Robert Piessens and Maria Branders
A SURVEY OF NUMERICAL METHODS FOR THE COMPUTATION
OF BESSEL FUNCTION INTEGRALS
1. Introduction.
We consider the numerical computation of
(1)
I(<t,p,P)=f
Jv(px)f(x)dx
•'o
where a is a positive real number or infinite, where J (x) is the Bessel function of the first kind and of real order v , and where p is a positive pàrameter.
Examples of this type of integrai are the Fourier-Bessel transforms and
the Hankel transforms. In most cases, analytical integration of (1) is impossible, and numerical integration is necessary. But integrals of type (1) are difficult to evaluate numerically if
(i)
the product ap is large
(ii)
a is infinite
(Hi) f(x)
shows a singular or oscillatory behaviour.
In cases (i) and (ii) the difficulties arise from the oscillatory behaviour of J (x) and they grow when the oscillations become stronger.
The purpose of this paper is to give a survey of numerical methods which
are especially suited for the evaluation of / (a, p, v) when ap is large or a
is infinite. We restrict ourselves to cases where f(x) is smooth or where
f(x) = xa g (x) where g (x) is smooth and a is a real number.
1980 Mathematics Subject Classification 65D30, 65D20
250
What is usually desired is not the value of an isolated integrai / (a, p, v) ,
but a whole family of such integrals for several values of p . This has to be
taken into account when different methods are compared with respect to their
efficiency.
2. Computation of Bessel function integrals over a finite interval
The integrai (1) can be written as
/ («, p, v) = <*a+ * I
(2)
xaJv (apx) g (ax) dx
We assume that oc + p > — 1 . If (a + v) is not integer, then there is an
algebraic singularity of the integrand at x = 0 . If ap i-s large, then the integrand is strongly oscillatory.
If (a + v) is integer and ap is small, classical numerical integration
methods, such as Romberg integration, Clenshaw-Curtis integration or GaussLegendre integration (see Davis and Rabinowitz [6]) are applicable. If (a + v)
is not integer, and ap is small, the only difficulty is the algebraic singularity
at x = 0 , and Gauss-Jacobi quadrature or IMT-integration (6,11 ] can be used.
If ap is large, special methods should be applied which take into account the
oscillatory r^ehaviour of the integrand. We describe here two methods.
2.1. Integration between the zeros of J (x)
We denote the s-th positive zero of J (x) by jv
and we set jv Q = 0.
Then
(3)
I(a,p,v)=
S ( - l ) * + 1 7, +
f
J(px)f(x)dx
where
(4)
Ik= I '
\Jv{px)\f{x)dx
and where N is the largest naturai number for which j
that N is large when ap is large.
v N
<ap . This means
251
Using a transformation due to Longman [13], the summation in (3) can be
written as
5= £
(-1)* + 1 / ^ = ^ / , - ^ 4 / , + \ A 2 / , +... + ( - 1 ) ^ 1 2-P AP-1 /,
+ ( - 1 ^ ^ + ^ / ^ + ^ / ^ + ... +2^A^/^+ 1j
+ 2~p ( - \ ? [Ap^-APh
+ APii-...
+ ( - ìyv-i-p A^rN_p]
Assuming now that N and /) are large and that high-order differences are
small, the last bracket may be neglected and
1
1
, ! ,
S~ - / - - A/, + - A2/, (5)
2
*
4
1
+ (-1)^-
l
* 8
- / + -A/ M , + - A 2 / w , +
2 N
A
N—1
o
N—2
The summations in (5) may be truncated as soon as the terms are small
enough. For the evaluation of Ik, k= 1, 2. ..., classical integration methods
(for example Lobatto's mie) can be used, but special Gauss quadrature formulas (see Piessens (19]) are more efficient. If the integrai /, has an algebraic
singularity at x = 0 , then the Gauss-Jacobi rules or the IMT-rule are recommended.
