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- Log ----------------------------------------------------------------- commit a40062c6860e8e778c1b15a48c2687c6cd2f5334 Author: Raymond Toy toy.raymond@gmail.com Date: Wed Mar 16 19:43:54 2011 -0400

Add gamma and log-gamma functions; work in progress.

oct.asd: o Add qd-gamma.lisp. The implementations need some work. The accuracy is less than desired because gamma(2.0) /= 1. It's close but not quite right.

rt-tests.lisp: o Basic tests of the gamma function. Accuracy is not as good as we would ike.

qd-gamma.lisp: o New file for implementation of gamma function.

diff --git a/oct.asd b/oct.asd index 89b56ee..e4011a9 100644 --- a/oct.asd +++ b/oct.asd @@ -65,6 +65,8 @@ :depends-on ("qd-methods" "qd-reader")) (:file "qd-theta" :depends-on ("qd-methods" "qd-reader")) + (:file "qd-gamma" + :depends-on ("qd-methods")) ))

(defmethod perform ((op test-op) (c (eql (find-system :oct)))) diff --git a/qd-gamma.lisp b/qd-gamma.lisp new file mode 100644 index 0000000..bb9004a --- /dev/null +++ b/qd-gamma.lisp @@ -0,0 +1,170 @@ +;;;; -*- 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*)) + +;; For log-gamma we use the asymptotic formula +;; +;; log(gamma(z)) ~ (z - 1/2)*log(z) + log(2*%pi)/2 +;; + sum(bern(2*k)/(2*k)/(2*k-1)/z^(2k-1), k, 1, inf) +;; +;; = (z - 1/2)*log(z) + log(2*%pi)/2 +;; + 1/12/z*(1 - 1/30/z^2 + 1/105/z^4 + 1/140/z^6 + ... +;; + 174611/10450/z^18 + ...) +;; +;; For double-floats, let's stop the series at the power z^18. The +;; next term is 77683/483/z^20. This means that for |z| > 8.09438, +;; the series has double-float precision. +;; +;; For quad-doubles, let's stop the series at the power z^62. The +;; next term is about 6.364d37/z^64. So for |z| > 38.71, the series +;; has quad-double precision. +(defparameter *log-gamma-asymp-coef* + #(-1/30 1/105 -1/140 1/99 -691/30030 1/13 -3617/10200 43867/20349 + -174611/10450 77683/483 -236364091/125580 657931/25 -3392780147/7830 + 1723168255201/207669 -7709321041217/42160 151628697551/33 + -26315271553053477373/201514950 154210205991661/37 + -261082718496449122051/1758900 1520097643918070802691/259161 + -2530297234481911294093/9890 25932657025822267968607/2115 + -5609403368997817686249127547/8725080 19802288209643185928499101/539 + -61628132164268458257532691681/27030 29149963634884862421418123812691/190323 + -354198989901889536240773677094747/31900 + 2913228046513104891794716413587449/3363 + -1215233140483755572040304994079820246041491/16752085350 + 396793078518930920708162576045270521/61 + -106783830147866529886385444979142647942017/171360 + 133872729284212332186510857141084758385627191/2103465 + )) + +#+nil +(defun log-gamma-asymp-series (z nterms) + ;; Sum the asymptotic formula for n terms + ;; + ;; 1 + sum(c[k]/z^(2*k+2), k, 0, nterms) + (let ((z2 (* z z)) + (sum 1) + (term 1)) + (dotimes (k nterms) + (setf term (* term z2)) + (incf sum (/ (aref *log-gamma-asymp-coef* k) term))) + sum)) + +(defun log-gamma-asymp-series (z nterms) + (loop with y = (* z z) + for k from 1 to nterms + for x = 0 then + (setf x (/ (+ x (aref *log-gamma-asymp-coef* (- nterms k))) + y)) + finally (return (+ 1 x)))) + + +(defun log-gamma-asymp-principal (z nterms log2pi/2) + (+ (- (* (- z 1/2) + (log z)) + z) + log2pi/2)) + +(defun log-gamma-asymp (z nterms log2pi/2) + (+ (log-gamma-asymp-principal z nterms log2pi/2) + (* 1/12 (/ (log-gamma-asymp-series z nterms) z)))) + +(defun log2pi/2 (precision) + (ecase precision + (single-float + (coerce (/ (log (* 2 pi)) 2) 'single-float)) + (double-float + (coerce (/ (log (* 2 pi)) 2) 'double-float)) + (qd-real + (/ (log +2pi+) 2)))) + +(defun log-gamma-aux (z