Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables

1059 Pages·1970·37.59 MB·English

by  Milton Abramowitz

Preface:
Thper esveonltui mse o aunt groofwC toahn feorneM nacteh eimcTaaatlb les
heladtC ambriMdagseos,n.S ,e pte1m-5b1e,16 r59 4u,n dtehrae u spoiftc hees
NaitonSacli eFnocueni doant tahMneads sachIunssettoitfTts eu cthen oTlhoeg y.
purpoofts hmeee etwiantsgoe valtuhanete eef dom ra theimctaaatlb ilnte hsle i ght
oft haev aitlyao bfli alrsigceac loem pumtaicnhgi Intew sas.t h ceo nseonfs us
opintihoiannts piotfte h ien creuassoeift n hngee mwa chitnhbeeas s niecef do r
tablesc ownotutiloend xu ies t.
Numertiacbaollfme ast heimcfaautln ctairioencn osn tidneumaablnys d c ien­
tisatnesdn gi.n egerresatero ffv uanrcitaeinthodyin gsah cecru orfta acbyu la­
A.
tioanrne o rwe quaisar r eeds oufsl cti eandtviafincac neds, esopfte hcei ailnl­y,
creaussioenfa g u tomcaotmipcu tIentr hsle.a tctoenrn ectthtieao bnl,e s serve
maifnolpryrelairsmyuirnvoepfyr so bbleefmposrr oeg rafmommraic nhgio npee ration.
Fotrh owsiet heoausatyc cteoms asc hisnuectsha, ba lreoesfc, o sueri,n disepe.n sabl
ConsequtehCneot nlfye,rr eecnocgetn hiatzthee dwr aeas p resnseiefndoga r
modervneirzseoidfto hcnel astsaibclafelus n cotofifJ o anhsnm kdee-T.Eo i mple­
mentthp er ojtehcNeta i,to nSacli ence FoundtahNteai iotonnB aurlre eqauue sted
ofS tandtaopr rdesp araev osluucmhee staanbdl iAsdHh oecd AadnCv oims­ory
mittweitePh,r o fPehsisMlo.ir pM oorfts hMeeas sachIunssettoitTftse u cthen ology
chaainrt,moa dvtihsseet aoffft hNea itonBaulr eoafSu t anddaurrditsnh ge
-as couorfis tpesr eparIanta idodni.t tihoCenh atior tmhaCeno ,m mictotnesei sted
ofA .E rdyeiMl,.C .G raNy.M, e troJp.oB l.Ri oss,sH e.r ,TC h.a chJerJr.o,,h n
ToddB, .T oC.m pkainnJds. , T Wu.k ey.
Thper imaairhmya bse etnoi nclamu adxei moufum s eifnuflo rmwaittihoinn
thlei moiftm soa d erlaatrevgloeyl uwmietp,ha rtiactultleantrtot i hnoeen e odfs
scieinnat lifilset lsdA sn. athtaebsme pemtna dteoc ovteheren tfiierolefsd p ecial
functTiooc nasro.ru ytt h geo saelft o rbtyth h Aed H oCco mmiithttae bsee ,e n
necestsosa urpyp lethmteea nbtbl yei sn clutdhmiean tgh eimcpaartlo petrhtaite s
ariem poritnca onmtp utatiaoswn e lawlsbo yrp kr,o vinduimnegrm iectahlo ds
whidcehm onstthrea atuneesd xe tenosfti htoean b les.
ThHea ndbwoaopskr epuanrdetedhrd e i reocftt'hil eoa nMt iel Atborna mowitz,
anIdr ene A.I tsSsut cecgheuasdnsse. p engdreedau tplotynh ceo operoaft ion
manmya thematTihceieiffaron trsot.gs e wtihttehhrc e o operoaftt hAiedo H no c
Commiatrtee e garpepartelcyTi haept aerdt.i ccounltarri obfut thieaosnneds
othienrd ivairdaeuc aklnso waltea dpgperdop plraiieantet hsete e xTth.se p onsor­
shiopft hNea tiSocniaelFn ocuen daftotirho penr eparoaftt himeoa nt eirsi al
graterfeuclolgyn ized.
Itih so ptehdta htvi osl uwminelo lot n lmye etthn ee eodfas lt la bulseeb rust
wiilnml a ncya saecsq uiatuissn etwr istn hef wu nctions.
Ar.LV.EA Ns TIN,
Director.
WashoinnD,g. tC.
m
/
Prefatcoet heN intPhr inting
Thee nthusiarsetciecp taicocno rdtehde" HandboookfM athematical
Functioinssl "i ttslheo rotf u nprecedeinntt ehdel onhgi stooryf m athe­
matictaalb ltehsa bte gawnh enJ ohnN apiepru blishhiesdt abloefsl oga­
rithmisn 1 614.O nlyf our aonnde -haylefa rasf tetrh efi rscto pyc ame
fromt hep resisn 1 964M,y ronT ributsh,e A ssistaSnetc retaorfCy o m­
mercfeo rS cienacned T echnolopgrye,s enttehde1 00,000ctohp yo ft he
Handbootko L ee DuBridgteh,e nS cienAcdev isotro t heP resident.
A.
Todayt,o tadli stribuitsia opnp roachtihneg1 50,0m0a0r ka ta scarcely
diminisrhaetde .
