jresv77Bn3-4p125

8/20/2019 jresv77Bn3-4p125

http://slidepdf.com/reader/full/jresv77bn3-4p125 1/8

JOURNAL

OF

RESE

A RCH

of

the Nationa l B

ure

a u of Standards - B.

Mathem

atic

al

Sciences

Vol. 77A , Nos.

3 4

, j u l

y-

Dec embe r 19

7

essel Functions

of

Real rgument and

Integer

Order

David

J.

Sookne

J

Institute for Basic Standards,

National

Bureau of

Standards, Washington

, D.C. 20234

(June 4,

1973)

A comp

ut

er program is desc ribed for

ca

lc ul ating Bessel Functions J x) and I x), for

x

real, and

n a

nonn

egative int

ege

r.

Th

e

method used

is th

at

of b

ackward

r

ec ur

sion , with st rict con trol

of

e

rror

.

Key word s: Bac

kward

recursion; Bessel func tions; difference equat ion; e rror

bound

s; M ll er al

go

rithms .

1. Method

G

iv

en a r

ea

l

number x

and a positive integer

NB, BESLRI

calc

ulat

es eith

er

I x),

n = 0, 1

.

.

.,

NB - 1

} n X),

n

=

0, 1, . , NB - 1

arit

hmetic.

The

method,

which is described in is based on algorithms of

[2]

and Miller

[3], app lied

to the d iffer e

nce equat

ion

2n

Y

- I

= -

Yll

-

SIGN

YII+ I

x

SIGN

is +

1 for}

s ,

- 1

for

J s

o

1)

Th e program sets MAGX

= [ Ix I ],

the

integer

part of

Ix

I

P

MAGX

=

0,

P MAGX +

I

=

1, and

en s uccess iv e ly calculat

es

2n

PII +I

=

N PII - SIGN - I

n=MAGX +

1,

MAGX +

2,

2)

e

nce

is strictly increasin

g.

The

prog

ram takes

N

to

be the least

n

such that

Pit

exce

eds a

TEST

defined in

sec

tions

2

and

3.

It th en sets yW) =

0,

~ ~ I =

1/p v,

and rec urs back ward

ing

1).

The computed sequence Y b

V

,

yl

N),

•

is th e recessive solut ion of

1)

which satis fi

es

the

ndary condition YM AG X

= 1.

From

this

solution, th e I s and} s

are

found by normali zing

n= 0 , 1 ,

.,

NB - }

n =

0,1 ,

.,

NB- }

AMS Sllbject Cfnss i

jinJl

i o

fl

; 68 A10.

I Pre

sent add r

ess:

W.U.J.S. Institute. Arad. lsraeL

: Fi gures in br ackets in dicate the lit er atur e ref er e lJces at the e

nd

uf this pap cc

125

8/20/2019 jresv77Bn3-4p125


where

[ .V/ t [

J-t = y V) + 2 2:

for } s

k = 1

for

[ s.

2.

Error ounds for the n

For n >

MAGX,

the truncation error

in

yi,-V)

is

' 1

T N )= y - = p .

L..

PrPr+

1

r=N

(3)

see [2], eqs 5.01) and 5 .02 ).

This

error is bounded

by using

Lemma 2 of [I], which states that for

s ~ r

p

= p d p s ~ min (Pr+I/Pr, r + l ) / l x l + v r + l ) t / l

x

I2- l ) .

The program

sets

(4 )

where L =

max MAGX

+

1, NB -

1)

,

and

NSIG is

the

maximum number

of

sign ificant

decimal

digits in a double· precision variable on

the

computer

being

used. Then N' is the least

n

such that

fJlI

> TEST

I. For} s, N

is

the least

N'

su

ch

that

{JII > TEST = - 0 - . TEST

1

.

p,

- 1

(5)

In

conseque

nce

of 4) and 5),

the relative

truncation error I T(;;) I

Y

I is less than . lO

- NSJ ;

for all

n

in

the range MAGX <

n

L; see [1,

sec.