2.2. Modified Clenshaw-Curtis quadrature.
The Clenshaw-Curtis quadrature method [5] is a well-known and efficient method for the numerical evaluation of an integrai I with smooth integrand This method is' based on a truncated Chebyshev series approximation of the integrand. However, when the integrand shows a singular or strongly oscillatory behaviour, the classical Clenshaw-Curtis method is not efficient
or even not applicable, unless it is modified in an appropriate way, taking into
account the type of difficulty of the integrand. We cali this method then a
modified Clenshaw-Curtis method (MCC-method). The principle of the MCCmethod is the following. The integration ' intérval is mapped onto
[—1, + 1 and the integrand is written as the product of a smooth function
g (x) and a weight function w (x) containing the singularities or the oscilla-
252
ting factors of the integrand, i.e.
+1
(6)
/= I
w(x)g(x)dx
The smooth function is then approximated by a truncated series of
Chebyshev polynomials
(7)
g(x)~
S ' e.
Tk(x),-Kx<ì
Here the symbol 2 ' indicates that the first term in the sum must be halved.
For the computation of the coefficients ck in (7) several good algorithms
are available (see Gentleman [10], Branders and Piessens [3]) .
The integrai in (6) can now be approximated by
(8)
'-Jó^*
where
Mlkh = I
w (x) T. (x) dx
-/..
are called modified moments (see Gautschi [8]).
The integration interval may also be mapped onto [ 0 , 1] instead of
[—1, 1 ] , but then the shifted Chebyshev polynomials T*(x) are to be used.
We consider now the computation of the integrai (2),
I=
(9)
l
X<X
Jv (ux)8
M
dx
If
(10)
g(xja
2 ' ck T*(x)
K
k=0
K
then
(11)
/ - JB'c k M
k=o
K
(u,i>,a)
K
where
(12)
M
k
(u> v> <*)= I x" Jv (">x) T* (x) dx
253
These modified moments satisfy the following homogeneous, linear, nine
terms recurrence relation:
2 i
co"
M
Té
^
+
(k + 3)(fc + 3 + 2a) + a '
+ [4(v2-a2)-2(k
(13)
~
_
_»2
-v
co
Mk + 2
4 -I
+2)(2a-l)]Mife + 1
2
2
3co:
2
2(fc -4)+6(y -a )-2(2a-l)-
8 J
M,
+ [4 {v2- a2 ) + 2 ( / 2 - 2 ) ( 2 a - 1)] Af^
co'
2
2
+ ( f e - 3 ) ( * - 3 - 2 a ) + (ce - *>
) Mk-2
4 .
+
co'
^
^,-4= °
The special case for a= 0 is already considered by Piessens and Branders [22,23].
Because of the symmetry of the recurrence relation of the shifted Chebyshev polynomials, it is convenient to define
T\-k (x) = T*(x),k=l,
2, 3,
and consequently
M.
(co, v, OL) =
M, (CO,
v, oc)
To start the recurrence relation with & = 0, 1, 2, 3, ... we need only
M0, Mi, M2 and M3 . Using the explicit expressions of the shifted Chebyshev polynomials we obtain
M0 = G (co, v, oi)
M j = 2 G (co, v,
OL
+ 1) - G (co, u, a)
M2 = 8 G (co, y, a + 2) - 8 G (co, i>, a) + G (co, u, a)
(14)
M3 = 32 G (co, i/, a + 3 ) - 4 8 G (co, i;, a + 2) + 18 G (co, v, a + 1) - G (co, i>, a)
where
(15)
G (co, v, a) = T * a yy (co*) </*
254
Since
co2 G (co, v, oc + 2) = \v2 - ( a + l) 2 ] G (co, v, oc)
(16)
+(«+rfl)Jy(w)-coJHM
we need only G (co, *>, tv) and G (co, i>, a + 1).
Luke [ 16 ] has given the following formulasi
(i)
a Neumann series expansion which is suitable for small co
/ v-a+1 \
2
G
(">"'")
=
< '
m
co(a + , + l )
£
+
*
r+ft
\
(17)
(ti)
+
1
> ( — )
+ 3x
^
+1
M
2 A
an asymptotic expansion which is suitable for large co
y+a + 1
2
(18)
G (w, i», a) = — i -l — ^ - r r '- 1 l/X
/ - ^ - (g,1 cose + g 2 sinfl)
^ja++ 1
/ y - Q ? + l •|
J —W
^
where
0 = CO - l>7T/2 + 7T/4
and
"*
(1/2-1¾ ( 1 / 2 + 1 ¾
2^fe!