Thes ucceosfts h eH andboohka sn ote ndeodu ri nterienst th es ubject.
Ont hec ontrawreyc ,o ntinouuerc loswea tcohv etrh eg rowinagn dc hang­
ingw orlodf c omputatainodnt od iscuwsist ho utsiedxep eratnsd a mong
ourseltvheesv ariopurso posfalosrp ossibelxtee nsoiros nu pplementation
oft hef ormulmaest,h odasn dt abltehsa mta keu pt heH andbook.
Ink eepiwnigt hp reviopuosl icay n,u mbeorf e rrordsi scovesriendc e
thel asptr intihnagv eb eenc orrectAesdi.d fer omt hist,h e mathematical
tablaesn da ccompanytienxgat r eu nalterHeodw.e vers,o men oteworthy
changheasv eb eenm adei nC hapterP hysicCaoln stanatnsd Conversion
2:
Factorpsp,,6 -8T.h et abloen p age7 hasb eenr evisedg ivteh ev alues
to
ofp hysiccaoln staonbttsa inienad recenrte evaluatainodpn a;g e6s a nd
8
haveb eenm odifietdo r efleccth angeisnd efinitiaonndn omenclatoufr e
physicuanli tasn di nt hev aluaedso ptfeodr t hea cceleradtuieto ong ravity
int her evisPeodt sdasmy stem.
Ther ecorodfc ontinuaicncge ptaonfct eh eH andbootkh,e p raisteh at
hasc omef roma lqlu artearnsd,t hef actth aitt i so neo ft hem ost-quoted
scientpiufbilci catiinro encs enyeta rasr ee videntchea tth eh opee xpressed
byD r.A stiinn h isP refaicseb einagm plyfu lfilled.
M.
LEWIS BRANSCOMDiBr,e ctor
NationBaulr eaouf S tandards
Novembe1r9 70
/
Foreword
This viotslh rueem seou ftl htce o opeerffaotorimftva enp ye rsaonndns u am ber
ofo rgaatniiozTnhseN. a itonBaulr eoafu Stahnadlsao rnbdges et nu rnoiuntg
mahtemattiacbaallne hdsa hsa udn der consiadtle era1as0tyt ie oatnrh,se , for
produocftc ioaom np enldiitkuhepme r esoenneDt.u riaCn ogn feorneT nacbel es,
calblyet dhN eB SA pplMiaetdh eimcDasit viosniM oany1 ,51 592D,r A.b ramo­
witozft haDti vimseinotni porneeldi minarsyu caphnul nadnesr tfbaoukrti ng,
inditchanete eefddo t re chandivcaianclde financial support.
The MathDeiimvsaitooiftnc h Nsea itonRaels eaCrocuhnh caiasll hsaoad n
actiinvteei rnte asb;tl ess1i49n3c heia ptsu blitshqheue adr tjeorluyr" nMaalt,h e­
maitcTaalb laensAd i dtsoC ompuitoan(t"M ATC ,)e, d itorialb esiunpge rvision
exerbcyia sC eodm miotftt heDeei vision.
Subseqtuote hnNetB SC onfeorneT nacbeli en1s 59 2t haet tenotfti hoen
NaitonSacli eFnocuen dwaatsi ondt rota hwdene siroaffib inlainatcyci tnigiv ni ty
tabplreo ductiiotnss.u ppaWo2 ir-ttdhC a oyn feornTe anbcwlea ecssa lltehde at
MassachIunssettoitftTs ue tceh noolnSo egpyt e1m5b-,e11 r596 4t,od isctuhses
neefdosr toafvb alreiksoi unsdT sw.e ntytp- eerisgahotntse nrdeepdr,e senting
scienatnids tesn ugsiintneageb ralssew se lalst abplreo ducTehricsso. n ference
reacchoends eonnss euvse croanlc luasnridoe ncso mmieonndsa,t wewrhseie cth
foritnth b peu bliRsehpeodrft t h Ceo nferTehnecwreae.gs e neargarlee netm,
�
foerx amp"lteht,ah tae d veonfht i -gshpceoemdp uetqiunigp cmheannttg heed
tasokft abmlaek ibnugdt e finidtinedol rtye motvhneee efdot ra bleIstw" a.s
also thaag"tra eoneu dt stnanedeifidson arHg andboofTo akb floetrshO ec cals iona
Compuwtiettrha, b olfue ssu ally efnucnocutanintoden rsae fdso ertma unlodaf s
tabfloeirsn terpaonlodat thiteoernc hnuisqeutfeouts lh o ec cascioomnpault er".
ThRee posrutg getshttaehtNde B S undtehrpetr aokdeu cstuicaoHh na nodfb ook
antdh tahtNe S Fc ontributaest safiinnscaTenh.cCe io anlf eerleencctee idt,s from
partictihpfeoa lnltoCswo,im nmgi tPt.M e.Me o:r (sCeh ai,rM m.Aa bnr)amtozw,i
J.H .C urtRi.Ws .sH ,a mminHg.,L eDh.mC e.rB ,.T ompkJi.Wn .sT ,u ketyo,
heilmp pletmheenste arnedc oomtmheenrd ations.