5]. For / s , N is taken to be N . Here a

glance

at I) shows

Pil i

P

- I > 2 for all

n

> MAGX, so the proof given in

[1,

sec. 5] for } s may also

be used

for [ s,

with

px repla

ced by 2 throughout.

In eq

1)

for

I s

2n

yIN)

and - SIGN· y N )

have the

same sign

since

if .x >

°

hen yIN) > 0 for

x

II 11+1 n

all n , and if x < 0, then

consecutive

~ ) S have

opposite

signs. Thus

relative

errors < E in Y ~ ~ I and

y \ ~ ) will pr

o

duce

a relative

error < E

in

Therefore, the

relative

truncation error

in all

/, ,(x),

o

x <

NB, is

bounded

by

1:

.

lO - NSJ ;.

2

For} I (x),

n MAGX,

the relative truncation

error

cannot

be

bounded, since

as = MAGX,

MAGX - 1, 1 in 1),

the

values} I (x)

oscillate,

and precision is lost owing to cancellation. In

this range

, JII(X) is accurate to

about

D

decimal places

,

where

D is the number

of decimals

in

J

MA(;X + 1 (x)

which

corresponds

to

NSIG

significant figures in

the same

quan

lity

[1, sec

. 5].

3. ormalization

and

Error ounds for

JA

The eqs 3)

were

chosen

to keep

cancellation under

contro

l. First,

1

I Jo x) 1 ~

I

J,,(x)

~ v 2

f o r n ~ l

(6)

126

8/20/2019 jresv77Bn3-4p125


] , so eac h te rm of the sum

3)

for

J

s is less than

V2

tim

es

the wh ole sum. S

in ce

6) is a

ca n

ce ll

ation is even less than this would indicate .

For

I

s, can

ce

llation is avoided altogether, sin ce all terms in the sum

(3)

for

I

s have the

sa

me

as

shown in sec tion 2.

Bes

id

es bo

undin

g th e

trun

cation e

rror

of th e algo rithm, the program pr

ov id

es an estimated

und for th e truncation e

rror

of th e normalization sum , de fin

ed

by eqs (3). For j s, this e rror is

'

=

- y ~ N + L ( Y 2 k Y ~ Z » )

1

For

2k

MAGX

,

a bound for

th

e error term yW - Y

2k

is un available.

For

MA

GX

<

2k

<

N ,

\ '

k - Y2

k I

<

PaP,

- 1)

[1

,

s

ec

.

5].

To avoid storing all the PH',

the

program allows only

N

rm

s for which 2k ~

N

Here = and Y2k = /h k L l / PrPr +, ). Th erefore

1 1.

1 { 1

1

+-

2

/J 2 k + 1 Pu

sec tion 2 above. Now let

e n

P

2 x

R

(N)

~

L

- 1

k [ ' ; I]

I

Y2 k

I .

1

p

{

1 1 }

1+-? +-+ . , .

p

- 1) J \'+

1

p"" p t,

hk+ 1

7)

For

J

s the program se ts

TEST

1 ~ 2 . 10

NS1G

• The normaliza

ti

on factor fL is 1/1MACX (x) ,

sec.

5],

1

5 .1 ) 1 I h 1 0

,S IC

is is a ra th er wea k uppe r bound , a nd the e rror

--; ;

turns out to be ess t a n 2' I -. '.

For I s, PII /

P

_ I > 2

so

the analog of (7) is

R (N)

-< 8 .

~ 9p y cosh x

127

8/20/2019 jresv77Bn3-4p125


this

is

derived by

substituting 2 for PN in (7), and

introducin

g

the

factor cosh x; see 3).

2·

10 -

NS

;

By setting TEST ~ exp(0.461 . MAGX) the

program

insures

that

Here, fL is -1 1 x), so the relative error

I

R N)/fL

I

s bounded by

MA

GX

I

R

(N)

I ,,; 4

IMAGx x)

exp(0.461 ·MAGX)

fL 9

cosh x .

lO

- NS (; .

Now Kapteyn s

inequality

[4,8.7] states that

lin nz)

I ,,; I

x p ~

I .

