特殊函数近似值大物实验报告s11abc cl05.pdfVIP

s–ApproximationsofSpecialFunctionss11abc

nagarcsinh(s11abc)

1.Purpose

nagarcsinh(s11abc)returnsthevalueoftheinversehyperbolicsine,arcsinhx.

2.Specification

#includenag.h

#includenags.h

doublenag_arcsinh(doublex)

3.Description

Thefunctioncalculatesanapproximatevaluefortheinversehyperbolicsineofitsargument,

arcsinhx.

For|x|≤1thefunctionisbasedonaChebyshevexpansion.

For|x|1

arcsinhx=signx×ln|x|+x2+1.

Thisformisuseddirectlyfor1|x|10k,wherek=n/2+1,andthemachineusesapproximay

ndecimalcearithmetic.

Forx≥10k,√x2+1isequalto|x|towithintheaccuracyofthemachineandhencewecanguard

prematureoverflowand,withoutlossofaccuracy,calculate

arcsinhx=signx×(ln2+ln|x|)

4.Parameters

x

Input:theargumentxofthefunction.

5.ErrorIndicationsandWarnings

None.

6.FurtherComments

6.1.Accuracy

Ifδand

aretherelativeerrorsintheargumentandtheresult,respectively,theninprinciple

x

√

|

|δ.

1+x2arcsinhx

Thatis,therelativeerrorintheargument,x,isamplifiedbyafactoratleast

x

√1+x2arcsinhx

intheresult.

Theequalityshouldholdifδisgreaterthanthemachineprecision(δduetodataerrorsetc.),

butifδissimplyduetoround-offinthemachinerepresentation,itispossiblethatanextrafigure