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- Log ----------------------------------------------------------------- commit 6cfb0ac4b6bcc1a25bc119e87fd2b57bfa1f4355 Merge: 8ec0d20 dd5c2a4 Author: Raymond Toy toy.raymond@gmail.com Date: Mon Apr 9 08:37:51 2012 -0700

Merge branch 'master' of ssh://common-lisp.net/var/git/projects/oct/oct

diff --cc qd-gamma.lisp index 410c315,f0e2418..fe79813 --- a/qd-gamma.lisp +++ b/qd-gamma.lisp @@@ -378,10 -372,13 +378,11 @@@ "Tail of the incomplete gamma function defined by:

integrate(t^(a-1)*exp(-t), t, z, inf)" - (let* ((prec (float-contagion a z)) - (a (apply-contagion a prec)) - (z (apply-contagion z prec))) + (with-floating-point-contagion (a z) - (if (zerop a) - ;; incomplete_gamma_tail(0, z) = exp_integral_e(1,z) - (exp-integral-e 1 z) + (if (and (realp a) (<= a 0)) + ;; incomplete_gamma_tail(v, z) = z^v*exp_integral_e(1-a,z) + (* (expt z a) + (exp-integral-e (- 1 a) z)) (if (and (zerop (imagpart a)) (zerop (imagpart z))) ;; For real values, we split the result to compute either the @@@ -531,20 -528,30 +532,36 @@@ ;; for |arg(z)| < pi. ;; ;; - (cond ((and (realp v) (minusp v)) - ;; E(-v, z) = z^(-v-1)*incomplete_gamma_tail(v+1,z) - (let ((-v (- v))) + (let* ((prec (float-contagion v z)) + (v (apply-contagion v prec)) + (z (apply-contagion z prec))) + (cond ((and (realp v) (minusp v)) + ;; E(-v, z) = z^(-v-1)*incomplete_gamma_tail(v+1,z) + (let ((-v (- v))) + (* (expt z (- v 1)) + (incomplete-gamma-tail (+ -v 1) z)))) + ((< (abs z) 1) + ;; Use series for small z + (s-exp-integral-e v z)) + ((>= (abs (phase z)) 3.1) + ;; The continued fraction doesn't converge on the negative + ;; real axis, and converges very slowly near the negative + ;; real axis, so use the incomplete-gamma-tail function in + ;; this region. "Closeness" to the negative real axis is + ;; teken to mean that z is in a sector near the axis. + ;; + ;; E(v,z) = z^(v-1)*incomplete_gamma_tail(1-v,z) (* (expt z (- v 1)) - (incomplete-gamma-tail (- 1 v) z))) - (t - ;; Use continued fraction for everything else. - (cf-exp-integral-e v z))))) + (incomplete-gamma-tail (+ -v 1) z)))) + ((or (< (abs z) 1) (>= (abs (phase z)) 3.1)) + ;; Use series for small z or if z is near the negative real + ;; axis because the continued fraction does not converge on + ;; the negative axis and converges slowly near the negative + ;; axis. + (s-exp-integral-e v z)) + (t + ;; Use continued fraction for everything else. + (cf-exp-integral-e v z))))

;; Series for Fresnel S ;; diff --cc qd-methods.lisp index f73c02b,4fac522..2a38e58 --- a/qd-methods.lisp +++ b/qd-methods.lisp @@@ -1139,4 -1132,4 +1146,4 @@@ the same precision as the argument. Th (pprint-logical-block (stream nil :per-line-prefix " ") (apply #'format stream (simple-condition-format-control condition) -- (simple-condition-format-arguments condition)))))) ++ (simple-condition-format-arguments condition))))))

commit 8ec0d2004ed4063fd568250b00d940b208573a92 Author: Raymond Toy toy.raymond@gmail.com Date: Sun Apr 8 10:14:37 2012 -0700

Define FLOATP, fix bugs in FLOAT.

qd-methods.lisp: * Define FLOATP * Fix bugs in FLOAT: * (FLOAT float nil) is an error * (FLOAT float) returns the float * (FLOAT rational) returns a single-float.

qd-package.lisp: o Export FLOATP, shadowing CL:FLOAT.

rt-tests.lisp: o Add a few tests for FLOAT.

diff --git a/qd-methods.lisp b/qd-methods.lisp index b54fb4b..f73c02b 100644 --- a/qd-methods.lisp +++ b/qd-methods.lisp @@ -255,6 +255,9 @@ (make-instance 'qd-real :value (rational-to-qd bignum)))

