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JOURN

AL

O F

RESE

ARCH of the Nat io nal Bureau of Standards - B. M a thema tica l Sciences

Vo

l

77B, N

os

. 3&4, Jul

y-De

cember 1973

and

J

of

Complex

rgument

and

Integer O rder

David

J. Sookne*

Institute for Basic

Standards,

National Bureau of Standards, Washington, D.C.

20234

(June

4, 1973)

A com

pu

ter prugram is d

esc

ribe d for c a lcula t i ng Bes sel func tion s J (z) a nd / (z) . fur z co

mpl

ex,

and

n

a nonn

eg

ative

in teg e r.

Th e method use d is th at of ba ck

wa

rd recursion, w

ith

strict c

ontrol f

e

rror

,

and op

ti

mum de te rmina tio n of the point at whic h to begin the r

ec

ur sion.

Key

wo

rds : B

esse

l func t

io

ns; bac kward rec

ur

s iun; error bo

und

s; Miller alg

orith

m;

diff

eren

ce

e

qu

ation.

1.

Method

G

iv

en a complex number z and a positive integer

NB,

BESLe I calculat

es

either

III(z),

n= O, l ,

. , NB-1

jll (Z) n= O,

1,

NB - 1

ng d ouble

-p

r

ec

ision arit hme

ti

c. Th e method u

se

d is described in [lJ 1 a nd is base d on al

go rithm

s

and

Miller [3], applied to the differen

ce

equation

2n

Y

- I

YII-SIGN ' Y +I,

z

SI GN is + l forj s , - l for I s .

(

1

Th e

pr

ogram

se

ts MAGZ = [Izl]' the integer

part

of z l, PM AGZ=

O ,

PM

AGZ +

1= 1,

and

then

ively calculates

PII+I= SIGN 2znpn-PII

_

I) ,

n=MAGZ 1 ,

MAGZ+2, .

(2)

e sequence is strictly in creasing in magnitude

[l

sec . 6].

Th

e program takes

N

to be the least

such that

lI

l

exceeds a number TEST defined in sec tion 2 and section 3. I t then

se

ts

)=O, y

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where

j s ,

1m z> 0

[N I

]

J t

= fON) +

2

2:

J s , lmz=O

n l

J.t=e-

Z

f

O

N

)

+2

Sf V

)

1/

.= 1

J s,

1m

z

0

r

I z]

J t

= f / ) + 2 2: - 1 ) I

I s ,Rez=O

n =

I s,

Re

z

MAGZ,

the

trun

ca

tion

error in y ~ : V

is

see

[2J; equations 5.01 and 5.02.

This erro

r is bounded by

using

the followin g

LEMMA:

For n > MAGZ, let

k

- Pn+' _

2n

Pn - '

-

- - - -

Pn

z Pn

and

let

n + 1 n

1

2

ff;

-

=-Izl+ zl )-1.

Let

Pn

=

min I k

n

, An)

Th e

n/or m ? n,

I

k

m

>

Pn

The

lemma , for re al z is lemma 2

of

[IJ.

The

proof for complex z is

essentially

the same.

The

prog

ram insure

s

that

TEST

? V2

lONS (;11 11

/, /,+ I

3)

4)

where L =

max (MAGZ

+

1,

NB

),

and NSIG

is

the

maximum number

of significant

decimal

digits in a

double-precisi

on variable on the computer

being used.

Then N is

the

least n such that

IplIl

> TEST

and N

is the l

east n ?N such that

IplI l > TEST =

~

TE

ST,

P

s

,-

112

5)

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nsequence of 4) and

(5), the

relative truncation

error 1 1 is

less than

t IO - s,C for

n

in

the rang

e MAGZ

< n

:s; L; see

[1

, sec. 5].

For

n

:s; MAGZ, it may be im possible to bound the relat ive truncation error in the above manner,

ing to loss

of

precision

due

to can ce llation in

1).

Experience indicates that

this

loss is negligible

ce

pt when the magnitud e

of ~ , N ) oscillates

as

n =

MAGZ, MAGZ - 1, .

.. 0

in the

back-recursion

calculating j s ,

with Re

Z 1m

z .

In

this ca

se,

th e

re

will

be about

D

de

c

imals of pre

cision

the

values

: V ) ,

where

D

is th e nu mber

of

dec i mals in JMA

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The

program sets TEST

l

2 . l'iSlG. The normalization factor fL is e

izfj

MA{;Z(Z) [ l sec. 5] , so

I

N I I I

= JMA ;Z Z)

2 p ~

:S; :S;

ps,+l) (p\ ,-1)2IfJII (Ps ,+l) Ps,-1)2

1

2

6)

Th

e bound

6)

holds for

the

first, third, fourth and sixth

equations

of 3);

the

derivations

are

the same . Similarly,

the second and

fifth

equations

yield

The se bounds

ar

e rather

weak,

and the error I5

Uv

/ fL I turns out to be less than

-N S

l(;.

4

eferences

[IJ

Olver

, F. W. J

.

and Sookne, D

].

A note on

the backward recurre

nce algorithms,

Mathematics

of

Computation

, Vo

l. 26,

No. 120,

October

1972. pp.

941-947.

[2J Olver. F. W. J., Numerica l Solutions of

Second

Ord

er

Difference

Equatio

ns, Nat. Bur.

Stand.

U .S.), 7lB (Math. and

Math. Phys

.)

. Nos. 2 3 ,

111

- 129 (1967).

[3] British

Assoc

. for the Advance ment of

Science.

Bessel Functions-Part 11.

Mathemati

ca l Tables. Vol.

10

(Cambridge

University Pr ess. Ca mbrid ge 1952).

[4] National Bureau of Standards , Handbook of Mathemati ca l

Functions.

Nat. Bur.

Stand.

(U.S.). Appl. Math. Ser. 55,

(1964).

(paper 77B3 4-385)

4