ThBeu reoafSu t arnddusan derttoo opkrt ohrdeeu ccoem mteanbdalenedtds h e
NatioSncaileF nocuen dmaatdifeou nna dvsa ilTaobp lreo.vt iedceh gnuiicdaaln ce
tot hMea thematicosft hBDeui rveiwashuii,coc anhr roiutethd we o rakn,d p rtoo­
vidtehN eS Fw itihn depejnuddegnmoten gn rtasfn ottrsh w eo rtkh,Ce o nference
Commiwtatser ee consatsit thuCeto emdm itontR eeev isoifMo ant hematical
Tabolfet shM ea theimcDasit viosfti hoNenat ioRneasle Caorucnhc Tihlia.sf ,t er
somceh anogfme esme brsbheicpa,tm hCeeo mmiwthtieicessh i gntihFniogsr eword.
Thper esveonltu meev iidtseh nCacoten ferceasnnoc meest riemaeccsoh n clusions
antdh tahte rierc ommensdoamteitgoienamtsce sto end.
v
/
VI FOREWORD
Actiwvoer kw ass tartaettd h eB ureaiun1 956T.h eo verapllla nt,h es election
ofa uthofrosrt hev ariocuhsa ptearnsd,t hee nthusiraesqmu irteobd e gitnh et ask
werec ontribuotfiD orn.sA bramowitSzi.n chei su nitmeldye atht,h ee fforhta s
continuuendd etrh eg enerdailr ectoifoI nr enAe. S tegunT.h ew orkeartst he
Bureau atnhdem emberosf tChoem mittheaev eh adm anyd iscussaiboonust
contenstt,y laen dl ayoutT.h ougmha nyd etahialvshe a dt ob ea rgueodu ta st hey
cameu p,t heb asiscp ecificaotfit ohnevs o lumhea ver emaintehdes amea sw ere
outlinbeyt dh Mea ssahcusetItnss titoufTt eec hnolCoognyf ereonfc1 e9 54.
TheC ommittweies hheesr teo r egisittecsro mmendatoifto hne m agnituadned
qualiotfty h et ask caorurtib eytd h es taofff t heN BS ComputiSnegc tiaonndt heir
expecrotl laboriantp olrasn nicnogl,l ecatnidne gd ititnhge sTea bleasn,d i tasp pre­
ciatiooftn h ew illingwnietswhsh iciht vsa riosuusg gestwieorniesn corporiantteod
thep lansW.e hopteh irse sultvionlgu mwei lblej udgebdyi tuss ertsob ea w orthy
memorialt hev isiaonnd i ndustorfyi tsc hife architMeicltt,o Anb ramowitz.
to
We regrheetd idn otl ivteos eeit psu blication.
P.M. MoRsE,
Chairman.
A.E RDELYI
M. GRAY
c.
METROPOLIS
N.c.
J.B .R ossER
H.C .T HACHEJRr,.
JoHNT onn
B.T oMPKINs
·c.
J. TUKEY.
w.
HandboookfM athemiactaFlu nctions
with
FormulGarsa,p hasn,d M atheamticTaalb les
EditbyeM di lton AbarnaIdmr oeAwn.Sie tt ezg un
I. Introduction
Thep resHeanntd bhoaobske edneg snied ttioo niam!p ortMaanncyne u.m erical examples
provsicdiee nitnivfiecs twiigtahtc oorams p raer­gei vteoin l lustthuresa oetft e h tea balneds
hensainsvdee l f-csounmtmaaoirftny hem eda thea­ltshoce o mputoafft uinocnvt ailoune sl iweh ich
matifcuanlc ttihoaanrtsi i snpe h ysaincedan lg io­utstihdereia rn gAett. h een d thotefe xitn
neerpirnogb lTehmesw .e ll-kTnaobwlonef s e ach cthhaeiprsates e hro britb lioggirvaipnhgy
FunctbiyEo .nJ sa hnakneFd . E mdhea bse enb ooaknspd a pienrws h ipcrho oofft shm ea the­
invaltuowa obrlkeie ntr hse fiseeli dnis tm sa ny matipcraolp esrttaiitenets dh c eh apmtaeybr e
editidounrsit1nh gep ashta lfr-cye.Tn hteu founAdl.sl oi sttehdbe i bliogarrtaehp eh ies
presveonltue mxet etnhwdeos r okft heasuet homrosr iem porntuina mnetr tiacballCe osm.p rehen­
by givinegx temnoasrniedv em aocrceu rastielv ies otfts a balregesi vientn h Ien dmeexn ­
numertiacbaallne ds ,g biyvl ianrgcg oeelrcl tiotniso naebdo vaen,cd urrent ionnfn oerwm ation
of hmeamtatpircoaple orftt iheets a bulattaebdil set sob ef ouinndt hNe t ai oRneas!e arch
functionnusm.b oeffTr uh nec tcioovnesr edC ohuansqc uialr tMeartlhye moaftC iocmsp utation
albseoe inn cr.e ased (formMeartlhye maTtaibclaaenlsOd t hAeird s
Thcel afiscsaitoifo nf unctions atnoCd o moprugtaantiizoant)i.o n
Thmea themnaottiactuaisloei ndns Htahnids­
of cthhaep itnet rhsiH sa ndbiosso ikm ilar to
booakr teh ocsoem moandloyp itnes dt andard
thaotfA nI ndoefxM athemaTtaibclbaeyls
texptasr,t icHuilgahTrerlray n sceFnudnecn­tal
A.F letJc.hC e.Pr .M, i llaenrLd., R oseenahd.2
tioVnosl,u 1m3e-,bs y A .E rdelyi, W. Magnus,
Ing enetrhacelh ,a pctoenrtnsau imne rtiacballe s,
F. OberhaentdFt .i TnGrg.ie cr(o MmciG raw­
grapphosl,y noomrri aatli aopnparlo ximHaitli1lo93,n5- s5. 5 S)omael taetrinnvoet athiaovnes
foaru tomcaotmipcu atnesdrt sa,t eomfet nhtes a lbseoel ni stedi.n trTohdoeunf ce tswiy mobno ls
princmiaptahle maticaolft hptera obpueh­ratbsie eeksne ptto mian imaunmad,n e ffohrats
ltaefdu nctpiaorntsi,ct uhlooasfrecl oym putbae-emna dteo atvhoueis odefc onflincottiantgi on.