+(defun floatp (x) + (typep x '(or short-float single-float double-float long-float qd-real))) + (defmethod qfloat ((x real) (num-type cl:float)) (cl:float x num-type))

@@ -299,8 +302,12 @@ x)

(declaim (inline float)) -(defun float (x &optional num-type) - (qfloat x (or num-type 1.0))) +(defun float (x &optional (other nil otherp)) + (if otherp + (qfloat x other) + (if (floatp x) + x + (qfloat x 1.0))))

(defmethod qrealpart ((x number)) (cl:realpart x)) diff --git a/qd-package.lisp b/qd-package.lisp index 7db5cd3..ed71347 100644 --- a/qd-package.lisp +++ b/qd-package.lisp @@ -180,6 +180,7 @@ #:decode-float #:scale-float #:float + #:floatp #:floor #:ffloor #:ceiling @@ -252,6 +253,7 @@ #:decode-float #:scale-float #:float + #:floatp #:floor #:ffloor #:ceiling diff --git a/rt-tests.lisp b/rt-tests.lisp index 173c528..eda6b0e 100644 --- a/rt-tests.lisp +++ b/rt-tests.lisp @@ -55,6 +55,22 @@

;;; Some simple tests from the Yozo Hida's qd package.

+(rt:deftest float.1 + (float 3/2) + 1.5) + +(rt:deftest float.2 + (float 3/2 1d0) + 1.5d0) + +(rt:deftest float.3 + (float 1.5d0) + 1.5d0) + +(rt:deftest float.4 + (= (float #q1.5) #q1.5) + t) + (rt:deftest ceiling-d.1 (multiple-value-list (ceiling -50d0)) (-50 0d0))

commit bd6d814332fdf6a831873bb33c9e4753f29d414f Author: Raymond Toy toy.raymond@gmail.com Date: Sun Apr 8 09:59:39 2012 -0700

Oops. The second arg to FLOAT is optional!

diff --git a/qd-methods.lisp b/qd-methods.lisp index c461ba3..b54fb4b 100644 --- a/qd-methods.lisp +++ b/qd-methods.lisp @@ -299,8 +299,8 @@ x)

(declaim (inline float)) -(defun float (x num-type) - (qfloat x num-type)) +(defun float (x &optional num-type) + (qfloat x (or num-type 1.0)))

(defmethod qrealpart ((x number)) (cl:realpart x))

commit c7fc989dad090ada2c046b28bf43406810474a77 Author: Raymond Toy toy.raymond@gmail.com Date: Sun Apr 8 09:57:12 2012 -0700

Fix typo in %big-a. Just use incomplete-gamma-tail in big-i.

diff --git a/qd-bessel.lisp b/qd-bessel.lisp index ee5b9ef..fbe9760 100644 --- a/qd-bessel.lisp +++ b/qd-bessel.lisp @@ -298,7 +298,7 @@ (cond ((and (= m v) (minusp m)) (if (< n m) (%big-a n v) - (let ((result (%big-a (+ n m) v))) + (let ((result (%big-a (+ n m) (- v)))) (if (oddp (truncate m)) result (- result))))) @@ -310,32 +310,18 @@ ;; ;; Use the substitution u=1+s to get a new integral ;; -;; integrate(exp(-t*z*s)*(1+s)^(-2*n-v), s, 0, inf) -;; = exp(t*z) * integrate(u^(-v-2*n)*exp(-t*u*z), u, 1, inf) -;; = exp(t*z)*t^(v+2*n-1)*z^(v+2*n-1)*incomplete_gamma_tail(1-v-2*n,t*z) -;; -;; The continued fraction for incomplete_gamma_tail(a,z) is -;; -;; z^a*exp(-z)/CF -;; -;; So incomplete_gamma_tail(1-v-2*n, t*z) is -;; -;; (t*z)^(1-v-2*n)/CF +;; integrate(exp(-t*z*s)*(1+s)^(-2*n-v), s, 0, inf) +;; = exp(t*z) * integrate(u^(-v-2*n)*exp(-t*u*z), u, 1, inf) +;; = exp(t*z)*t^(v+2*n-1)*z^(v+2*n-1)*incomplete_gamma_tail(1-v-2*n,t*z) ;; -;; which finally gives +;; Thus, ;; -;; integrate(exp(-t*z*s)*(1+s)^(-2*n-v), s, 0, inf) -;; = (t*z)^(1-v-2*n)/CF +;; I[n](t, z, v) = z^(v+2*n-1)*incomplete_gamma_tail(1-v-2*n,t*z) ;; -;; and I[n](t, z, v) = exp(-t*z)/CF (defun big-i (n theta z v) - (/ (exp (- (* theta z))) - (let* ((a (- 1 v n n)) - (z-a (- (* theta z) a))) - (lentz #'(lambda (n) - (+ n n 1 z-a)) - #'(lambda (n) - (* n (- a n))))))) + (let* ((a (- 1 v n n))) + (* (expt z (- a)) + (incomplete-gamma-tail a (* theta z)))))