limit nterms) + (let ((precision (float-contagion z))) + (cond ((minusp (realpart z)) + ;; Use reflection formula if realpart(z) < 0 + ;; log(gamma(-z)) = log(pi)-log(-z)-log(sin(pi*z))-log(gamma(z)) + ;; Or + ;; log(gamma(z)) = log(pi)-log(-z)-log(sin(pi*z))-log(gamma(-z)) + (- (apply-contagion (log pi) precision) + (log (- z)) + (apply-contagion (log (sin (* pi z))) precision) + (log-gamma (- z)))) + (t + (let ((absz (abs z))) + (cond ((>= absz limit) + ;; Can use the asymptotic formula directly with 9 terms + (log-gamma-asymp z nterms (log2pi/2 precision))) + (t + ;; |z| is too small. Use the formula + ;; log(gamma(z)) = log(gamma(z+1)) - log(z) + (- (log-gamma (+ z 1)) + (log z))))))))) + +(defmethod log-gamma ((z cl:number)) + (log-gamma-aux z 9 9)) + +(defmethod log-gamma ((z qd-real)) + (log-gamma-aux z 26 26)) + +(defmethod log-gamma ((z qd-complex)) + (log-gamma-aux z 26 26)) + +(defun gamma-aux (z limit nterms) + (let ((precision (float-contagion z))) + (cond ((minusp (realpart z)) + ;; Use reflection formula if realpart(z) < 0: + ;; gamma(-z) = -pi*csc(pi*z)/gamma(z+1) + ;; or + ;; gamma(z) = pi*csc(pi*z)/gamma(1-z) + (/ (float-pi z) + (sin (* (float-pi z) z)) + (gamma-aux (- 1 z) limit nterms))) + (t + (let ((absz (abs z))) + (cond ((>= absz limit) + ;; Use log gamma directly: + ;; log(gamma(z)) = principal part + 1/12/z*(series part) + ;; so + ;; gamma(z) = exp(principal part)*exp(1/12/z*series) + (exp (log-gamma z)) + #+nil + (* (exp (log-gamma-asymp-principal z nterms + (log2pi/2 precision))) + (exp (* 1/12 (/ (log-gamma-asymp-series z nterms) z))))) + (t + ;; 1 <= |z| <= limit + ;; gamma(z) = gamma(z+1)/z + (/ (gamma-aux (+ 1 z) limit nterms) z)))))))) + +(defmethod gamma ((z cl:number)) + (gamma-aux z 9 9)) + +(defmethod gamma ((z qd-real)) + (gamma-aux z 39 32)) + +(defmethod gamma ((z qd-complex)) + (gamma-aux z 39 32)) + diff --git a/rt-tests.lisp b/rt-tests.lisp index c624f4a..59d798f 100644 --- a/rt-tests.lisp +++ b/rt-tests.lisp @@ -1141,3 +1141,57 @@ when result append (list (list (list k m) result))) nil) + +(rt:deftest gamma.1.d + (let ((g (gamma 0.5d0)) + (true (sqrt pi))) + ;; This should give full accuracy but doesn't. + (check-accuracy 51 g true)) + nil) + +(rt:deftest gamma.1.q + (let ((g (gamma #q0.5)) + (true (sqrt +pi+))) + ;; This should give full accuracy but doesn't. + (check-accuracy 197 g true)) + nil) + +(rt:deftest gamma.2.d + (loop for k from 0 below 100 + for y = (+ 1 (random 100d0)) + for g = (abs (gamma (complex 0 y))) + for true = (sqrt (/ pi y (sinh (* pi y)))) + for result = (check-accuracy 45 g true) + when result + append (list (list (list k y) result))) + nil) + +(rt:deftest gamma.2.q + (loop for k from 0 below 100 + for y = (+ 1 (random #q100)) + for g = (abs (gamma (complex 0 y))) + for true = (sqrt (/ +pi+ y (sinh (* +pi+ y)))) + for result = (check-accuracy 196 g true) + when result + append (list (list (list k y) result))) + nil) + +(rt:deftest gamma.3.d + (loop for k from 0 below 100 + for y = (+ 1 (random 100d0)) + for g = (abs (gamma (complex 1/2 y))) + for true = (sqrt (/ pi (cosh (* pi y)))) + for result = (check-accuracy 44 g true) + when result + append (list (list (list k y) result))) + nil) + +(rt:deftest gamma.3.q + (loop for k from 0 below 100 + for y = (+ 1 (random #q100)) + for g = (abs (gamma (complex 1/2 y))) + for true = (sqrt (/ +pi+ (cosh (* +pi+ y)))) + for result = (check-accuracy 196 g true) + when result + append (list (list (list k y) result))) + nil)

commit a666f3923b82be0eb435ddd3e9b20697097fafe0 Author: Raymond Toy toy.raymond@gmail.com Date: Wed Mar 16 19:41:33 2011 -0400

QCOMPLEX takes two required args now.

