71~651179b··005 -.4244914324-006. 6. -.1085114509+005 -.7507824251+003. 7 .47802S770g"006 -.3323221144-007. 7 -.1399557851+006 -.1111091993+ ...
AD-767 223 AD-767 223 A METHOD COMPUTING SPHERICAL SPHERICAL BESSEL BESSEL METHOD OF COMPUTING FUNCTIONS COMPLEX ARGUMENT FUNCTIONS OF COMPLEX ARGUMENT WITH WITH TABLES TABLES
W. J. Lentz Lentz Army Electronics Electronics Command Command
Prepared Prepared for: Atmospheric A tmospheric Sciences Sciences Laboratory Laboratory September 1973 1973 SE::ptember
SEE CORRECTIONS AT THE END!
DISTRIBUTED DISTRIBUTED BY:
Service National Technical Information Senite
U.S. S.DEPARTMENT COMM!IRCE U. DEPARTMENT OF Of COMft' Jf:RCE
5285 Port Royal Road, Springfield Va. 22151
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DIS CLAIMER NO DISCLAIMER NOTICE TICE THI S DO THIS DOCUMENT CUMENT IS BES BEST QUALITY T QU ALITY PRA CTICABLE. THE "CO PRACTICABLE. COPY FURNISHED PY FUR NISHED TO OTI C CO DTIC CONTAINED SIGNIFICANT NTAINED A SIG NIFICANT , NU MBER OF PAG NUMBER PAGES WHICH NOTT ES WH ICH DO NO REPRODUCE , ,RE PROD'UCE LEG LEGIBLY. IBLY. I
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Report s Contro Symboll Reports Controll Symbo OSO·1 36t, OSD- 1366
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RESEARCH AND DEVELOPMENT TECHNICAL REPORT RESEARCH AND DEVELOPMENT TECHNICAL REPORT ECOM ECOM-5509 ·5509
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,AANEW METH OD OF COM METHOD COMPUTING SPHERICAL PUTING SPHE RICAL :to ARGUMENT COMPLEX FUNCTIONS BESSEL EL FUNC TIONS OF COM PLEX ARGU MEN • : I;ESS T :~ • WITH TABL TABLES ES
f
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:= :c i
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lex Ten-place tables of spherical Bessel f'tnctions; increments for 100 consecutIve consecu.tive orders in varying Increments Imaginary values from 0.001 lo ~o 1000 In rrethod used to ~o~erate based on a a continued Isbased i'itarate these tables Is are presented. The method This of consecutive order. This fraction representation of tho ratio of Besse'l functions of Bessel functIons rrethod retains a machine, Incorporates Incorporates method a high degree of accuracy In na finite digit machIne, Is fast fast and 10 I 111> Ie, and Is versatile wide a terminating condItion, and eflople, and Is versatile enough to find wide condition, Is app II cation as a method for conti nued fractions , n genera I. application as a method for continued fractions Ingeneral.
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APPENDIX APPENDI X..