(defun sum-big-ia (big-n v z) )

commit 95aa580ce0dd62113535c30c4c0c82f8656c2d86 Author: Raymond Toy toy.raymond@gmail.com Date: Sun Apr 8 09:37:22 2012 -0700

T is not a variable. Use a different name.

diff --git a/qd-bessel.lisp b/qd-bessel.lisp index 6d153a7..ee5b9ef 100644 --- a/qd-bessel.lisp +++ b/qd-bessel.lisp @@ -320,19 +320,18 @@ ;; ;; So incomplete_gamma_tail(1-v-2*n, t*z) is ;; -;; (t*z)^(1-v-2*n)*exp(-t*z)/CF +;; (t*z)^(1-v-2*n)/CF ;; ;; which finally gives ;; -;; integrate(exp(-t*z*s)*(1+s)^(-2*n-v), s, 0, inf) -;; = CF +;; integrate(exp(-t*z*s)*(1+s)^(-2*n-v), s, 0, inf) +;; = (t*z)^(1-v-2*n)/CF ;; -;; and I[n](t, z, v) = exp(-t*z)/t^(2*n+v-1)/CF -(defun big-i (n t z v) - (/ (exp (- (* t z))) - (expt t (+ n n v -1)) +;; and I[n](t, z, v) = exp(-t*z)/CF +(defun big-i (n theta z v) + (/ (exp (- (* theta z))) (let* ((a (- 1 v n n)) - (z-a (- z a))) + (z-a (- (* theta z) a))) (lentz #'(lambda (n) (+ n n 1 z-a)) #'(lambda (n)

commit dbc1e376add1d492dc35c37b3098c2ffb796d6f1 Author: Raymond Toy toy.raymond@gmail.com Date: Sun Apr 8 09:36:46 2012 -0700

Build qd-bessel now. (qd-bessel is a work in progress.)

diff --git a/oct.asd b/oct.asd index 3f8d70c..0ca30e5 100644 --- a/oct.asd +++ b/oct.asd @@ -67,7 +67,8 @@ :depends-on ("qd-methods" "qd-reader")) (:file "qd-gamma" :depends-on ("qd-methods" "qd-reader")) - )) + (:file "qd-bessel" + :depends-on ("qd-methods"))))

(defmethod perform ((op test-op) (c (eql (asdf:find-system :oct)))) (oos 'test-op 'oct-tests))

commit 6cd96249b94dc29962cecda996a5799d8ac27271 Author: Raymond Toy toy.raymond@gmail.com Date: Sun Apr 8 09:35:35 2012 -0700

Define macro WITH-FLOATING-POINT-CONTAGION.

qd-methods.lisp: o Define the macro

qd-gamma.lisp: o Use it.

diff --git a/qd-gamma.lisp b/qd-gamma.lisp index eb8c55b..410c315 100644 --- a/qd-gamma.lisp +++ b/qd-gamma.lisp @@ -108,10 +108,11 @@ ;; 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)))) + (let ((p (float-pi z))) + (- (log p) + (log (- z)) + (log (sin (* p z))) + (log-gamma (- z))))) (t (let ((absz (abs z))) (cond ((>= absz limit) @@ -377,9 +378,7 @@ "Tail of the incomplete gamma function defined by:

integrate(t^(a-1)*exp(-t), t, z, inf)" - (let* ((prec (float-contagion a z)) - (a (apply-contagion a prec)) - (z (apply-contagion z prec))) + (with-floating-point-contagion (a z) (if (zerop a) ;; incomplete_gamma_tail(0, z) = exp_integral_e(1,z) (exp-integral-e 1 z) @@ -409,9 +408,7 @@ "Incomplete gamma function defined by:

integrate(t^(a-1)*exp(-t), t, 0, z)" - (let* ((prec (float-contagion a z)) - (a (apply-contagion a prec)) - (z (apply-contagion z prec))) + (with-floating-point-contagion (a z) (if (and (< (abs a) 1) (< (abs z) 1)) (s-incomplete-gamma a z) (if (and (realp a) (realp z)) @@ -539,19 +536,12 @@ (let ((-v (- v))) (* (expt z (- v 1)) (incomplete-gamma-tail (+ -v 1) z)))) - ((< (abs z) 1) - ;; Use series for small z + ((or (< (abs z) 1) (>= (abs (phase z)) 3.1)) + ;; Use series for small z or if z is near the negative real + ;; axis because the continued fraction does not converge on + ;; the negative axis and converges slowly near the negative + ;; axis. (s-exp-integral-e v z)) - ((>= (abs (phase z)) 3.1) - ;; The continued fraction doesn't converge on the negative - ;; real axis, and converges very slowly near the negative - ;; real axis, so use the incomplete-gamma-tail function in - ;; this region. "Closeness" to the negative real axis is - ;; teken to mean that z is in a sector near the axis. - ;; - ;; E(v,z) = z^(v-1)*incomplete_gamma_tail(1-v,z) - (* (expt z (- v 1)) - (incomplete-gamma-tail (- 1 v) z))) (t ;; Use continued fraction for everything else. (cf-exp-integral-e v z)))) diff --git a/qd-methods.lisp b/qd-methods.lisp index 0eef82d..c461ba3 100644 --- a/qd-methods.lisp +++ b/qd-methods.lisp @@ -79,6 +79,17 @@ (complex (coerce (realpart number) precision) (coerce (imagpart number) precision)))))

+;; WITH-FLOATING-POINT-CONTAGION - macro +;; +;; Determines the highest precision of the variables in VARLIST and +;; converts each of the values to that precision. +(defmacro with-floating-point-contagion (varlist &body body) + (let ((precision (gensym "PRECISION-"))) + `(let ((,precision (float-contagion ,@varlist))) + (let (,@(mapcar #'(lambda (v) + `(,v (apply-contagion ,v ,precision))) + varlist)) + ,@body))))

(defmethod add1 ((a number)) (cl::1+ a))

commit c86217a5f7191f1a29566d6ee24cb57344dd546d Author: Raymond Toy toy.raymond@gmail.com Date: Sun Apr 8 09:04:18 2012 -0700

Fix some comments.

diff --git a/qd-gamma.lisp b/qd-gamma.lisp index be832bc..eb8c55b 100644 --- a/qd-gamma.lisp +++ b/qd-gamma.lisp @@ -249,13 +249,18 @@ :format-control "~<Continued fraction failed to converge after ~D iterations.~% Delta = ~S~>" :format-arguments (list *max-cf-iterations* (/ c d))))))))

-;; Continued fraction for erf(b): +;; Continued fraction for erf(z): ;; -;; z[n] = 1+2*n-2*z^2 -;; a[n] = 4*n*z^2 +;; erf(z) = 2*z/sqrt(pi)*exp(-z^2)/K +;; +;; where K is the continued fraction with +;; +;; b[n] = 1+2*n-2*z^2 +;; a[n] = 4*n*z^2 ;; ;; This works ok, but has problems for z > 3 where sometimes the -;; result is greater than 1. +;; result is greater than 1 and for larger values, negative numbers +;; are returned. #+nil (defun cf-erf (z) (let* ((z2 (* z z))

commit 015558a3df8fae2686c7b6e2f049da3f4b1a2cd8 Author: Raymond Toy toy.raymond@gmail.com Date: Sat Apr 7 23:28:08 2012 -0700

Fix a divide by zero error in s-exp-integral-e for v = 1. We need to skip the first term in the series.

diff --git a/qd-gamma.lisp b/qd-gamma.lisp index e353ff5..be832bc 100644 --- a/qd-gamma.lisp +++ b/qd-gamma.lisp @@ -257,7 +257,7 @@ ;; This works ok, but has problems for z > 3 where sometimes the ;; result is greater than 1. #+nil -(defun erf (z) +(defun cf-erf (z) (let* ((z2 (* z z)) (twoz2 (* 2 z2))) (* (/ (* 2 z) @@ -504,10 +504,13 @@ (- (psi v) (log z))) (loop for k from 0 for term = 1 then (* term (/ -z k)) - for sum = (/ (- 1 v)) then (+ sum (let ((denom (- k n-1))) - (if (zerop denom) - 0 - (/ term denom)))) + for sum = (if (zerop n-1) + 0 + (/ (- 1 v))) + then (+ sum (let ((denom (- k n-1))) + (if (zerop denom) + 0 + (/ term denom)))) when (< (abs term) (* (abs sum) eps)) return sum))) (loop for k from 0

commit a5a4c7acd95d1946e7cf426f9a927a918c9e2afc Author: Raymond Toy toy.raymond@gmail.com Date: Sat Apr 7 09:28:16 2012 -0700

Add more parts of the exp-arc algorithm. Needs lots of work, but it seems that bessel_j(n,z) mostly works.