2. Accuraocfyt heT ables
Thneu mboefsr i gnififigcuagrnietvsi e nen a chp reciisn iotinhn etierr pmoalyoa btteatsih ne m
tabhladese pentdose odme ex toennt thn eu mbebry u soef h igheri-notredreprpo rloacteidounr es.
avaiilnea xbilsett aibnugl aTthieohrnaebss e. e nd escrbiebleodw .
noa ttetmomp atk ietu niftohrrmo ugthhoeu tI nc erttaaibmnla ensy -fifguunrcevtdai loune s
Handbwohoikwc,oh u hladv bee enc osaat nldy argei vaetin r reignutleairrnv t ahalers g eunmt.
laboruinoduesr .t akmIionns gtta baltel se asAtn e xamippslr eo vbiydT eadb9 l4.e.T hpeu r­
fivsegi nifificgaunrthe asv e pbreoevnia dnedd p,o soeft hetsaebi ltseo fs u rn"iksevhya lufeosr"
thtea builnatre hravvagele sn erbaeleclnhy o setnhc eh ecokpfir nogg froaarmu st omcaotmipcu ters;
toe nstuhrlaeit n ienatre rpwoilylailte filoodun.r n­oq uesotfii notne rpaorli.as teison
orfi ve-fiagcucruer wahciysc,uh ffi cienms o st Thmel txiemnudm- fiegrurrooer"r t,o lerance"
physiacpapll icUasteirrosen qsu.i hriignhgei rnt htea bilnet sh is Hains%do obfo 1 ou kn it
everywhtehrceea soefit nh eel emefnutnacr­y
Them ostr ecentth,e s ixtwhi,t hF .Lo escha ddedas co-authwoars, tioanns1du, n iintt h cea osfte h hei gfhuenrc tions
putb lishIend1 96b0y McGraw-HiUl.lS,. Aa.,n dT eubnerG,e rmany. excienapf te cwa swehsei rhtea bse epne rmitted
Thes econedd itiwoint,h L .J .C omriaed deda sc o-authwoarsp, u blished
In•t wov olumeIsn 1 962b y Addison-WesUl.eSy.,A .a,n dS cientiCfiocm ­ tor itso2eu nits.
putinSge rviLcted .G,r eaBtr itain.
IX
X INTRODUCTION
3. AuxiliFauryn ctiaonndsA rguments
Oneo ft heo bjecotfts h iHsa ndbooiks pro­ Thel ogaritshimnigcu laprirteyc luddierse icntt er·
to
vidtea bloersc omputimnegt hodwsh icehn ablep olatinoena rx =O. Thef unctiEoin(s-x I)n
:t
theu setro e valuattheet abulated fuonvcetri aonnds x-1[E-il(nx )x -'Y],h owevera,r e well­
complertaen goefsr eavla luoefst hepiarr ameterbse.h aveadn dr eadiilnyt erpoliantb hliers e gion.
Ino rder achietvhei osb jecftr,e queunsteh as Eithewril ld oa sa na uxilifaurnyc titohnel; a tter
to
beenm adeo fa uxilifauryn ctionrse movteh e wasi nf acste lecatesid t y ielsdlsi ghhtilgyh er
to
infinite opfat rhteo riginal fuantct thieoinrs a ccurawchye nE i(xi)sr ecoverTehde.f unction
singularaintdi easu,x iliary taocr ogpuwmeie tnht sx -1[Ei(-xln) x-'Y]h asb eent abulatteodn ine
infinirtaen gesA.n examplwilel maket hep ro­ decimaflosrt her ange0� x�;! . Forl � x� 2,
ceducrlee ar.
Ei(xi)ss ufficienwtellly- behatvoea dd mit direct
Thee xponenitnitaelg orfap lo sitairvgeu ment
tabulatbiuotnf ,o r largero fvx a,il tuesex sp o­
isg ivebny
nenticahla racptreerd ominatAe ssm.o othaenrd
fx
e" morer eadiilnyt erpolfaubnlcet ifoonrJ argxe i s
Ei(x)==-d u ze-xEit(hzi)hs;a sb eent abulaftoerd2� z� 10.
