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oct
oct
Commits
76ea9b13
Commit
76ea9b13
authored
Apr 12, 2012
by
Raymond Toy
Browse files
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Merge branch 'master' of
ssh://common-lisp.net/var/git/projects/oct/oct
parents
25129063
5daefcc4
Changes
2
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+208
-17
qd-bessel.lisp
qd-bessel.lisp
+71
-17
rt-tests.lisp
rt-tests.lisp
+137
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qd-bessel.lisp
View file @
76ea9b13
...
...
@@ -156,7 +156,8 @@
(
do*
((
k
0
(
1+
k
))
(
bk
(
bk
0
p
)
(
bk
k
p
))
(
ratio
v
;; Compute g[k](p)/(2*k)!, not r[2*k+1](p)/(2*k)!
(
ratio
1
(
*
ratio
(
/
(
+
v2
(
expt
(
1-
(
*
2
k
))
2
))
(
*
2
k
(
1-
(
*
2
k
))))))
(
term
(
*
ratio
bk
)
...
...
@@ -169,7 +170,7 @@
(
format
t
" ratio = ~S~%"
ratio
)
(
format
t
" term = ~S~%"
term
)
(
format
t
" sum - ~S~%"
sum
))
(
*
sum
#c
(
0
2
)
(
/
(
exp
p
)
q
)))
(
*
sum
4
(
exp
p
)))
(
when
*debug-exparc*
(
format
t
"k = ~D~%"
k
)
(
format
t
" bk = ~S~%"
bk
)
...
...
@@ -182,18 +183,18 @@
(
defun
integer-bessel-j-exp-arc
(
v
z
)
(
let*
((
iz
(
*
#c
(
0
1
)
z
))
(
i+
(
exp-arc-i-2
iz
v
)))
(
cond
((
=
v
(
ftruncate
v
))
(
cond
((
and
(
=
v
(
ftruncate
v
))
(
realp
z
))
;; We can simplify the result
(
let
((
c
(
cis
(
*
v
(
float-pi
i+
)
-1/2
))))
(
let
((
c
(
exp
(
*
v
(
float-pi
i+
)
#c
(
0
-1/2
)
))))
(
/
(
+
(
*
c
i+
)
(
*
(
conjugate
c
)
(
conjugate
i+
)))
(
float-pi
i+
)
2
)))
(
t
(
let
((
i-
(
exp-arc-i-2
(
-
iz
)
v
)))
(
/
(
+
(
*
(
cis
(
*
v
(
float-pi
i+
)
-1/2
))
(
/
(
+
(
*
(
exp
(
*
v
(
float-pi
i+
)
#c
(
0
-1/2
)
))
i+
)
(
*
(
cis
(
*
v
(
float-pi
i+
)
1/2
))
(
*
(
exp
(
*
v
(
float-pi
i+
)
#c
(
0
1/2
)
))
i-
))
(
float-pi
i+
)
2
))))))
...
...
@@ -213,13 +214,13 @@
(
defun
alpha
(
n
z
)
(
let
((
n
(
float
n
(
realpart
z
))))
(
/
(
cf-
incomplete-gamma
(
1+
n
)
(
/
z
2
))
(
/
(
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
)))
(
/
(
-
(
incomplete-gamma
(
1+
n
)
(
/
z
2
))
(
incomplete-gamma
(
1+
n
)
(
/
z
-2
)))
(
expt
z
(
1+
n
)))))
;; a[0](k,v) := (k+sqrt(k^2+1))^(-v);
...
...
@@ -375,21 +376,74 @@
(
format
t
" f = ~S~%"
f
)
(
format
t
" term = ~S~%"
term
)
(
format
t
" sum = ~S~%"
sum
))))))
;;
;; TODO:
;; o For |z| <= 1 use the series.
;; o Currently accuracy is not good for large z and half-integer
;; order.
;; o For real v and z, return a real number instead of complex.
;; o Handle the case of Re(z) < 0. (The formulas are for Re(z) > 0:
;; bessel_j(v,z*exp(m*%pi*%i)) = exp(m*v*%pi*%i)*bessel_j(v, z)
;; o The paper suggests using
;; bessel_i(v,z) = exp(-v*%pi*%i/2)*bessel_j(v, %i*z)
;; when Im(z) >> Re(z)
;;
(
defun
bessel-j
(
v
z
)
(
let
((
vv
(
ftruncate
v
)))
(
cond
((
=
vv
v
)
;; v is an integer
;; Clear the caches for now.