diff --git a/qd-bessel.lisp b/qd-bessel.lisp index e85d2d1..6d153a7 100644 --- a/qd-bessel.lisp +++ b/qd-bessel.lisp @@ -39,12 +39,12 @@ ;; = 1/2^(k+3/2)/p^(k+1/2)*integrate(t^(k-1/2)*exp(-t),t,0,p) ;; = 1/2^(k+3/2)/p^(k+1/2) * g(k+1/2, p) ;; -;; where g(a,z) is the lower incomplete gamma function. +;; where G(a,z) is the lower incomplete gamma function. ;; -;; There is the continued fraction expansion for g(a,z) (see +;; There is the continued fraction expansion for G(a,z) (see ;; cf-incomplete-gamma in qd-gamma.lisp): ;; -;; g(a,z) = z^a*exp(-z)/ CF +;; G(a,z) = z^a*exp(-z)/ CF ;; ;; So ;; @@ -183,6 +183,177 @@ i-)) (float-pi i+) 2))) + +;; alpha[n](z) = integrate(exp(-z*s)*s^n, s, 0, 1/2) +;; beta[n](z) = integrate(exp(-z*s)*s^n, s, -1/2, 1/2) +;; +;; The recurrence in [2] is +;; +;; alpha[n](z) = - exp(-z/2)/2^n/z + n/z*alpha[n-1](z) +;; beta[n]z) = ((-1)^n*exp(z/2)-exp(-z/2))/2^n/z + n/z*beta[n-1](z) +;; +;; We also note that +;; +;; alpha[n](z) = G(n+1,z/2)/z^(n+1) +;; beta[n](z) = G(n+1,z/2)/z^(n+1) - G(n+1,-z/2)/z^(n+1) + +(defun alpha (n z) + (let ((n (float n (realpart z)))) + (/ (cf-incomplete-gamma (1+ n) (/ z 2)) + (expt z (1+ n))))) + +(defun beta (n z) + (let ((n (float n (realpart z)))) + (/ (- (cf-incomplete-gamma (1+ n) (/ z 2)) + (cf-incomplete-gamma (1+ n) (/ z -2))) + (expt z (1+ n))))) + +;; a[0](k,v) := (k+sqrt(k^2+1))^(-v); +;; a[1](k,v) := -v*a[0](k,v)/sqrt(k^2+1); +;; a[n](k,v) := 1/(k^2+1)/(n-1)/n*((v^2-(n-2)^2)*a[n-2](k,v)-k*(n-1)*(2*n-3)*a[n-1](k,v)); + +;; Convert this to iteration instead of using this quick-and-dirty +;; memoization? +(let ((hash (make-hash-table :test 'equal))) + (defun an-clrhash () + (clrhash hash)) + (defun an-dump-hash () + (maphash #'(lambda (k v) + (format t "~S -> ~S~%" k v)) + hash)) + (defun an (n k v) + (or (gethash (list n k v) hash) + (let ((result + (cond ((= n 0) + (expt (+ k (sqrt (float (1+ (* k k)) (realpart v)))) (- v))) + ((= n 1) + (- (/ (* v (an 0 k v)) + (sqrt (float (1+ (* k k)) (realpart v)))))) + (t + (/ (- (* (- (* v v) (expt (- n 2) 2)) (an (- n 2) k v)) + (* k (- n 1) (+ n n -3) (an (- n 1) k v))) + (+ 1 (* k k)) + (- n 1) + n))))) + (setf (gethash (list n k v) hash) result) + result)))) + +;; SUM-AN computes the series +;; +;; sum(exp(-k*z)*a[n](k,v), k, 1, N) +;; +(defun sum-an (big-n n v z) + (let ((sum 0)) + (loop for k from 1 upto big-n + do + (incf sum (* (exp (- (* k z))) + (an n k v)))) + sum)) + +;; SUM-AB computes the series +;; +;; sum(alpha[n](z)*a[n](0,v) + beta[n](z)*sum_an(N, n, v, z), n, 0, inf) +(defun sum-ab (big-n v z) + (let ((eps (epsilon (realpart z)))) + (an-clrhash) + (do* ((n 0 (+ 1 n)) + (term (+ (* (alpha n z) (an n 0 v)) + (* (beta n z) (sum-an big-n n v z))) + (+ (* (alpha n z) (an n 0 v)) + (* (beta n z) (sum-an big-n n v z)))) + (sum term (+ sum term))) + ((<= (abs term) (* eps (abs sum))) + sum) + (when nil + (format t "n = ~D~%" n) + (format t " term = ~S~%" term) + (format t " sum = ~S~%" sum))))) + +;; Convert to iteration instead of this quick-and-dirty memoization? +(let ((hash (make-hash-table :test 'equal))) + (defun %big-a-clrhash () + (clrhash hash)) + (defun %big-a-dump-hash () + (maphash #'(lambda (k v) + (format t "~S -> ~S~%" k v)) + hash)) + (defun %big-a (n v) + (or (gethash (list n v) hash) + (let ((result + (cond ((zerop n) + (expt 2 (- v))) + (t + (* (%big-a (- n 1) v) + (/ (* (+ v n n -2) (+ v n n -1)) + (* 4 n (+ n v)))))))) + (setf (gethash (list n v) hash) result) + result)))) + +;; Computes A[n](v) = +;; (-1)^n*v*2^(-v)*pochhammer(v+n+1,n-1)/(2^(2*n)*n!) If v is a +;; negative integer -m, use A[n](-m) = (-1)^(m+1)*A[n-m](m) for n >= +;; m. +(defun big-a (n v) + (let ((m (ftruncate v))) + (cond ((and (= m v) (minusp m)) + (if (< n m) + (%big-a n v) + (let ((result (%big-a (+ n m) v))) + (if (oddp (truncate m)) + result + (- result))))) + (t + (%big-a n v))))) + +;; I[n](t, z, v) = exp(-t*z)/t^(2*n+v-1) * +;; integrate(exp(-t*z*s)*(1+s)^(-2*n-v), s, 0, inf) +;; +;; Use the substitution u=1+s to get a new integral +;; +;; integrate(exp(-t*z*s)*(1+s)^(-2*n-v), s, 0, inf) +;; = exp(t*z) * integrate(u^(-v-2*n)*exp(-t*u*z), u, 1, inf) +;; = exp(t*z)*t^(v+2*n-1)*z^(v+2*n-1)*incomplete_gamma_tail(1-v-2*n,t*z) +;; +;; The continued fraction for incomplete_gamma_tail(a,z) is +;; +;; z^a*exp(-z)/CF +;; +;; So incomplete_gamma_tail(1-v-2*n, t*z) is +;; +;; (t*z)^(1-v-2*n)*exp(-t*z)/CF +;; +;; which finally gives +;; +;; integrate(exp(-t*z*s)*(1+s)^(-2*n-v), s, 0, inf) +;; = CF +;; +;; and I[n](t, z, v) = exp(-t*z)/t^(2*n+v-1)/CF +(defun big-i (n t z v) + (/ (exp (- (* t z))) + (expt t (+ n n v -1)) + (let* ((a (- 1 v n n)) + (z-a (- z a))) + (lentz #'(lambda (n) + (+ n n 1 z-a)) + #'(lambda (n) + (* n (- a n))))))) + +(defun sum-big-ia (big-n v z) + ) + +(defun bessel-j (v z) + (let ((vv (ftruncate v))) + (cond ((= vv v) + ;; v is an integer + (integer-bessel-j-exp-arc v z)) + (t + (let ((big-n 100) + (vpi (* v (float-pi (realpart z))))) + (+ (integer-bessel-j-exp-arc v z) + (* z + (/ (sin vpi) vpi) + (+ (/ -1 z) + (sum-ab big-n v z)))))))))  (defun paris-series (v z n) (labels ((pochhammer (a k)