,
k
b
*o = l
i
*+ i
,
2(fe+l)(or*-l/2)
(i^-fe-l/2)(^+^ +1/2)
*
255
If a and
v are integers, the following formulas are useful (1 ]
f1
r1
J2(ux)dx=\
2 v-i
2 J
J0(cox)dx
(co)
For the evaluation of
I Jo (ux)
Jo
dx
Chebyshev series approximations are given by Luke [17]. We discuss now
the numerical aspect of the recurrence formula (13). The numerical stability
of forward recursion depends on the asymptotic behaviour of M, (co, v, a)
and of 8 linearly independent solutions yik ,i = 1, 2, ... 8, k -+°° . Using
results of Denef and Piessens [7] and Branders [2] and using the asymptotic
theory of Fourier integrals, we find
{
y2,k ' - *~4
\yìk i ~ r 2 ( a + 1 ) - 2 i ;
(19)
Iy4,fe I ~fc-2<«+n+2p
~ k-2(oc+1)tnk
if
u=£0
if p= 0
co \k
co
and
Mk (co, vt a) ~ - — Jv (co) fc-2 +
(20)
3
2
( - 1)* 2 - ^ ^
1
co"
— cos[7r(a+i)]r(2o;+2)r 2 a - 2 i ; - 2
r(^+l)
The asymptotically dominant solutions are y7 k and yg k . The
asymtotically minimal solutions are y5k
and y6k . We may conclude that
256
forward and backward recursion are asymptotically unstable. However, the
instability of forward recursion is less pronounced if k <co/2. Indeed, practical experiments demonstrate that Mk(co, v, oc) can be computed accurately
using forward recursion for k < co/2 . For k > co/2 the loss of significant
figures increases and forward recursion is no longer applicable. In that case
Oliver's algorithm [18] or Lozier's algorithm [15] has to be used. This means
that (13) has to be solved as a boundary value problem with 6 initial values
and 2 end values. The solution of this boundary value problem requires the
solution of a linear system of equations having a band structure.
In the following numerical examples we show that the MCC-method may
be much more efficient thanDQAGS,whichis the best automatic integrator
of QUADPACK (251 Moreover, an important advantage of the MCC-method
is that the function evaluations of g , needed for the computation of the
coefficients e, of the Chebyshev series expansion, are independent of the
value of co . Consequently, the same function evaluations may be used for
different values of co , and have to be computed only once.
Example 1
f.
x
Ju3wx)e
"*"
^7(900+«*)»«
The results are presented in table 1
x
a
1
10
100
1000
absolute error
ofMCC-method
(TV = 2 4 )
exact integrai
0.15358717785
0.27332776101
0.73802062269
0.40052916650
x
x
x
x
10~3
IO" 3
IO - 5
IO" 8
0.57
0.15
0.26
0.17
Table 1
x
x
x
x
IO" 11
IO - 1 1
IO" 10
10~ 12
DQAGS
absolute
number of
error
function eval
0.13
0.28
0.10
0.65
x
x
x
x
IO-10
IO" 10
IO" 1 0
IO - 7
105
105
147
567
257
Exatnple 2
/
* 5 / 4 . / 1 / 4 {ax){\-x2)ll2dx=2112
T(9f2)a-9l2J19fA
(a)
The results are presented in table 2.