-coU
Finalltyh,er ang1e0s zsoo isc overbeydu seo f
thei nverasreg umexnt- 1T•we nty-oennet rioefs
xe-"'Eic(oxrr)e,s pondizn-g1 . (1- .005s)u0f,­
to
ficet op roduacnei nterpotlaabbllee.
4. Interpolation
Thet ablienst hiHsa ndbooakr en otp rovided Letu ss upposteh awte wisht oc omputteh e
witdhi fferenocroe tsh eari dtso i nterpolbaet­ionv,a luoef x e:eEf1o(rxx =)7. 95f2ro7m t hitsa ble.
causietw asf eltth atth es pacteh erye qiuerc ould We descriinbt eurn thea pplicaotfti hoemn e thods
beb etteemrp loybeydt het abulatoifao dnd itionaolfl ineianrt erpolLaatgiroann,ag ned Ai tkena,n d
functioAndsm.i tteadildycs o ulhda vbee egni ven ofa lternamteitvheo dbsa seodn d ifferenacnesd
withocuotn sumienxgt ra sbpyai cnec reatsihneg Taylors'esr ies.
intervoaftl asb ulatbiuottn ,h is wouldc ohna­ve (1L)i near interpTohleaf toiromnu.fl oart his
flictewdi tthh er equiremtehnaltti near interpproolcaei­ssgs i vebny
tioinsa ccurattofe o uro rfi vefi gures.
Fora pplicatiinow nhsi clhi neianrt erpolation
isi nsufficienatclcyu raittei si ntendtehda t
Lagrangef'osr muolraA itkenm'est hoodf i tera­wherje0 j, arec onsecuttaibvuel vaarl ueosft he
1
tivlei neianrt erpolabteiu osne d.T o helpt he functicoonrr,e spondtiona gr gumenXtos, re­
3 x11
usetrh,e riesa s tatemaetnt th ef oootfm ostta bless pectivpeilsty h;eg ivefnr actoifot nh ea rgument
oft hem a:hlmum erroirn a lineianrt erpolaitnet,e rval
andt.h en umbero ff unctivoanl uense edeidn p= (x-zo)/(zl-Xo)
Lagrangfeo'rsm uolraA itkenm'est hod inter­
to
polattoef ultla bulaacrc uracy. andf pt her equiriendt erpolIantt eh.e p resent
ane xamplceo,n sidtehref ollowienxgt racti nstanwceeh ,a ve
As
fromT abl5e. 1.
}o=. 89717 430.28 9872131 3 p=.527
7:1: 5. .8x 9e2�76E88t5 (4x8 )X.0 .8x 9e8t•7x2E(1)3 1 3
7.6 .8 9368341 28 1. . 897982878 Them ostc onveniweanytt oe valuattheef ormula
7.7 .8 9499676 68 .2 .9 0072390 6 ona deskc alculatmiancgh iniest .o s efto a ndf t
7.8 .8 9680783 78 .3 . 906102'>93 int uronn t hek eyboaarndd, caorruytt hem ulti­
7.9 .8 9741370 28 .4 .9 0242679 5 plicatbiyo1 n-s pa nd/c umulativae playr;t ial
checiks t henp rovidbey them ultipldiiearl
reading unWiet yo.b tain
f.=m (1-.527)(.483907+21. )75 27(.78191832)3
Then umberisn t hes quarber ackemtesa nt hat
= .89747430 3.
them aximume rrori na lineianrt erpoilsa te
3 1X0 -a6n,dt hatto i nterpotlota htefe ultla bular
accuracfiyv ep oinmtuss tb eu sedi nL agrange'sS incieti skn own thatth eriesa possiebrrloer
andA itkemne'tsh ods. of3 X 1 0-i6nt hel inefaorr mulwae,r ounodff t his
resutlot. 8977T3he.m aximump ossiebrlreoi rn
o1uA tt. hC eu. sA eio ttdk ie.tfOnen,r I enncestP,er rpooeEl.da ibtnyiIb otuner Mrgaahtt ohifSoc.po r.no 3p,o rt5&-ip71o93a6nr2a t)wils(t. ,h · thiasn sweirsc omposoefdt hee rrorc ommitted
INTRODUCTION
XI
byt hlea rsotu ndtihnaigts., , 4XI4 5,0p0l -u3sT hneu mbientrs h teh iarndfd o ucrotlhu amrnes
30X-,aI6n d cesrot caainnneloxytc .eXe80d-I •5 thfier satn sde codnidff eroeftn hcveeas l oufe s
(L2ag)r'a fsno gremuIlnta h.ei xsa mplex,e "1(tE xh()e s beeel ;o wt)shmea llonfte hssese cond
relefvoarnmituts lh ,ae'} -poongiein,vt be yn differpernocveic dheeoscn tk ah t eh rienet erpola­
tionTsh.er equviarleiudsne o wo btaibnye d
f=-A2(f-p2+)-At(f- pt+A) oJo+(i(Afppt)) linienatre rpolation:
+2(Af2p )
Taboltfeh sce o efficAi(eknaptrgsei) vi ecnnh apter
fp=3(..8+97(.7.708239 779977)45 7)
2f5o�r h rea npge= .OOlW(e)e vlal.tuha et e
formfuolpra . =.5a52n.d3, i5 nt u4rAn g.a in, =8.9797.127 3
ine acehv aluwaea� cicounm tuhlAea( tkiepnt h)e
multipliseirn tcrheees giuirmis sut neirtW ye.