commit 1d404aca1e6652ad2456ba14e8bbdf5d329c06a0 Author: Raymond Toy toy.raymond@gmail.com Date: Fri Apr 6 19:28:23 2012 -0700

Add some comments, rename bessel-j-exp-arc to integer-bessel-j-exp-arc.

diff --git a/qd-bessel.lisp b/qd-bessel.lisp index 95a61c0..e85d2d1 100644 --- a/qd-bessel.lisp +++ b/qd-bessel.lisp @@ -51,6 +51,22 @@ ;; B[k](p) = 1/2^(k+3/2)/p^(k+1/2)*p^(k+1/2)*exp(-p)/CF ;; = exp(-p)/2^(k+3/2)/CF ;; +;; +;; Note also that [2] gives a recurrence relationship for B[k](p) in +;; eq (2.6), but there is an error there. The correct relationship is +;; +;; B[k](p) = -exp(-p)/(p*sqrt(2)*2^(k+1)) + (k-1/2)*B[k-1](p)/(2*p) +;; +;; The paper is missing the division by p in the term containing +;; B[k-1](p). This is easily derived from the recurrence relationship +;; for the (lower) incomplete gamma function. +;; +;; Note too that as k increases, the recurrence appears to be unstable +;; and B[k](p) begins to increase even though it is strictly bounded. +;; (This is also easy to see from the integral.) Hence, we do not use +;; the recursion. However, it might be stable for use with +;; double-float precision; this has not been tested. +;; (defun bk (k p) (/ (exp (- p)) (* (sqrt (float 2 (realpart p))) (ash 1 (+ k 1))) @@ -157,7 +173,7 @@

