2) sqrt-pi z)) + (* #c(0 1) + (erf (* #c(1/2 -1/2) sqrt-pi z))))))) + (rfs (z) + ;; When z is real, recall that erf(conjugate(z)) = + ;; conjugate(erf(z)). Then + ;; + ;; S(z) = 1/2*(realpart(erf(c*z)) - imagpart(erf(c*z))) + ;; + ;; But for large z, we should use erfc. Then + ;; + ;; S(z) = 1/2 - 1/2*(realpart(erfc(c*z)) - imagpart(erf(c*z))) + (if (> (abs z) 2) + (let ((s (erfc (* #c(1/2 1/2) sqrt-pi z)))) + (- 1/2 + (* 1/2 (- (realpart s) (imagpart s))))) + (let ((s (erf (* #c(1/2 1/2) sqrt-pi z)))) + (* 1/2 (- (realpart s) (imagpart s))))))) + ;; For small z, the erf terms above suffer from subtractive + ;; cancellation. So use the series in this case. Some simple + ;; tests were done to determine that for double-floats we want + ;; to use the series for z < 1 to give max accuracy. For + ;; qd-real, the above formula is good enough for z > 1d-5. + (if (< (abs z) (ecase prec + (single-float 1.5f0) + (double-float 1d0) + (qd-real #q1))) + (fresnel-s-series z) + (if (realp z) + ;; FresnelS is real for a real argument. And it is odd. + (if (minusp z) + (- (rfs (- z))) + (rfs z)) + (fs z))))))

(defun fresnel-c (z) "Fresnel C: diff --git a/rt-tests.lisp b/rt-tests.lisp index 5f4cdd5..ecb401a 100644 --- a/rt-tests.lisp +++ b/rt-tests.lisp @@ -1000,7 +1000,7 @@ for epi = (elliptic-pi n phi 0) for true = (/ (atan (* (tan phi) (sqrt (- 1 n)))) (sqrt (- 1 n))) - for result = (check-accuracy 47.5 epi true) + for result = (check-accuracy 46.5 epi true) unless (eq nil result) append (list (list (list k n phi) result))) nil) @@ -1059,7 +1059,7 @@ for epi = (elliptic-pi n phi 0) for true = (/ (atanh (* (tan phi) (sqrt (- n 1)))) (sqrt (- n 1))) - for result = (check-accuracy 206 epi true) + for result = (check-accuracy 202 epi true) ;; Not sure if this formula holds when atanh gives a complex ;; result. Wolfram doesn't say when (and (not (complexp true)) result) @@ -1075,7 +1075,7 @@ for m = (random 1d0) for t3 = (elliptic-theta-3 0 (elliptic-nome m)) for true = (sqrt (/ (* 2 (elliptic-k m)) (float-pi m))) - for result = (check-accuracy 51 t3 true) + for result = (check-accuracy 50.5 t3 true) when result append (list (list (list k m) result))) nil) @@ -1213,15 +1213,74 @@ nil)

(rt:deftest gamma-incomplete-tail.1.q - (let* ((z 5d0) + (let* ((z #q5) (gi (incomplete-gamma-tail 2 z)) (true (* (+ z 1) (exp (- z))))) - (check-accuracy 212 gi true)) + (check-accuracy 207 gi true)) nil)

-(rt:deftest gamma-incomplete-tail.1.q +(rt:deftest gamma-incomplete-tail.2.q (let* ((z #q(1 5)) (gi (incomplete-gamma-tail 2 z)) (true (* (+ z 1) (exp (- z))))) (check-accuracy 206 gi true)) nil) + +(rt:deftest gamma-incomplete-tail.3.d + (let* ((z -5d0) + (gi (incomplete-gamma-tail 2 z)) + (true (* (+ z 1) (exp (- z))))) + (check-accuracy 52 gi true)) + nil) + +(rt:deftest gamma-incomplete-tail.3.q + (let* ((z #q-5) + (gi (incomplete-gamma-tail 2 z)) + (true (* (+ z 1) (exp (- z))))) + (check-accuracy 206 gi true)) + nil) + + +;; Fresnel integrals. +;; +;; For x small, Fresnel +;; +;; S(z) = %pi/6*z^3*(1 - %pi^2*z^4/56 + %pi^4*z^8/2040 - ...) +;; +(defun fresnel-s-series (z) + (let* ((fpi (float-pi z)) + (z^3 (expt z 3)) + (z^4 (* z^3 z))) + (* fpi 1/6 z^3 + (+ 1 (/ (* fpi fpi z^4) + -56) + (/ (* (expt fpi 4) (expt z^4 2)) + 7040))))) + +(rt:deftest fresnel-s.1d + (let* ((z 1d-3) + (s (fresnel-s z)) + (true (fresnel-s-series z))) + (check-accuracy 52 s true)) + nil) + +(rt:deftest fresnel-s.2d + (let* ((z #c(1d-3 1d-3)) + (s (fresnel-s z)) + (true (fresnel-s-series z))) + (check-accuracy 52 s true)) + nil) + +(rt:deftest fresnel-s.1q + (let* ((z #q1q-20) + (s (fresnel-s z)) + (true (fresnel-s-series z))) + (check-accuracy 212 s true)) + nil) + +(rt:deftest fresnel-s.2q + (let* ((z #q(1q-3 1q-3)) + (s (fresnel-s z)) + (true (fresnel-s-series z))) + (check-accuracy 212 s true)) + nil)

commit 70745f486e3ee012f02c080de0cf9daf00f37375 Author: Raymond Toy toy.raymond@gmail.com Date: Thu Mar 24 14:25:53 2011 -0400

qd-gamma depends on qd-reader.

diff --git a/oct.asd b/oct.asd index e4011a9..7df866c 100644 --- a/oct.asd +++ b/oct.asd @@ -66,7 +66,7 @@ (:file "qd-theta" :depends-on ("qd-methods" "qd-reader")) (:file "qd-gamma" - :depends-on ("qd-methods")) + :depends-on ("qd-methods" "qd-reader")) ))

(defmethod perform ((op test-op) (c (eql (find-system :oct))))

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

Summary of changes: oct.asd | 2 +- qd-gamma.lisp | 103 +++++++++++++++++++++++++++++++++++++++++++++++------- qd-methods.lisp | 11 ++++++ rt-tests.lisp | 73 +++++++++++++++++++++++++++++++++++---- 4 files changed, 167 insertions(+), 22 deletions(-)

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