(
an-clrhash
)
(
%big-a-clrhash
)
(
cond
((
and
(
=
vv
v
)
(
realp
z
))
;; v is an integer and z is real
(
integer-bessel-j-exp-arc
v
z
))
(
t
;; Need to fine-tune the value of big-n.
(
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
)
(
sum-big-ia
big-n
v
z
)))))))))
(
if
(
=
vv
v
)
0
(
*
z
(
/
(
sin
vpi
)
vpi
)
(
+
(
/
-1
z
)
(
sum-ab
big-n
v
z
)
(
sum-big-ia
big-n
v
z
))))))))))
;; Bessel Y
;;
;; bessel_y(v, z) = 1/(2*%pi*%i)*(exp(-%i*v*%pi/2)*I(%i*v,z) - exp(%i*v*%pi/2)*I(-%i*z, v))
;; + z/v/%pi*((1-cos(v*%pi)/z) + S(N,z,v)*cos(v*%pi)-S(N,z,-v))
;;
;; where
;;
;; S(N,z,v) = sum(alpha[n](z)*a[n](0,v) + beta[n](z)*sum(exp(-k*z)*a[n](k,v),k,1,N),n,0,inf)
;; + sum(A[n](v)*I[n](N+1/2,z,v),n,0,inf)
;;
(
defun
bessel-y
(
v
z
)
(
flet
((
ipart
(
v
z
)
(
let*
((
iz
(
*
#c
(
0
1
)
z
))
(
c+
(
exp
(
*
v
(
float-pi
z
)
1/2
)))
(
c-
(
exp
(
*
v
(
float-pi
z
)
-1/2
)))
(
i+
(
exp-arc-i-2
iz
v
))
(
i-
(
exp-arc-i-2
(
-
iz
)
v
)))
(
/
(
-
(
*
c-
i+
)
(
*
c+
i-
))
(
*
#c
(
0
2
)
(
float-pi
z
)))))
(
s
(
big-n
z
v
)
(
+
(
sum-ab
big-n
v
z
)
(
sum-big-ia
big-n
v
z
))))
(
let*
((
big-n
100
)
(
vpi
(
*
v
(
float-pi
z
)))
(
c
(
cos
vpi
)))
(
+
(
ipart
v
z
)
(
*
(
/
z
vpi
)
(
+
(
/
(
-
1
c
)
z
)
(
*
c
(
s
big-n
z
v
))
(
-
(
s
big-n
z
(
-
v
)))))))))
(
defun
paris-series
(
v
z
n
)
(
labels
((
pochhammer
(
a
k
)
...
...
rt-tests.lisp
View file @
76ea9b13
...
...
@@ -1542,3 +1542,140 @@
(
check-accuracy
219.1
e
true
))
nil
)
;; Bessel J tests for negative order
(
rt:deftest
bessel-j.neg-order.d.1
(
let
((
b
(
bessel-j
-1d0
2d0
))
(
true
-0.5767248077568734d0
))
(
check-accuracy
50.2
b
true
))
nil
)
(
rt:deftest
bessel-j.neg-order.d.2
(
let
((
b
(
bessel-j
-1d0
1.5d0
))
(
true
-0.5579365079100996d0
))
(
check-accuracy
50.5
b
true
))
nil
)
(
rt:deftest
bessel-j.neg-order.d.3
(
let
((
b
(
bessel-j
-1.5d0
2d0
))
(
true
-0.3956232813587035d0
))
(
check-accuracy
50.59
b
true
))
nil
)
(
rt:deftest
bessel-j.neg-order.d.4
(
let
((
b
(
bessel-j
-1.8d0
1.5d0
))
(
true
-0.251327217627129314d0
))
(
check-accuracy
49.98
b
true
))
nil
)
(
rt:deftest
bessel-j.neg-order.d.5
(
let
((
b
(
bessel-j
-2d0
1.5d0
))
(
true
0.2320876721442147d0
))
(
check-accuracy
51.89
b
true
))
nil
)
(
rt:deftest
bessel-j.neg-order.d.6
(
let
((
b
(
bessel-j
-2.5d0
1.5d0
))
(
true
1.315037204805194d0
))
(
check-accuracy
52.37
b
true
))
nil
)
(
rt:deftest
bessel-j.neg-order.d.7
(
let
((
b
(
bessel-j
-2.3d0
1.5d0
))
(
true
1.012178926325313d0
))
(
check-accuracy
50.01
b
true
))
nil
)
;; Bessel-J tests for positive order
(
rt:deftest
bessel-j.pos-order.d.1
(
let
((
b
(
bessel-j
1.5d0
1d0
))
(
true
0.2402978391234270d0
))
(
check-accuracy
51.83
b
true
))
nil
)
(
rt:deftest
bessel-j.pos-order.d.2
(
let
((
b
(
bessel-j
1.8d0
1d0