;; This currently only works for v an integer. ;; -(defun bessel-j-exp-arc (v z) +(defun integer-bessel-j-exp-arc (v z) (let* ((iz (* #c(0 1) z)) (i+ (exp-arc-i-2 iz v)) (i- (exp-arc-i-2 (- iz ) v)))

commit 4c1ed0f45e79bbe467f7d4694f417ff92ae46adb Author: Raymond Toy toy.raymond@gmail.com Date: Fri Apr 6 19:21:46 2012 -0700

First cut at Bessel functions. Needs lots of work.

diff --git a/qd-bessel.lisp b/qd-bessel.lisp new file mode 100644 index 0000000..95a61c0 --- /dev/null +++ b/qd-bessel.lisp @@ -0,0 +1,186 @@ +;;;; -*- 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) + +;;; References: +;;; +;;; [1] Borwein, Borwein, Crandall, "Effective Laguerre Asymptotics", +;;; http://people.reed.edu/~crandall/papers/Laguerre-f.pdf +;;; +;;; [2] Borwein, Borwein, Chan, "The Evaluation of Bessel Functions +;;; via Exp-Arc Integrals", http://web.cs.dal.ca/~jborwein/bessel.pdf +;;; + +(defvar *debug-exparc* nil) + +;; B[k](p) = 1/2^(k+3/2)*integrate(exp(-p*u)*u^(k-1/2),u,0,1) +;; = 1/2^(k+3/2)/p^(k+1/2)*integrate(t^(k-1/2)*exp(-t),t,0,p) +;; = 1/2^(k+3/2)/p^(k+1/2) * g(k+1/2, p) +;; +;; where g(a,z) is the lower incomplete gamma function. +;; +;; There is the continued fraction expansion for g(a,z) (see +;; cf-incomplete-gamma in qd-gamma.lisp): +;; +;; g(a,z) = z^a*exp(-z)/ CF +;; +;; So +;; +;; B[k](p) = 1/2^(k+3/2)/p^(k+1/2)*p^(k+1/2)*exp(-p)/CF +;; = exp(-p)/2^(k+3/2)/CF +;; +(defun bk (k p) + (/ (exp (- p)) + (* (sqrt (float 2 (realpart p))) (ash 1 (+ k 1))) + (let ((a (float (+ k 1/2) (realpart p)))) + (lentz #'(lambda (n) + (+ n a)) + #'(lambda (n) + (if (evenp n) + (* (ash n -1) p) + (- (* (+ a (ash n -1)) p)))))))) + +;; exp-arc I function, as given in the Laguerre paper +;; +;; I(p, q) = 4*exp(p) * sum(g[k](-2*%i*q)/(2*k)!*B[k](p), k, 0, inf) +;; +;; where g[k](p) = product(p^2+(2*j-1)^2, j, 1, k) and B[k](p) as above. +;; +;; For computation, note that g[k](p) = g[k-1](p) * (p^2 + (2*k-1)^2) +;; and (2*k)! = (2*k-2)! * (2*k-1) * (2*k). Then, let +;; +;; R[k](p) = g[k](p)/(2*k)! +;; +;; Then +;; +;; R[k](p) = g[k](p)/(2*k)! +;; = g[k-1](p)/(2*k-2)! * (p^2 + (2*k-1)^2)/((2*k-1)*(2*k) +;; = R[k-1](p) * (p^2 + (2*k-1)^2)/((2*k-1)*(2*k) +;; +;; In the exp-arc paper, the function is defined (equivalently) as +;; +;; I(p, q) = 2*%i*exp(p)/q * sum(r[2*k+1](-2*%i*q)/(2*k)!*B[k](p), k, 0, inf) +;; +;; where r[2*k+1](p) = p*product(p^2 + (2*j-1)^2, j, 1, k) +;; +;; Let's note some properties of I(p, q). +;; +;; I(-%i*z, v) = 2*%i*exp(-%i*z)/q * sum(r[2*k+1](-2*%i*v)/(2*k)!