Fixes issues like (complex 1/2 #q1), which was signaling an error.

qd-class.lisp: o Update defgeneric for QCOMPLEX for two required args.

qd-methods.lisp: o Update existing QCOMPLEX methods to take two args. o Add methods to QCOMPLEX to handle the missing cases.

diff --git a/qd-class.lisp b/qd-class.lisp index 2134ef7..43cc2e9 100644 --- a/qd-class.lisp +++ b/qd-class.lisp @@ -203,8 +203,8 @@ (defgeneric qexpt (x y) (:documentation "X^Y"))

-(defgeneric qcomplex (x &optional y) - (:documentation "Create a complex number with components X and Y. If Y not given, assume 0")) +(defgeneric qcomplex (x y) + (:documentation "Create a complex number with components X and Y."))

(defgeneric qinteger-decode-float (f) (:documentation "integer-decode-float")) diff --git a/qd-methods.lisp b/qd-methods.lisp index fa27ce2..9341311 100644 --- a/qd-methods.lisp +++ b/qd-methods.lisp @@ -507,13 +507,21 @@ underlying floating-point format" (return nil))) (return nil))))

-(defmethod qcomplex ((x real) &optional y) - (cl:complex x (if y y 0))) +(defmethod qcomplex ((x cl:real) (y cl:real)) + (cl:complex x y))

-(defmethod qcomplex ((x qd-real) &optional y) +(defmethod qcomplex ((x cl:real) (y qd-real)) + (qcomplex (make-qd x) y)) + +(defmethod qcomplex ((x qd-real) (y qd-real)) + (make-instance 'qd-complex + :real (qd-value x) + :imag (qd-value y))) + +(defmethod qcomplex ((x qd-real) (y cl:real)) (make-instance 'qd-complex :real (qd-value x) - :imag (if y (qd-value y) +qd-zero+))) + :imag (make-qd-d y)))

(defun complex (x &optional (y 0)) (qcomplex x y))

commit a93aca9a6cfdb5b1305cb2570eea23e6f44b51c0 Author: Raymond Toy toy.raymond@gmail.com Date: Tue Mar 15 13:14:05 2011 -0400

Move float-contagion stuff to qd-methods.

diff --git a/qd-elliptic.lisp b/qd-elliptic.lisp index 385f00f..475763b 100644 --- a/qd-elliptic.lisp +++ b/qd-elliptic.lisp @@ -29,59 +29,6 @@

(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)))) - max-type)) - -(defun apply-contagion (number precision) - (etypecase number - ((or cl:real qd-real) - (coerce number precision)) - ((or cl:complex qd-complex) - (complex (coerce (realpart number) precision) - (coerce (imagpart number) precision))))) - ;;; Jacobian elliptic functions

(defun ascending-transform (u m) diff --git a/qd-methods.lisp b/qd-methods.lisp index 814672c..fa27ce2 100644 --- a/qd-methods.lisp +++ b/qd-methods.lisp @@ -1048,4 +1048,58 @@ the same precision as the argument. The argument can be complex.")) (float pi (realpart z)))

(defmethod float-pi ((z qd-complex)) - +pi+) \ No newline at end of file + +pi+) + +;; 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)))) + max-type)) + +(defun apply-contagion (number precision) + (etypecase number + ((or cl:real qd-real) + (coerce number precision)) + ((or cl:complex qd-complex) + (complex (coerce (realpart number) precision) + (coerce (imagpart number) precision))))) +

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

Summary of changes: oct.asd | 2 + qd-class.lisp | 4 +- qd-elliptic.lisp | 53 ----------------- qd-gamma.lisp | 170 ++++++++++++++++++++++++++++++++++++++++++++++++++++++ qd-methods.lisp | 72 +++++++++++++++++++++-- rt-tests.lisp | 54 +++++++++++++++++ 6 files changed, 295 insertions(+), 60 deletions(-) create mode 100644 qd-gamma.lisp

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