))
(
true
0.1564953153109239d0
))
(
check-accuracy
51.97
b
true
))
nil
)
(
rt:deftest
bessel-j.pos-order.d.3
(
let
((
b
(
bessel-j
2d0
1d0
))
(
true
0.1149034849319005d0
))
(
check-accuracy
51.87
b
true
))
nil
)
(
rt:deftest
bessel-j.pos-order.d.4
(
let
((
b
(
bessel-j
2.5d0
1d0
))
(
true
0.04949681022847794d0
))
(
check-accuracy
47.17
b
true
))
nil
)
(
rt:deftest
bessel-j.pos-order.d.5
(
let
((
b
(
bessel-j
-2d0
1.5d0
))
(
true
0.2320876721442147d0
))
(
check-accuracy
51.89
b
true
))
nil
)
;; Bessel J for half integer order and real args
(
rt:deftest
bessel-j-1/2.d.1
(
loop
for
k
from
0
below
100
;; x in [1,1+pi/2] because we don't want to test the Bessel
;; series and we don't want to test near pi because sin(pi)
;; = 0, where we will lose accuracy.
for
x
=
(
+
1
(
random
(
/
pi
2
)))
for
b
=
(
bessel-j
0.5d0
x
)
for
true
=
(
*
(
/
(
sin
x
)
(
sqrt
x
))
(
sqrt
(
/
2
pi
)))
for
result
=
(
check-accuracy
48.42
b
true
)
when
result
append
(
list
(
list
(
list
k
x
)
result
)))
nil
)
(
rt:deftest
bessel-j-1/2.d.1.a
(
let*
((
x
2.3831631289164497d0
)
(
b
(
bessel-j
0.5d0
x
))
(
true
(
*
(
/
(
sin
x
)
(
sqrt
x
))
(
sqrt
(
/
2
pi
)))))
(
check-accuracy
48.42
b
true
))
nil
)
(
rt:deftest
bessel-j-1/2.q.1
(
loop
for
k
from
0
below
10
;; x in [1,1+pi/2] because we don't want to test the Bessel
;; series and we don't want to test near pi because sin(pi)
;; = 0, where we will lose accuracy.
for
x
=
(
+
1
(
random
(
/
(
float-pi
#
q1
)
2
)))
for
b
=
(
bessel-j
#
q0.5
x
)
for
true
=
(
*
(
/
(
sin
x
)
(
sqrt
x
))
(
sqrt
(
/
2
(
float-pi
#
q1
))))
for
result
=
(
check-accuracy
169.45
b
true
)
when
result
append
(
list
(
list
(
list
k
x
)
result
)))
nil
)
(
rt:deftest
bessel-j-1/2.q.1.a
(
let*
((
x
#
q1.1288834862545916200627583005758663687705443417892789067029865493882q0
)
(
b
(
bessel-j
#
q0.5
x
))
(
true
(
*
(
/
(
sin
x
)
(
sqrt
x
))
(
sqrt
(
/
2
(
float-pi
#
q1
))))))
(
check-accuracy
182.92
b
true
))
nil
)
(
rt:deftest
bessel-j-1/2.q.1.b
(
let*
((
x
#
q1.1288834862545916200627583005758663687705443417892789067029865493882q0
)
(
b
(
bessel-j
#
q0.5
x
))
(
true
(
*
(
/
(
sin
x
)
(
sqrt
x
))
(
sqrt
(
/
2
(
float-pi
#
q1
))))))
(
check-accuracy
173.28
b
true
))
nil
)
(
rt:deftest
bessel-j-1/2.q.1.c
(
let*
((
x
#
q1.0360263937639582798798376485114581552570020473846457752365459851056q0
)
(
b
(
bessel-j
#
q0.5
x
))
(
true
(
*
(
/
(
sin
x
)
(
sqrt
x
))
(
sqrt
(
/
2
(
float-pi
#
q1
))))))
(
check-accuracy
169.45
b
true
))
nil
)
;; Bessel J for complex args
(
rt:deftest
bessel-j-complex.pos-order.d.1
(
let
((
b
(
bessel-j
0d0
#c
(
1d0
1
)))
(
true
#c
(
0.9376084768060293d0
-0.4965299476091221d0
)))
(
check-accuracy
50.73
b
true
))
nil
)
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