*B[k](-%i*z)) +;; +;; Note thate B[k](-%i*z) = 1/2^(k+3/2)*integrate(exp(%i*z*u)*u^(k-1/2),u,0,1) +;; = conj(B[k](%i*z). +;; +;; Hence I(-%i*z, v) = conj(I(%i*z, v)) when both z and v are real. +(defun exp-arc-i (p q) + (let* ((sqrt2 (sqrt (float 2 (realpart p)))) + (exp/p/sqrt2 (/ (exp (- p)) p sqrt2)) + (v (* #c(0 -2) q)) + (v2 (expt v 2)) + (eps (epsilon (realpart p)))) + (when *debug-exparc* + (format t "sqrt2 = ~S~%" sqrt2) + (format t "exp/p/sqrt2 = ~S~%" exp/p/sqrt2)) + (do* ((k 0 (1+ k)) + (bk (/ (incomplete-gamma 1/2 p) + 2 sqrt2 (sqrt p)) + (- (/ (* bk (- k 1/2)) 2 p) + (/ exp/p/sqrt2 (ash 1 (+ k 1))))) + ;; ratio[k] = r[2*k+1](v)/(2*k)!. + ;; r[1] = v and r[2*k+1](v) = r[2*k-1](v)*(v^2 + (2*k-1)^2) + ;; ratio[0] = v + ;; and ratio[k] = r[2*k-1](v)*(v^2+(2*k-1)^2) / ((2*k-2)! * (2*k-1) * 2*k) + ;; = ratio[k]*(v^2+(2*k-1)^2)/((2*k-1) * 2 * k) + (ratio v + (* ratio (/ (+ v2 (expt (1- (* 2 k)) 2)) + (* 2 k (1- (* 2 k)))))) + (term (* ratio bk) + (* ratio bk)) + (sum term (+ sum term))) + ((< (abs term) (* (abs sum) eps)) + (* sum #c(0 2) (/ (exp p) q))) + (when *debug-exparc* + (format t "k = ~D~%" k) + (format t " bk = ~S~%" bk) + (format t " ratio = ~S~%" ratio) + (format t " term = ~S~%" term) + (format t " sum - ~S~%" sum))))) + +(defun exp-arc-i-2 (p q) + (let* ((sqrt2 (sqrt (float 2 (realpart p)))) + (exp/p/sqrt2 (/ (exp (- p)) p sqrt2)) + (v (* #c(0 -2) q)) + (v2 (expt v 2)) + (eps (epsilon (realpart p)))) + (when *debug-exparc* + (format t "sqrt2 = ~S~%" sqrt2) + (format t "exp/p/sqrt2 = ~S~%" exp/p/sqrt2)) + (do* ((k 0 (1+ k)) + (bk (bk 0 p) + (bk k p)) + (ratio v + (* ratio (/ (+ v2 (expt (1- (* 2 k)) 2)) + (* 2 k (1- (* 2 k)))))) + (term (* ratio bk) + (* ratio bk)) + (sum term (+ sum term))) + ((< (abs term) (* (abs sum) eps)) + (* sum #c(0 2) (/ (exp p) q))) + (when *debug-exparc* + (format t "k = ~D~%" k) + (format t " bk = ~S~%" bk) + (format t " ratio = ~S~%" ratio) + (format t " term = ~S~%" term) + (format t " sum - ~S~%" sum))))) + + +;; This currently only works for v an integer. +;; +(defun bessel-j-exp-arc (v z) + (let* ((iz (* #c(0 1) z)) + (i+ (exp-arc-i-2 iz v)) + (i- (exp-arc-i-2 (- iz ) v))) + (/ (+ (* (cis (* v (float-pi i+) -1/2)) + i+) + (* (cis (* v (float-pi i+) 1/2)) + i-)) + (float-pi i+) + 2))) + +(defun paris-series (v z n) + (labels ((pochhammer (a k) + (/ (gamma (+ a k)) + (gamma a))) + (a (v k) + (* (/ (pochhammer (+ 1/2 v) k) + (gamma (float (1+ k) z))) + (pochhammer (- 1/2 v) k)))) + (* (loop for k from 0 below n + sum (* (/ (a v k) + (expt (* 2 z) k)) + (/ (cf-incomplete-gamma (+ k v 1/2) (* 2 z)) + (gamma (+ k v 1/2))))) + (/ (exp z) + (sqrt (* 2 (float-pi z) z)))))) +

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Summary of changes: oct.asd | 3 +- qd-bessel.lisp | 358 +++++++++++++++++++++++++++++++++++++++++++++++++++++++ qd-gamma.lisp | 50 +++++--- qd-methods.lisp | 26 +++- qd-package.lisp | 2 + rt-tests.lisp | 16 +++ 6 files changed, 428 insertions(+), 27 deletions(-) create mode 100644 qd-bessel.lisp

hooks/post-receive