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ACKNOWLEDGEWvNTS . ACKNOWLEDGE~NTS • . . . . . . . . . . . . . . . . LITERATURE CITED. CITED. . LITERATURE
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INTRODUCTION I NTROOOCT ION
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Spherical Bessel Bessol functions of complex argument argurrent find application in computations ranging from electronic circuit analysis In scientific .;:omputatlons analysis to the understanding of nuclear and Mle scattering. Yet tables of of mited to spherical Bessel functions Bessel functions are presently presently'lI Imlted functIons argum~nt and to nonspherlcal nonspherical Bessel functions of complex of real argurrment order. Tables Tab les of of the former forrrer may be found In argument of 0 and I order argument Abramowitz tables Abrarro\',ltz EI], [IJ, and tab les of the latter latter In [2J. The The In L.owan Lowan E2]. generation of spherical Bessel functions of complex corq:>lex argument argurrent is Is not not a trivial problem, problem, even with modern high-speed computers, because because of difficulties d I ff I cu 11'1 es associated as soc I ated with each of of the methods used for their the i r geMre.r ion, ion. generat It bezame To surmount surrroun't these difficulties, It be'.:arre necess'ry neces:s~lry Ic tc develop a rachine computation, method rrethod which whi ch would wou I d be ssimple I rrp I e and accurate accu rate for 1ll;1ch I ne .comp u tat Ion, "he result was and which would incorporate Incorporate a terminating condition. condition. fhe of a rrethod method based on a continued fraction representation of the ratio of Bessel functions consecutive order. order. Bessel functions of of consecutive The purpose Is primarily primarl ly to present purposa of this report Is presant the new method rrethod of calculatIng calculating continued fractions, fractions, and secondar! secondarily Iy to present present Bessel functions of complex argument tables of spherical I39ssel argurrent against against which method and others may be checked. checked. The devtlloprrent development wh I ch this th I s rrei'hod Is placed In In perspective by reviewing the advantages and disIs dlsadve,ni'ages oJiher methods and algorithms which might be used. advantages of the o'her SuggesTlons are mode made relative to the extension of the rrethod method to Sug~stlons to other functions of complex argurrent. argument. RECURSION RELATIONS RELATIONS
Recursion relations simplest common rrethod method of of Recu rs ion roe I at Ions are the simp I est and mst fli)st comrron Is generating Bessel functions. fundions. The fundamental fundarrental recursion relation Is
~~\~ >'
~,~
~, It. -r
jv+l(X (
+ Jv-1IW = 2v/x Jv(x)
(() I)
''Ii
"r ;
0 ,,. 'l~
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'I
is the Bessel function of order vv and complex argument ~x where JJ Is = m+1/2 m+I/2 where n is is an Integer. integer. This relation Hlay may be used with and vv = two consecutive orders to generate all the necessary orders by starting at an appl'oxlmate approximate high at 10\', Io%; order and recurring upward or starting ai' orner order and recurring downward. ser~lous ilimitations. methods have serJous Unfortunate I y both of these rrethods I ml tat Ions. Lowan Unfortunately Is [2J E2] has noted that the upward recursions are unstable when 2v/x Is of greater than unity, consecutive orders of unity, because the values of con:Hlcut'lve
,.I 33
Preceding page blank
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Bessel functions functions have comrrKJn common Initial initial digits. Gautchl Gautchl [3J [3] has shown Bessel that the the upward recur'~I(~n recursion has no no digit accuracy after only seven that = 1.0. recirsions with with 32-blt 32-bit capacity capacity for xx = Therefore, even using rec'Jrslons 1.0. Therefore, double precision arl+hmetlc, ari+hmetlc, the procedure will not work for rrore more double a very few cons'3cut consecutive I ve orders. than a
. "