1

+ Z2

Ixl

.

Let

z = MAGX so for large x, z = 1, approximately.

Then

and

,,; I z

exp

IM A ;X.

1 ~

For

large

x,

the

approximate bound

for

M Gx x ) is

(

exp

V2

) MA(;X

1 + V2 < exp (0.533 . MAGX).

Substituting

this

in

(8), we obtain

I

N

I

<

4 .

exp

(0.533 MAGX) . exp (0.461 . MAGX)

fL 9

cosh

x

.

lO

-

NSI

.

<

. . lO

- NS G < 1 .

lO

-N

S«;

9 2

Besides this approximate upper bound,

the

strict

bounds

I

R .V) I

-;;- < 7.6 .

lO

-

NS

(;

Ixl

1

I

V )

I

<

0.565 . lO

- NS G

Ixl <

1

may

be derived

by using

Kapteyn s

inequality

to obtain a

strict

bound

for

IIMAGx+, x) I.

128

8)

8/20/2019 jresv77Bn3-4p125


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4. Appendix: Algorithm

BESLRI

WELT

BESLR

1 '1.730222 .

sR730

C

C

C

C

C

5UBI{OIJTINE

uESLR

I IX, Nij , rZE , Q,NC ALC)

THIS

ROU TI NE CALCULA

T

ES

'IESSEL

FUNC

TI

ONS

I J

OF

REAL

ARGUMENT

~ N D

I NT

EGER ORDER

.

EXPlANITI

ON OF

VIRIAALES I N THE

C

LLI

NG

SEQUENCE

C x

C

DOUU LE PllEc rSION HEAL A R G U ~ E N T FOR WHICH I S OR J S

I .E

TO BE CllCUlATED . IF

1

5

ARE

TO RE COLCULATlJ .

O

S (X)

MUST BE LE SS

T

HON

EX

PARG (WH

I

CH SEE BE

L

Ow ).

I NT

EGER TY PE

. 1 + HI

GHES

T ORDER TO

BE CALCULA

T

CO

.

IT

MUS

T

BE

POS

IT I

VE.

C

C NO

C IZE

I NTeGER

TY

PE . ZERO I F J . s ARE TO BE·CILCUlATED . 1

IF 1.5 ARE TO BE

CALCULATE D.

C B

C

DOUDLE PREC

I S

IO

N

VEC

T

OR

OF lENG TH

NR

.

NE

ED

NO

T

HE

I

IH

TI ALllED BY USER .

IF

T

HE ROU

T

H)E

T

ERMHIA

T

ES

NORMAll

Y (

NC

AlC=

NR)

.

IT RETURNS

J(OR

l ~ R O

T

HflOUGH

J(OR

I - SUB

-

NB

- MI

NUS

-

ONE OF x

I N THIS

VE

CT

OR

.

C

C

C

NCAlC

I NTEGER

TY PE

.

NEED

BE I N

IT I AL

I ZED ny USER.

BEFORE US l

t\J

THE

RESULTS

, THE USEH SHOULD

CHECK

T ri

l

NCALC =

NO

.

I.

E.

~ l L O R D E ~ S

HAVE A E E ~

C A l . C U l ~

TO

T

HE DES

I RED

ACC URACY

. SEE

ERROR

RE TURIJS

R ~ L O .

C

C

C

EIPllNA TI

ON OF MACH

I

NE-DEPENDEN

T CONST'NTS

C

NS

I G

DEC IMAL S I GN I FI

CANCE DES

I RED .

SHOU

LD RE

SE

T

TO

I FIX ALOGlnC2) +rJQIT + . f HERE 'Jf1 IT IS

TH

E -IUr

1HEH OF

B

IT

S IN

TH

E

MANTISSA OF

A

DOURLE

PRECISION

V A I < I A U l ~

SETTING

NS

IG

lOWER WIll

RESULT

I N

nECREASED

A

CCUkACY

WdlLE

o;ETTHJG

NS

IG

HIGHER

WILL

I

NCREASE CPU e

WI

THOUT

I

NCREAS

I

NG

_CCURACY.