absolute error
ofMCC-method
(N=24)
exact integrai
1
10
100
1000
0.59425444454 x
-0.10797817102 x
-0.76243591476x
0.58592034700 x
IO" 1
IO" 2
IO" 8
IO - 1 3
0.27
0.78
0.61
0.13
x
x
x
x
IO" 11
IO - 1 2
IO - 1 2
IO - 1 2
DQAGS
absolute
•number of
error
function eval
0.47 x
0.14 x
0.46 x
O.lOx
10~ 9
IO - 9
10~ 10
IO-9
63
105
315
2499
Table 2
3. Computation of Bessel function integrals over an infinite interval
In this seQtion we consider four methods for the computation of
(21)
Kp,v)=
l
ro
Jv{px)f{x)dx
3.1. Integration between the zeros of J (x\
and convergence acceleration
We have
(22)
Ufi,v)=
f , ( - l ) * + 1 Ik
*=i
where
(23)
rjv.k/p
\JV (px)\ f(x)dx
258
Using Euler's transformation [6], the convergence of the series (22) can
be accelerated
(24)
- Ir~ - A/, + - A2 /,
Hp,v)=
K^)
v
> /
2
i
ì
4
ì.
g
:
A3 /, +...
i
1 6
It is not always desirable to start the convergence acceleration with lx ,
but with some later term, say / , so that
i
Hp,v)=
f
( - i r
* '
Jv(px)f(x)dx
1
! - /
[ 2
+
- - A/ + - A 2 / -...
w
m
4
"" 8
Other convergence accelerating methods, for example the e-algorithm
[26 ], are also applicable (for an example see [25 ]).
3.2. Transformation imo a doublé intergral
Substituting the integrai expression [1]
(25)
V
'
J (x) = 2
v
V ^
1=\ ( l - ' 2 r 1 / 2
Tiy +1/2)^
JQ
cos(xt)dt
into (21) and changing the order of integration, we obtain
(26)
Kp,i>)=
- . 2 ^
- f
r {y + 1/2) V7T J0
(l-t*r1/2F(t)dt
where
(27)
xv f (x) cos (pxt) dx
F(t)=l
•'o
We assume that the integrai in (27) is convergent. If we want to evaluate
(26) using a iV-point Gauss-Jacobi rule, then we have to compute the Fourier
integrai (27) for N values of t . Since F (t) shows a peaked or even a singular behaviour especially when f(x) is slowly decaying, N has to be
choosen large enough.
259
This method is closely related to Linz's method [12], which is based on
the Abel transformation of / (p , v) .
3.3. Expansion methods
If f(x)
(28)
can be approximated by a truncated series expansion
f(x)^
S e 0 (x)
k=o * *
then
(29)
/(p,*0-
S e *
(p,v)
where
oo
(30)
<ì>k (p ,v)= f
A good choice of the sequence
Jv (px) <j)k (x) dx
p0 , 0A , ... is important. The series expan-
sion
oo
/ ( * ) = 2 e. (j>.(x)
must be rapibly converging. An algorithm for the computation of e,
of $^ (p , v) must be available.
As an example, we consider Laguerre series expansions.
If f(x) has a rapidly converging expansion
f(x) = xv+1 e-**
(31)
and
lQCkL^(x2)
where L(i;) (x) is the generalized Laguerre polynomial of degree k , then
k
-P2,A
(32)
N
Cu
HP.»)*—
(see Gabutti [9] and Cavanagh and Cook [4]).
/i)\2k
P
l0i( j)
+V
260
3.4. Truncation of the infinite interval
If a is an arbitrary positive real number, we can write
(33).
/(p,i>) = I Jv(px)f(x)dx
+R(p,a)
where
(34)
R(p,a)=f
Jv{px)f{x)dx
*<*
The first integrai in the right side of (33) can be computed using the
methods of section 1.
If a is sufficiently large and / is strongly decaying, then we may neglect
R(p,a).
If
(35)
/ ( * ) - ^ - + ~ + ...
x
xz
is an asymptotic series approximation which is sufficiently accurate in the
interval [a , «> ) , then
(36)
R(p,.a)~
1 e
f
Jv {pX)
dx
Longman [14] has tabulated the values of the integrals in (36) for some
values of v and ap.
Using Hankel's asymptotic expansion [1 ], for x "* °°
(37)
Jv (px)~
1/
[Pv (px) c o s x - Qv (px) sinx]
where X =px—(vll +1/4) ir , and where Pv{x) and Q (x) can be
expressed as a well-known asymptotic series, R (p , a) can be written as
the sum of two Fourier integrals.