Millerts downward r(~curslol1 recursion scheme [4J [4] has solved the problem problem of of Miller's accuracy at the expanse of machine time and storage. By sti:~r1'lng aq::uracy startling with an approximate value for high orders, one can recur downward with onl onlyy roundoff roundoff error. error. Since downward recurs recursiurts a very I vII:' start at a high order order for for rrodf=rate moderate arguments, arguments, this rrethod method is Irrpraci"lcal impractical high when only only aa few few orders orders are are needed. needed. This This technlaue technique also suffers when from the the same problem that the the forward recurs recursion once from same probl em that 10il has -- Cince Is lost in in aa term, term, all the following -terms terms haVf~ have lost the accuracy Is same amount amount of of accuracy. In In this case, roundoff roundoff error may accumulate same this case, so that that the the more more significant smaller orders are very much rrore more so Inaccurate than ~the the approximate Initial initial high order. Inaccurate INFINITE SERIES INF INITE SERI ES The infinite infinite series series for for Bessel Bessel functions functions Is seldom used because the The series coMpLtatlonal ly divergent divergent for for rroderate moderate x and small orders. orders. series Is c01Tpl,tatlonally When the serle!; serie.s does converge, it it converges it is is difficult COn\ffJq~es slowly, and it difficult program. I.,flnlte lifinite series represen'l"c.,tlons represenieations of Bessel functions are to program. therefore interesting but not very practical for therefore mathematically matrematlcally Interesting for computat Ion. Ion. corrputat
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CONTINUED IONS CONTINUED FRACT FRACTIONS Continued fractions have the advantage of being the advantage program, Cont i nued fr;3ct Ions have be i n9 ssinple, i rrp I e, easy to program, and accurate accurate for for machine machine corrpui'atlon. computation. Unfortunately, have been and Unfortunately~ they havo neglected In In favor favor of of recursion recursion relations because 01 oi difficul1'les difficulties In in neglected determining the accuracy of a given number of terms ina in a coni'lnlled continued fraction. The following development of continued fractions is fraction. folloWing developrrent is Intended weaknesses in in available to review their basic properties, point out the wec.>,knesses avai lable alqorithms, a method of generating Sessel Bessel functions which algorithms, and to develop a is free of these weaknesses. is
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Basic Properties Bas Ie Properti as A continued fraction may be written ~1S ,s
f= aOO +-I' b I f =a
----,-c I + b2 C.2 + b 3
3
2
I3 ~4
(2) (2)
...
4
,
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which can can be be rewritten rewritten as as which
~
f= a +
'KThe
~
(3)
th The nnthapproximate approximate convergent convergent to to the the continued continued fraction fraction representation representation Of of ff is is written written as as b
n fn =a 0 + cbbn .. ++c n(4
(4)
n
Continued Continued fractions fractions my may be be generated from from a three-term three-term recursion recursion relation in a simrple simple way: way: relation J
2v v-I SV1 2v
2v 2v
v+1
jJ v+i
=---=
x
J J v
2(v+ I) 2(v+l)
x x
J J v
J Jvv+2
--
x
42v
I
2v
- -.xx
J v+1
22(vY±l (v+ I) xx
-2(v+2) -x- x
(5
..
So So J
Iv
v-I V-1
-I1
2v = 2vx
r;-
=
1 ______
-I
22(v+i)x (v+ 1)x
-I - 2(v+2)x 2v2x
...
'
(6) (6)
Th Is form s one i rrp Ie cont i nued ffraction ract i on wh i ch may i ned as This form Iis one of of aa ssimrple continued which may bEl bet def defined as
f(x)
=
f~)=
aI
+ _I a
a+ 2
1
I
+ a + 8 + ••• + an + +a3 + a4 + + +
...
(7)
(7)
55
..
where 8a # # 00 and the ats a's may my be negative. negative. where 1 Equation (7) (7) may may be be written written In in the the nore rore convenient convenient notation notation of of Equation f(x) f(x) =
[al, a2, a3, .. a , a , ••• J a2, 2 3
(] (8) (8)
[ai'
th convergent The n nth convergent is is written written In In Ilike manner: The Ike manner:
I
fn (x)
[al,, a2, a 3, ... , an]
(9) (
has been been shown shown by by Wa Wall [5] that that aall continued be IIt t has II [5J I I cont I nued ffractions ract Ion 5 may be expressed as as simple simple continued continued fractions fractions of of the the above above form. form. In In expressed addition, Watson [6J [6] has shown that the form of the continued fraction addition, fraction for Bessel functions is is unique, unique, requiring that for Bessel functions that all the nth convergents of the continued fraction fraction in in whatever whatever form form have have the same nurrerlcal numerical the continued the sarre of value. For example, all of the following are equivalent equivalent In in the the nth lowing are nth value. F(lr example, all of the fol convergent: convergent: , ' ~
J
2 1/4 x /(v[v+l]) /(v[v+IJ)
1/2 x/v
Jv
=-.-
J _
I
v 1
2 1/4 i/([v+I][v+2]> x /([v+l][v+2])
I
.
,
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JI
Jv
-J-
v-I
I
---=r 2vx
...