nowh avtehf eo lloswuibntga ble. Inc ases wchoerrroeer cdottefth r heL e a grange
polyniosnm oiktan lo wonn,oe ft hper eliminary
X xe>E), (x interpomlaayht aivtoeonb sep erfowrimtehd
7.952. 89797725 7 polynoomfti waoolr ms o rdei fferenatsa orders
10622 cheocntk h eaidre q.u acy
7.953.8 977043 79 -2 (A3it)kme ent'hosoif dt erlaitniievnaetr e rpola­
10620
7.954. 89707959 9 tioTnh.se c hefmocera rroyuittnh gpi rso cess
int hper eseexnatmi pasls fe o llows:
n x,. y,.=uzE,x)( Y.o.. Yot.. .. Yo1...2.. Yo1..2,.. s. x-,.x
8.0 .8 982731 13 .0 473
01 7.9 . 819747 302.8 977434 034 -.0 527
2 8.1 .8 992778 88.8 9747 4 826.48 977731 499 .1 473
3 7.8 .8 960887 37 90220 2394. 897717933 8 -.1 527
4 8.2 .9 002793 06 4� 98773 1216 16 8977731 930 .2 473
5 7.7 .8 949976 66 2 35221 2706 43 30 -.2 527
Here Xo fo
8ftt2
Yo... = 1 'Yo XoX-� Xt 82ft
x,.-xoy ,. x-,.x It 8fat2 88/a/2
X2 fz tJ2fz 8"/2
1 !Yot. 8fst2 88/s2/
Yot..ro= 2
y X a fa 8fa
�"" "- � ....1. O,n 8/7t2
1 !Yo.t. m-l.mX -mX x-.
Yo...t. .m.- t.m.n=Y · · ·• m-l,n.x ,.-xI Here "'
�" "" -� .c,"' 0,1•, • ,•
Ift hqeu antitaineXdsm -Xa rues eads 8ftt2=f8,f-afto2,= f2-fh
multiwphleifneo:z: ,.rr-s:z:m t ihncegr ososd-uopcnrta 1'8=t 8f· s8· tf·2= t-' /21· -22 ftf+o
desmkac htihneaeic,rc umu(lxa,t.-ixo)n- (xm-x) 88f321=82 /2-82/=tf a-3f23+f- tf o
int hmeu ltipliietsrh deri evgitisobso eutr se erd
84=/2 88fs-t8233//=2 h -4fa+ 6fa-4f+t fo
att hastt agAene. x tdreac ipmlaailcsu es ualalnyd o ns.o
carriinte hdie n termiendtieartpteoo l astaefsIe nt­ hper eseexnatmt phlreee lepvaaornfttt h e
guard aacgucamiunlsattr ioounn edorirfrns og. differteanbiclasees f olltohdweis ff,e rbeenicnegs
Thoer dienr wthhiteca hb uvlaalru es awrrei tuitsnuee nndi o tfts h lea st dpelcaoicftme ha el
isi mmatteosr oimaeelx tebnuttto,a chievef utnhceat sii ocsnu ,s tomTahresym .a llonfte hses
maximruamto efc onveragneadnt tc hese a meh igdhi fferpernocveaisc d heesoc ntk h e function
time miancicmuimzuoelf ar toiuonnde irnr,go rvsalues
web egaisn ,t ihnie sx amwpliett,hh tea bular X xe•t(EX) 82f B"f
arugmennte arteots htge i vaernug mentth,e n
7.9 8.9 71473 02 -22 754 -34
taktehn ee aroefts htre e maitnaibnuaglr agru ­
8.0 8.9 827313 1 -2 2036 -39
mentasn,sd o o n.
Then umrb oeft abuvlaalrur eesq uitroe d
Applyfionergx ,a mEpvleeetr,t i'nst erpolation
achiae gviev perne ciesmeirognnea st urianl floyr mula
thceo uorfts heie t eraTthiuoisnnt s hp.er esent
examspivlxae l ueuss weeedvr,ete nh ouigwtha s f,=(-1p)foE+2(p)82foE+-.(pf)oa+."
knowinna dvatnhcafietv w eo usludffi cTeh.e +PitF+2 CP· W+· f Ft, (pf)ta+"
extrroacw o nfitrhmceso nveragnepdnr coev ides
av alucahbelcek . antda ktihnnegu merviaclaoulfet shi en terpola­
(D4iff)er feonurclmea Wse.u steh cee ntrtailoc no efficEi2e(npEt),s,p() ,F 2(apnd)F () ,p
differneontcaet ion2 5,() chapter froTma b2l1e5 w,efi. n tdii.a t
XII INTRODUCTION
109fs.2'1=. 473(849370127.) 0 6119267(5-24 .)012(34)c anb eu sedW.e firscto mpuatsem anyo ft he
+
+ .52(87982731 1+3 ).0 6343290(326 .)0 12(39d)e rivaptni>v eass a res ignifiacnadnt th,e n
(Xo)
=8977731 9.3 evaluatthee s erifeosrt heg ivevna luoefx .