THE TRUNC

ATI

ON

ERWOR

IS LIN

ITED

TO T=. 5

+I

O

NSIG

FO

R J . S OF ORDER LESS

THAN

A P G U ~ f N AND

TO A

RELA

TIVE ERROR OF T FOR

r+

s A

Nn

T

H€

OTHER

J 5 .

f'JTE J LARGEST I NTEGER K SllCH

THAT

t o•• K I S MI\CHINE -

REPRE5DJTA LE

D O L I ~ L E

PRECISION .

LARGO UPPEH OI J T

HE M ~ G N

L J D E OF X. nEAR

IN

M I;U

THAT

IF ABSIX) =N. THEN AT

LEAST

N IT ERA TI

ONS OF THE

BACK ,A

RD

RECUR SI

ON

WI

LL BE

EXECU T

ED

.

EXPORG

L

ARGES

T

DOURlE PREC

I SI

ON ~ R G U M E N

THAT T

HE

LI

BR HY

DEXP ROUTI NE

CAN

H ' ~ D L E

ERROR U R ~ S

LET

G

nENOTE

EIT HER I

OR

J

I N

CASE OF AN ERROR

,

NCALC

.NE.NR ,

AN

D

ALL

G*S

ARE

C ~ L C U l A

TO THE

DESIRED

ACCURACY.

IF

IKALC .L l .

O.

N

ARGU" EN

T IS

OUT OF RANGE

.

NB

.

Ll

. O

OR I ZE IS NE IT

HER

0

NOR

J

OR

I

ZE=

l

ND

A n S I X ) E . r A k ~ .

IN THI S

CASE

. THE U- VECTOR I S NOT CALCUL

AT

ED . A

ND

NCALC

IS

SET

TO MI

NO

( N8 . 0

)-1 50 NCA

l C.NE .

NB

.

129

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CUR

IF

~ ~ G X ~ N D

SU:3-N f3- 0F- X/ G - SUf3 - \1AGX - ( )F - X) • LT. l O. *. (NTF.N/?), I E . Nti

IS ~ U C ~ G R [ ~ THAN \1AGX . IN THIS CASE , R{NI IS CA L

CU

L ~ T E D

TO T

HE DES

IRE

D

ACCURACY

F

DR

N.LE .

NCALC

.

BU

T

FON

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T. N. LE.N . PRECISI ON

IS

L

OS

T. IF

N C A L C NU

A U S ( R ( ~ C A L C I /

I ) ) . E Q

~ . - ~ . THEN

OML Y T

H[

FIRST NSIG-K

SI G

N IFICA NT

FIGU

RES OF ~ ( ~ )

BE IF

T H ~ USER

WlSH£:S

TO ( , \LC ULATE R (NI T O HIGHER ACC:URr.CY, HE

SHOULD

USE

AN

/\Sy\1f>T

OTIC

Fl.W

1ULA FO R

LARGE ORDER

.

DOUdlE

PRECI S

IOn

1

T E ~ P A

T E ~ P 9

E M P C

X P A R G

O V E H ,

2 PLA S T, r OLD , rSAVE , PS AVEL

~ I \ 1 E

\ j S I O ~ J

[JOJB I

DATA

L A R G ~ X , E X P A R G / 1 3 0 7 , 1 0 0 0 0 0 7 . D 2 /

E M p r = D . \ ~ l S ( X )

~ A G ~ = l F I Y ( S N ~ E M P A ) )

IF GT . 0 . A tD . ·..,AGX . Lt: A ~ G E X AND . { 17E . EO . (l . OR.

E ~ P A E X P A R G ) ) GO TO 1

E H H 3 ~

R E T U f f ~ - - X.