2(v+ I)x-I
tIl)
,
v "
,,
J
'
JvI
l.... /. ~
v-I
=XJv ~I 2v 2v
2 x x
- 2(v+ I) -2(v+l)
2xI_2v.~_
I
X
2(v+2)6
-2(v+2)
...
;
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A Igorl i"hms Algorithms It is it is now possible to generate the ratio of Bessel functions by starting with the last value on the right of Eq. (9) (9) and working backwards to a al'. However, the minimum minimum necessary value of of n must be specified befoie the computation starts, since adding one term means the rebefore computation of the entire fraction. fraction. Gautchl [3J [3] has made an estimate of the value of n for determining Bessel functions to a given accuracy. is and is fractions and continued fractions general to In general apply In does not This method T11s rrethod does not apply to continued only an approximate estimate which is difficult to apply. Moreover, onl y approxl mate est imate wh ich is d iff Icu It appl y. tvbreover~ the tai tail-to-head I-to-head method suffers somewhat sorrewha'~ from roundoff error when when the difference between two va values is near Iues Is neal" zero. Another method of comput computing logically i n9 J v./Jv II J follows fo I lows 109 Ica II y from f rom the relations relations v- v
f~ •
Pn
anPn-I + Pn-2
qn
ann-I n
+
n-2
(13)
"
( 13)
I
so so
. I
Pn ••• , a ] -- , a2 =.n n
t
((14) 14)
t h partial numerator and q the n t h partial where p is Is the n nth nth denomlngtor of ~he convergent. n denominator he nth nth convergent.
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Wall [5] has shown Wal I [5J shown that these these relations are generally true for for any fraction. Starting with the il1ltial initial values of pl al, continued fraction. PI = al' q, = I, p = I, I, and and q 0, one one can generate successive approximants approximants 0 = ql = I, Po q = = 0, can generate successive to the exact infiniteoc-ntinued fraction until the desired accuracy is to the exact inflniteOc~:',tinued fraction unt! I the desired accuracy Is obtained. is II illustrated in Te Teichroew Abramowitz Th i s method Is I ustrated in i chroew [7] [7J and Abrarro'tlitz obta Ine~. This [I]. The former has noted that the values of pn [IJ. p and q q may grow so so Iow in I ve negative overflow in the corrputer. computer. MJ?eover Moreover,t n for negat large as to cause overf cancel, causing loss of accuracy number of digits may cancel, a an initial Initial number am i~ the mth mtr convergent in in exactly the same way as the original original in recursion (I)). This forward ratio method leaves (Eq. (I». recursion relation (Eq. much to be desired. desired.
i
':. .' -'f
-
j
00
(x)
sln(x) J 1/2 (x) = =jk 2Li~x x j
1/2
Fix
jn+1/2
I + / j b,\ = n I 2 J n j/-1/2 ,~-1/2 J
-n-I/2 -I2 Yn (x) = (x J ~-n+1/ -n+I/22
0o
x
J
__n-1/2
n-I/2 J n--3/2 jn-. 3/ 2
J
v!;x2x
()== y (x)
-J
(x) () = --cos -cos(x) J_-1/2 (x) / xx I2
,.11' ... J 1/2 / 2x 12,2
-n+l12 2 ...