Ana dviscahbelcoekn t hceo mputveadl uoefts h e
We may notiicnep assitnhga tE verett's
derivaitsit vore esp rodtuhceae d jacteanbtu lar
formulsah owtsh atth eer rionar l ineianrte rpolate
valubeyes v aluatthisene gr fioexrs= -x1a ndx 1•
isa pproximately
Int hper eseexnatm pwleeh ,a ve
E2(PW!oF+2( p)I!J'"2t"'/l£ Ez+( Fp2)( ]p[2/)lio 11+2 /tl
f(x)
Sintcheme a ximuvma loufeI E2(p2())p+ilFn t he (x=)x(e +1•x B-t1)(-fX1() x )
ranOge< p<iI%s ,t hmea ximuemr rionar l inear ff'"( x==) (xl)f()
interpiosal paptreo ximately f"'(=x 1l()+ + x--l')xf--"x2(x-x-2)2f f('x()x )
+2z-3f(x).
1 1 WitXQh= 7.a9n xd -0x=.052o7urc omputations
16ii!J2/o+I!J2tfhattls ,,1i 6 1/-zf-tfo+-ftl· araes f olloawnes x;t dreac imhala sb eerne tained
int hvea luoefts h tee rimnst hsee riessa feguard
(5T)a ylosre'rsi Iensc .a swehse rteh seu ccesa­gaiancsctu muloaftr ioounn deirntorg o rs.
sivdee rivaotfit vheets a bulaftuendc tciaonbn e
computfeadi rly Teaayslioelrxy'p,sa nsion k /(kx>o()/k! kfCkl(!x o)fk
(x-:no)
0 .89741370 2 .89741370 2
f('xu ) (x)o 1 .01007646 9 .00065063 33
(x-x)oJ: "!+( x-x!"0 )Z 2 -.00171632 1 -.00030105 59
f(x)=f(xo)-1- �
3 .00011928 7 .00000001 97
aflll)( xo
-1-(x-o)x +
3! .89777139 4
5. InverIsnet erpolation
Witlhi neiarn terpotlhaetiriseno ond ifferenTchee d esiriestd h erefore
x
inp rincbieptlweed einr eacntdi nveirnstee rpola­
tionI.nc a sewsh erteh lei nefaorr mulpar ovides =8.1. 70835=78(..117)05 873
X=:to-1-P(Xt-1--Xa)
ani nufsficienatclcyu raantsew tewro,m ethoadrse
availabWlee m.a yi nterpodliarteec ftolry , Toe stimatpeo stshieebr lreio nrt hiasn swer,
exampblyeL ,a granfgoerm'uslt aop repaanr eew wer ectahlaltt h mea ximuemr roofrd irelcitn ear
tabaltea fi nei nterivnta hlne e ighboorfht ohoed i nterpoilnat thiitosan b liesllf =3X1 �0. An
approxivmalauteea, n dt hena pplayc curataep proxivmaaltufeeo dr j/idsx trhaeto ifot he
invelrisnee ianrt erpolatthieso unb tabulafirtsedtdi fferetnotc heae r gumeinntt er(vcahla pter
values. Altewrnetoa m taiyvu esleAy i,t ken'25s) i,n t hicsa s.e10 .0 Hencteh mea ximum error
methoodr e venp ossitbhleyT aylosre'rsi eisnx iasp proxim3aX1t-e06l(/y.1 )00t,h aits. ,0 003.
methowdi,tt hh reo loefsf unctainodan r gument( iSiu)b tabulmaettihoondT .o improtvhee
interchanged. approxivmaaltuoeef jusotb tainweeid n,t er­
x
Iti si mporttaorn eta ltihzaett h ea ccuroafc yp oladtiree ctfolpry= 7.0.,17 a nd. 7w2i tthh aei d
ani nveirnstee rpmoalyab teve e rdyi fferent offLr aogmr an5g-ep'osfi onrtm ula,
thaotfa direicntt erpoTlhaitisesp . a rticularly
xezEt(x)
truien r egiownhse rteh ef unctiisos nl owly X � fj2
8.1 70 .8 999396 83
varyifnogerx,a mplnee,a ar maximourmm i ni­
1 0151
mum.T hem aximump reciastitoani nianab nl e 8.1 71 .9 000308 34 -2
inveirnstee rpcoalbnae et set imwaittehd atihde 1 0149
thfeo rmula of 8.1 72 .9 000319 83
dj
��f).jj Inverlsien eianrt erpoilnat thieno enw t able
dx gives
.9-.893969893
in whicllhji st hmea ximupmo ssiebrlrieon rt he p .6223
.00000115 1
functviaolnu es.
ExamplGei.v exn� E1(:efi)n dx f rotmh e Hencx=e8 .7106223 .
tabolnep age
X.
(iI)n verlsien eianrt erpolTahteif oonr.m ula estimoaftt heme a ximuemr riontr h irse sult
An
fopri s is
p=
(jp-fo)/(jl-Jo).
Int hper eseexn1:1tm pwleeh ,a ve
.-9.8992778 88 722 112 (iiAii)t kemne'tsh oTd.h iiscs a rroiueitdnt he
708357
p .90072390 6-.8979828781 019 418· . sammea nnearsi nd ireicntt erpolation.