N8

,

OR

I 1E IS

OU

T OF RANGE

N C A L ( ) - l

H . E T U Q ~ J

S I G,J = Dt iLE(FLOA T ( 1- 2 *I

lE

)

NC

ALC=N

:3

USE

2

-T l. ASCENnING

SERIES

FOR

X

IF(T

fViP

A.*

l

T

•• I 0 0 · . ~ J S I G }

GO

T O 30

INITI ALIZ

E

THE C U L A OF

POS

NRViX=iJR

- /llAG ;-t.

N= ·1AGX+ l

PLA,T=I . DO

r = :JJL[ (FLOAT(2*hl )

)/TE\1PA

CA L CU

LATE

E H A L

SIGNIFICANCE

TES T

TEsr=2. Df *I .

Jl

NSIG

IFII ZE . LO.

I.ftNO.2·.A

G

.GT.soNSIG)

TES T=OSQR TITES TO P)

E Q . 1 . A N ~ A G X . L E ~ S I G ) T E S T = T E S T / 8 S * *

ol =n

IF P>lf \'-1X . L l . j ) GO T O 4

C 0 L ~ T P *S u ~ T r N H - l . CHECK FOR POSSI BLE

O V E R F L O ~ .

TOV ER

= l. JI

• • (N T EN -

NSIG)

T A f n =

~ 1 A G X

N E N l J ~ J t i

DO 3 S T A R T N E N O

P

OLU

=rLAS T

PlI\S T=P

P : OR LE(FLOAT ( 2 *N) ) . PLA ST/TEMPA - SIGN*POLO

IF(P-

TOVER)

3 , 3 , 5

CONT I JUE

= ~ E I D

CALC U

LAT

E SP ECIA L SIGNIF I

CANCE

TEST FOR >lRMX.G T.

2.

~ A X 1 (

D S Q H T

T · l . I G ) · n S Q R

T ( 2 . D O ·

C

L

CU

L

ftTE

pos

UNTI

L SIGNIFICANCE

TE

ST

PASSES

4

i J =Nt

1

POLD =PL ST

r L I I ~ T : p

P=DllLE IFLOA T (2 0N )° LftST I TE MP - S IGN OPOLO

IF

P . LT . TESTl GO TO l

IFII7E

.

E0

ol . 0 R. M

EO ol) GO

TO 12

C

FOR JOS

• •

STRONG

V R I

N

T OF

THE

TE ST IS NECESSARY .

130

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ND

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lI ·JT l L

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, NE'ID

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1

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DE ALL

[J

n y

SUV1 .

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7E

.

EQ

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00 2(,

-1 =1 ' ~ H

26

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F<Elu ... U

T

WO-

T

ERV1

ASCENU I NG SER I ES FOR SMALL X

3 0 TE '1pl\= 1. no

r E v p n =

D O · X .

( t ) :: 1 • On TE '}PR

II="(,J ;

.E J

. l l GO TO

32

')0 3 1 ~ J l 1

T E V ) A = T [ ~ P A

X / O ~ L [ ( ~

31

~ ( ~ ) T E . P A E ~ P A

L E I F ) ) )

32

RETURN

[r ID

END CUR LCC 11

02

- 0038 L8

S. References

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[11

Olver, F. W. J . and

Sookne

, D ./.. A note on

ba

ckward

recurrence

algorithm

s,

Mathema

ti

cs

of

Computation. Vo

l. 26,

No.

120 (O

c

t. 1972).

pp.

941 - 947.

[2J Olver. F. W. J..

Numer

ical solu

ti

on of second-order linear difference eq uations.

J. Re

s. Nat. Bur. Stand. (U .S.). 71B,

(Math. and Math. Phys .). Nos 2&3. 111 - 129 (Apr.- Se p

t.

1967).

[3] British Assoc. f

or

the Advancement

of

Sc ien ce. Bes se l Functions - Part ll . Mathematical Tables. Vol. lO

(C

am bridge

Un iv er s

it

y Press, Cambridge, 1952).

[4] Watson. C. N .. A

Tr

e

atis

e on the Th eory of Bessel Func

tions

(Second Edition.

Camhrid

ge Univers

il

y

Pr

ess, Cambrid ge,

1944

).

(

Paper

77B3&4-387)

132

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