·iT J 1-
-1/2
J--n+3/2** n+312
V2x
(22)
ACCURACY
7"
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A conparison comparison between'the 64-bit and A between tho accuracy accuracy of aa double precision precl~lon 64-blt single single precision 32-bit 3Z-blt program was made for the real values of of argume~nt in Abramowitz [13, The agreement argument tabulated in Abrarrowltz [IJ. agrt:Jement between the the ~double drld the exact11 for c a MI double precision precision rouiitiu rou.i illt:! alio the harndboo'K hClndbook was eXact tOi 10 s~gn1f-I significant digits comrparing the single precision digits for all arguments. By comparing It Is Is os+irnated ~s+irr~ted that the version to the double precision version, It su.:e"l i'e-rirac;flne last tour of the 16 digits would "'/ould be su;.:::er>·',,·le raciline roundoff. Therefore, 1,c diow i-he table tab leiis s reproduced to 10 I'e 'II ow for the Therefore, the additional possibility of roundoff roundofaf err ,I, xi arguments. possibility of err -.1. ,( arguments. additional OTHER FUNCTIONiS FUI\ICTICJ!'iS Many known funet functions ~~ny kno'/ln Ions have been exprest.ti expres!::l\ d ~ n"\ '\. i. . "r r, r. of cont inued I nued fractions, and they may be evaluatai evaluate!j /)y Eqv :17). Ahrarrowitz [l3 [lJ has has )yEq.. 17). Abrarrowitz I sted one form of acontinued tn,c;:tiol1 tiction rej resentation for most listed a-continued re,rcsentation rrost of the
10 10
functions In his handbook. handbook. The The form functions In his form be changed to ;nat of Eq. (7) by the the be changed to :nat of Eq. (7) by
bI 2 f(x) -+ c f(x) =a 0 + c I 2
+
... ...
of any any continued continued fraction fraction may may of following transformations: following transformations:
d b
= a 0o+ dC - I
d d2 b2 + d2 c2
d2 d3 b3 + d3 c3
...
+
The only requirement a be be nonzero; zero, th this The on I y requ i rement Iis s that that a nonzero j iif f aa Iis s zero, Is requirement be met ret by by takig the rec reciprocal 8f the the cont continued requ I rement may may be tak I~g the I proca I 8f i lilled fraction. fract ion. The continued continued fraction technique wi II converge converge with with fewer terms terms The fraction technique wi II than any other current method Is specifically than any other current method and and Is specifically adapted to machine computation. Although Although mathematically mathematically equivalent equivalent to to recursion recursion computation. relations, the continued fraction i'echnlque technique gains gains accuracy accuracy from the relations, the continued fraction fact that the product product of numbers In in the the "orrputer -omputer does does not lose fact that the of two two numbers accuracy, as does the difference of two numbers. numbers. accuracy, as does the difference of two ACKNOWLEDGEMENTS would like like to to express express my lo Mr. G. B. B. Hoidale for his II would my thanks i0 Mr. G. his numerous suggestions and recommendations numerous suggestions and recomrrendatlons on this manuscript. manuscript. I would like to to thank thank Mr. Mr. G. G. Ethridge Ethridge for his exhaustive would especially especially like checking of the tables and formulas in in this paper. checking of the tables and the the formulas this paper.
,v
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LITERATURE CITED
•
I. Abramowitz, M., and I. I. A. Stegum, ed., 1966, "Handbook "Handbook of MathI. Abramowi tz, M., A. Stegum, ematical Functions," Commerce, National Bureau of of erootlcal Functions," US Department D:3pnri'rrent of Commerce, Standards, Washington, Washington, D. 20402. Standards, D. C. C. 20402. I,
, ' ,
)'"
. ,
2. I. A., A., 1950, "lables (z) diid and Y I (z) 2. Lowan, Lowan, I. "1ables of Bessel Functions Y Yo(z) (z) for Corrplex Complex Arguments," Arguments," Computation Computation Laboratory, Laboratory, Rational National Bureau of Standards, Washington, D. D. C. C. 20402. 20402 • 3. Gautchi, "Computational Aspects of Three-Term 3. Gautchi .• Walter, 1967, "Corrputational Three-Term Recurrence Relations,!! Relations," Slam Review, Review, 2. 9 (I), 24-82. (1),24-82. 4. 4. British Association for the Advancement of Science, Science, 1952, Mathematical Tables. Vol X: Bessel Functions; Part II: Functions Order, University Press, Press, Cambridge. of Positive Integer Ordec,
G-
,
r '~
I ri
)
5. 5. Wall, \~all, H. H. S., 1948, Analytic Theory of Continued Fractions, Fracticns, D. D. Van Nostrand Company, York. Nostrand Corrpany, Inc., Inc., New New York.