INTRODUCTION
xm
n y=,.xe•1(xE) x,. y-,.y
Xo,n Xo.n, l Xo1,,2,n Xo,,2l, 3,n
0 '9070320986 2. .0 0072390 6
1 '8979828788 . 81.07 8537 12 -.0 0027121 2
1
2 .9 01620938 33. 81.07 21305 58 1.7 6019 521 .0 0162093 3
3 . 87918128303. 8 1.7 181034 3 25 94881. 7 026224 4 -.0 0127868 7
4 . 94062928547. 81.96 9924 37 17 335 41851. 7 026321 8.0 0242679 5
5 .8 97413707 2. 81.917 4043 82 28 142 231 265- .0 0258G2f! 8
Thee stimattheme a oxfi meurmr ionr thdiiss creipnat nhcheyi ghest initnte hripso lates,
resiustl htse a maesi nt he tsaubbulmaettihoond c.a se and
Ani ndicoaftt hieeor nr iosarl spor ovibdyte hdeI Xo ,2,ta,,4, Xo,t,3,2s,.
6.B ivariate Interpolation
Bivariinatteer piogsle anteiromanol slty simipnltpyoe lratiisto hnec na rroiuetd tihnse e cond
perfoarsm esdae queonfuc nei variate diinrteecrtpioolna.­
tionWse. c aroruy tt hei ntetripooinlno a ne Ana lterpnraotcievidentu hrceea sef uonfc tions
dirieocbnty,o noeft hmeet hoadlsr eadye dd,e osfca cr oimbpvlaerxi iastb olu est eh Tea ylsoerr'ise s
fosre vetraabluv laalruo efts h see coanrdg umenetx panspiroonv,it dheasdtu ccedsesriivvea tives
int hen eighboorfh ogoiidtv sev na luTeh.e oft hfeu nctciaobnn ec ompuwtietdh mouucth
interpaorlead tieffse reansc ecdah eck, dainffidc ulty.
7. GeneratoifFo unn ctiforonms Recurrence Relations
Manyo ft hsep elcm iaathaetmifcuanl ctio(nitsih ide)i recitnwi hointc hhre e curirsbe enicneg
which doenp peaarn adm etcearlt,lh eedii nrd exa,p pliEexda.m palreaessf owlslo.
ordoerrd egrseaet,i sfy dia ffelrieenqneucaaer­
n
Stab-iilnictryeasing
tio(nor re currreelnactewi iotnh) rteost pheicst
P,.,( 7:xx)P()
parametEexra.m palrefesu rnibsyht ehdLe e ­
gendfruen ctPni(oxnt) h,Be e ssfeuln ctJin(oxn) Q(,.x,)Q (':x)x <1()
andt heex peonntiinatle Egnr(axfl)o ,wr h icwhe Y(,.x)K,,.x )(
havteh ree spievrceet currreelnactei ons J- nx-)I'A,- (n (x-)'A
E,.x()(< nx)
(+n1 ),.P-+(l2ln)+x P,.=+0n P,.-1
2n n
J,.I-+J,.+1J=,.0- Stability-decreasing
-:;
P(,.x,)P :(x)(x <)l
nEI+,.x+E=,.-e�. Q,.x(,)Q (-;:x)
J,.x)(l,,.x( )
Particfuolarau rtloym waotr,irk ce currence re­
latipornosv aindi em poratnadpn otw ercfou­ml J+,.'Ax,()I +n'Ax()
putitnogo Ilf.t hvea luoefF'.s n (xo)rJ n(ax)r e E,.x)(( n>x)
knowfno trw oc onsecvuatliuovefes o rE n(x) F,.(.,,
(Coulwoamvbfe u nction)
is knfoowrn onoef vtahletunhef en u ,n ction may p)
bec ompuftoeordt rhv ena, luoefs b ys uccessiIvlel ustorfta htgeie onnesr oaftf iuonnc tfiroonms
applicoaftt hireoe nlsoa nt.Si inn gceen eraitsi tohner ierc urrreelnacateri geoi nvsie ntn h pee rtti nen
carroiuetpde rfowrictreho undveadl ues, cihta ptiesr s. It tihsae tv aeilnns c oa sessh own
vitakln othwoo we rromrasyb ep ropagiant ewdh erer etchuer prreonccieesus nss taibtml aey,
three curprorecnecseIs ft. h eer rdoorn so gtr ow stibleul s ewdh etnh set arvtailnugaer ksen own
relattoit vhese i zoeft hwea ntfeudn cttihoen t,os ufficiaecnctu racy.
prociesss asi tdob es tabIlfeh,.o wevtehre, Mentimounsa tl bseom adhee roefar efinement,
relaetrirvoegr rso awn dw ilelv teunalolvye r­duteo J .C .P .M iller, whair cehc uernraebnlcees
whehtnhw ea ntfeudn cttihpoern o,c ieussn ss tabplreo.c wehsisci hss tabflodere creastiobn eg
Iti si mporttoar neta ltihzasett abimlaiyt ya pplwiietdh aonukytn owloefds gtean r vtailnuge s
'
depeonn(d i t)hp ea rtiscoulluaotrfti hodeni fferf­olra rgeM ilsl aelrgtohrmiw,h icihsw ell­
enceeq uatbieoicnnmo gp ut;e( ditih)ve a luoefs s uitte.anod.u o tmatwiocr, ik sd escriinb ed
19.28,
oro thpearr ameitnte hrdesi ffereeqnucaet ion;
x Examp1l.e