特殊功能近似大物实验报告s21bcc cl05.pdfVIP

s–ApproximationsofSpecialFunctionss21bcc

nagellipticintegralrd(s21bcc)

1.Purpose

nagellipticintegralrd(s21bcc)returnsavalueofthesymmetrisedellipticintegralofthesecond

kind.

2.Specification

#includenag.h

#includenags.h

doublenag_elliptical_integral_rd(doublex,doubley,doublez,NagError*fail)

3.Description

Thisfunctioncalculatesanapproximatevaluefortheintegral

RD()=3∞dt3

20(t+x)(t+y)(t+z)

wherex,y≥0,atmostoneofxandyiszero,andz0.

Thebasicalgorithm,whichisduetoCarlson(1978)andCarlson(1988),istoreducethearguments

recursivelytowardstheirmeanbytherule:

x0=x,y0=y,z0=z

µn=(xn+yn+3zn)/5

Xn=1−xn/µn

Yn=1−yn/µn

Zn=1−zn/µn

λn=√xnyn+√ynzn+√znxn

xn+1=(xn+λn)/4

yn+1=(yn+λn)/4

zn+1=(zn+λn)/4

Fornsufficientlylarge,

n=max(|Xn|,|Yn|,|Zn|)∼1/4n

andthefunctionmaybeapproximatedadequatelybya5th-orderpowerseries

n−14−m

RD()=3m=0(zm+λn)√zm

4−n3(2)1(3)3(2)23(4)3(2)(3)3(5),

√

+31+7Sn+3Sn+22(Sn)+11Sn+13SnSn+13Sn