\' i
6. Watson, G. N. N., I 1952, 1952, A ATreatiseon 6. Watson, G. Treati se on 'the the Theory of Bessel Functions, Funct io~, Press, Cambridge. University Press,
", , ,
7. Teichroew, "Use of Cont Continued Fractions High Speed 7. Te ichroew, D., 1952, IIUse I nued Fract ions In I n Hi gh Sp,ged Corrmputing," Cormut., 6, 127-133. Corrputing," Math Tables Aids Comput., 6, 127-133.
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APPENDIX APPENDIX AA
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USE USE OF THE THE TABLES TABLES
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The following following tables tables contain ct:.ntaln 100 100 orders orders of of spherical spherical Bessel Bessel functions functions of of the the first and and second second Kind I'd nd for for the same sarro argument argument on on the the same sarre page. page. The The real real and and imaginary Imaginary parts parts of of the the spherical spherical Bessel Bessel functions functions of of the first /XJ n+ first kind jn (x) (x) = ~/xJ +1 (x) (x) are presented presented on the left I eft side s I de n
{
,
!" 1',
;
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ct
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N 2 3 4 5
-.12189A8248-2~2
...........
= -.1000000000-002 SECOND KIND Cz
z~' 0
W.4'0 0'
0
r* 4 0 Cc AI
004
if) c0
-.84026A9331-174
0
41-
-.844470279~-169
0
.4 .4 lJ- '.
.
-.3457733971-131 -.972487697H-128
0*
4
Nr-fs0'\0. -
.76887A081~-077
0
N- V..4..4
N0.4 o
)Xn*I-
.106082966~-030
.179015007H-026 .1253271274-071
N
.
coo t
-.127260973b-271 -.1034642070-276
OWWON)N4.0'o)0U).-41'-.-4Ul)o0'4LC
0a 4Wo-tillo--#% IIn
.11834A374~-230 -.11834~372!l-227
4.Uo
Nl-i
.40425~5131-064
.1155010069-068 .160084551!l-080 -.1600645474-077 -.7809002501-082 -.2918533262-135 .291853320H-132 .. 162267694~-17q. .1955032491-179
BESSEL
.3333333333-006 -.3333332667-003 -.1333333333-006 -.1904762116-010 -.38480038 4 8-021 .3848003552-018 .5920005920-022 .3946670845-026 .2443758894-037 -.2443758777-034 -.2327389423-038 -.1011908485-042 -.2998247245-054 .2998247140-051 .2067756719-055 .6670183168-060 .1155010038-071 -.1155010006-068 -.6243297500-073 -.1600845552-077 -.1816047096-089 .1838675950-127 .2918533298-132 • 1955032q.78-182 -.19550324q.5-179 0.04U
N N.N 4-44 NAl)% %0W4V. 4 vq
0-t
.101190844~-O45
-.1011908404-042 -.8095267557-047 -.2998247347-051 -.6670182971-063 .6670182764-060
I~hGINAPY
oIIIIIIIIIIIIIII 111
)
-.3946670614-029 .39466703dl-026 .464314189ts-030 .24437!:i901U-034
0oM
4 I .rc4)In
4Mll
-.423280423~-014 -.~848004144-018
IMAGINARY X
0
o-4 ti.
on
-.lq0476169~-OI0
+-
-
0 0 4. M) 4 0) f .4 100 C- .N . 01)4 ri0 . -0 N" vit 4c)4.. cc4 M0 L 4. 5". na~ 0n)") 03 c'- X)' f- 0',') ') M) .4 t)04e0to ccMN 4t 4' ) I ). -0 If4Illcc0 U) 4-t) 0ZVL) 00 it %0 to ) KU o t- 0 (: 4)) %DtP-0 4.U)4'Nr- -4 0C O0 n00U)UN 0 0% r%'%Q'M-o f t- t-0 0t- 0\ N 4T 70' 0 4 -J, 0t to to to .-4 c4c40 k t-0C )nr 4'coN LIDt M 0) N Go) C In i o 4')1 o 0
0
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