Especially, if a = (8 + vii + 1/4) ir/p , we have
R(p ,a)= \
1/
+
Pv (p(u +a))f(u
+a)cospudu
(38)
u +tìf
v
~ JI ly/ —
, • v Q„(p(
))/(« +a) sinpw du
np(u+a)
0
261
If f
is a smooth function on
[a , °° ) with an asymptotic behaviour
~ x'13 , x
f{x)
oo
for some real (3>0 , then the Fourier integrals in (3.8) can be computed
efficiently, using the MCC-method described in [24].
Numerical example
oo
., . , Jo (*) dx = Ix + / 2 + / 3
^o
V
where
/l=
I2 =
Jrh?JoMdx
/'
/
,
70 v^ + i
and
00
7
3 = "" /
Jn
¢ 1 (#)
COSA; ^x
..
;—TTTTTi
iU *2 M sin* d*
(A; + \)
with
T
/ x
*i(*) =
- F
2x
+ 2
X+33TT/4
1/—7—TTT—;po(^+337r/4)
V 7T ([*
; + 33 ir/4) °
V 1 + (*
+
33 7T/4)2
and
$2
, _i ^/?
= (x+l)3/2
ì/
2
1 / —(x—+—33;
~~
ir/4)
; Qo (^
+
,
X + 3 3 7T/4
33 7T/4)
V
1 +(*
+ 33 TT/4)2
Using the MCC-method described in section 2, we obtain, for
/ i =0.364134451873.
(the underlined figures are correct).
Using the MCC-method, described in [24], we obtain, for N~ 48
/ 2 = 0.0029944411
and
/ 2 = 0.000750548992
N= 50,
262
so that
/^0.367879441965
The exact value of the integrai is
e'1 =0.36787944117.
4. MCC-quadrature for integrals with other weight functions
The numerical examples in sections 2 and 3 show the efficiency of MCCintegration.
The study of the MCC-method described in this paper and in ref [22] and
[24] is a part of a research project for the construction of numerical software
for the computation of the modified moments for the following weight functions:
( 1 +xf
i-xr ( 1 + #)0'exp (—ax)
W\ M = ( i-xr
V>2 (x) = (
w3 (#)=.( i-xr
w4 (x) = ( 1
1V5 ( # ) = (
( 1 + xf
ex? (~ ax) Sin ((,1 +x)/2)
+ x)^exp (—ax2)
-xr (1
i-xr ( 1 + xfexpi-
w6 O) = ( i-xr < 1 +xfexp(ra/(x
a(x + 1) 2 )
+1))
1 +#y*exp (- a/x1)
w1 (#) = ( i-xr
(
W | W = ( l-x)a
( 1 + xfzxp(-al(x
w9 (x) = ( l-x)a
( 1 +xY 2w((l +x)/2)
wì0 (x) = a-xn
ì +X)13 Sin ((1 +x)/2) Sin ((1 -x)/2)
ivn (x) = 1 x — aa 1
wì2 (x) = ! x — a a1sign (x — a)
wl3 (x) = ! x — a
+1)2)
a
Sin \ x — a !
263
wl4 (,x) = I x — a la Sin I x—a I sigw (#-tf)
w 1 5 U ) = ( l - * ) a (1 + *)" I
w16 (x) = (l-x)a
wxl(x) = [(x-b)2
x-a\y
(1 +x)P \x — a\y%n \x — a\
+ *2]-1
WigU> = ( l + *)"./„<*(* + D / 2 )
In zi/., i =2, 3, ..., 8 the parameter a may be complex. The results of
this research project will be published elsewhere.
Acknowledgment.
This research is supported by the 'Onderzoeksfonds' of the Catholic University of Leuven.
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ROBERT PIESSENS- MARIA BRANDERS,Department of Computer Science Catholic
University of Leuver^Celestijnenlaan 200A, B-3030 Heverlee, Belgium.