A stopping criterion for Richardson s extrapolation scheme. under finite digit arithmetic.

Transcription

1 A stoppg crtero for cardso s extrapoato sceme uder fte dgt artmetc MAKOO MUOFUSHI ad HIEKO NAGASAKA epartmet of Lbera Arts ad Sceces Poytecc Uversty Hasmotoda,Sagamara,Kaagawa JAPAN Abstract: We propose estmators of a roud off error cotaed a approxmato for cardso s extrapoato sceme uder fte dgt artmetc We aso propose a stoppg crtero, based o cosderato of te roud off error, for cardso s extrapoato sceme Usuay te error of a approxmato s evauated by a trucato error However, we ca accuratey estmate te beavor of ts error utzg bot trucato ad roud off errors uder fte dgt artmetc We empaszes tat te stoppg crtero proposed s depedet of toerace Key-Words: Numerca aayss;error aayss; Extrapoato sceme;oud off error 1 Itroducto Uder fte dgt artmetc, a approxmato cotas a accumuated roud off error ad a accumuated trucato error e accuracy of te approxmato depeds o tese errors We ow cardso s extrapoato sceme as a sceme tat reduces a accumuated trucato error[1 We we use ts sceme for te actua umerca cacuato, t ca be doe so tat ts accumuated trucato error may ot exst uder fte dgt artmetc s meas tat te sze of te accumuated trucato error smaer ta te mace epso by cardso s extrapoato sceme erefore, a accuracy of te approxmato depeds o oy roud off error We adopt suc stuato as stoppg crtero for cardso s extrapoato sceme 2 Premary Cosderato We cosder to te foowg ta vaue probem of ordary dffereta equato dy 1 y x = y, = fx, y x I dx,were I ad yx are a terva of defto, te accurate souto,respectvey A dscrete approxmato η x, depeds o te step sze caracterzed as foows for y x s ca be

2 I =, s were s s a assocated sequece for We cosder tat te foowg cardso's extrapoato sceme s apped to 1 2 = ,were s put wt η x, It s we ow tat s gve by te foowg expaso 3 = yx α 1-1 L - K erefore, te trucato error τ cuded by 4 τ α L K = We empaszes tat te frst term of τ -1 s emated f -1 ad te sceme are substtuted for Next, we cosder about te fte dgt artmetc It decdes to be carred out uder fte dgt artmetc we te sceme 2 s apped to te umerca cacuato of te practce e, we ca get te vaue wc vad dgts egt was decded as Here, te artmetc of te p foatg pot dgt s carred out uder fte dgt artmetc We troduce te foowg symbo to express uder fte dgt artmetc [ Furtermore, te ed dgt of cacuato s expressve ofε = p I geera, cude a accumuated roud off error fte dgts It ca be expressed as foows we ts quatty s sow wt te = [ erefore, [ tat appeared fte dgts s sow wt [ sows te roud off error taε s't cuded by ts vaue fte dgts Smaer quatty 3 e evauato of roud off error We cosder about a accumuated roud off error we te sceme 2 s apped s roud off error s formed by te cacuato of, ad determe of by te cacuato of te recursve sceme 2 We gve a estmato of for te foowg formua by usg te recursve sceme 2 = ,were sows te oca roud off error wc appear by te cacuato of te sceme 2If wt 5 s smaer ta, = Cosequety, we ca sow 6 were amout, = =, =,, -1 ca be sow wt s gve to t by te foowg - = = 1 - -

3 erefore,, - -1, - = > = - - e estmato of s gve to t wt, max I geera, f a assocated sequece fufs oeptz s codto[2, eac as foows max, =, s, From 6, we suggest te foowg formua to estmate 7 [3 s suggest as foows Uder fte dgt artmetc, t ca t get te accuracy of beyod te accuracy of [yx 4 A stoppg crtero of te extrapoato sceme Uder fte dgt artmetc, we we oo for approxmato wt te sceme 2, [, =, 1,Kwc as te reay same vaue appears We empoy ts vaue as a approxmato for 1 Evetuay, we t 8 [ = [ 1 = K = [, we adopt [ as a approxmato for te ta vaue probem 1 We cosder te stoppg crtero tat we propose Here, te foowg crtero s cosdered for te approxmato[4 9 < ε -1 We suppose tat satsfes te crtero 1 From te sceme 2, We ca coduct te foowg formua = α -1 L < ε - 1 K Here, f we suppose te foowg approxmate formua τ -1 τ α -1 L We ca get te foowg approxmate formua -1 τ < ε Hece, τ of s sow as foows τ < ε We ote tat te formua 11 s te assumpto tat te foowg assumpto s equa to It s deoted tat τ << τ -1 ca't be adopted as a approxmato we te assumpto of te formua 11 s't satsfed te crtero 1 Actuay, te probem 1 tat te assumpto of te formua 11 s't satsfed s te probem, wc as a d of sguarty We cotro terva I so tat step sze ca fuf 11 for suc probem Next, we cosder te roud off error for te crtero 1 by usg te sceme 2, -1 are sow as foows < ε 1 ad We uderstad te foowg fact from te recursve formua < ε

4 erefore, we obta te foowg cocuso by usg te recursve formua 5 If s satsfed te crtero 9, [ [ -1 5 Our Stoppg crtero e stoppg crtero tat we propose s te mode of te crtero 9 We cosder t about ts crtero 9 as a vaue uder fte dgt artmetc -1 be sow as foows [ uder fte dgt artmetc ca -1 [yx -1-1 = τ From te cosderato of te precedg paragrap, we -1 satsfes te crtero 9, τ -1 s ot preset te cacuated dgts erefore, t [ -1 [yx -1 = From te expaso 4, t τ < τ -1 ad [ = [ -1 We ca get te formua 8 from tese tgs Because, f a extrapoato sceme s apped, > τ > τ ad tey are ot cuded fte dgts > 1 [ O te oter ad, Next, we cosder te crtero 9 for te sceme 2 As for τ cuded te formua 11 aga If α < s supposed toward te coeffcet formua 11 ad -1 L - 1 fuy sma, t < ε - -1 α te s made erefore, t τ -1 ε f te formua 13 s fufed s ca aceve te above assumptos by mag terva I sma Bacward, ow I s made sma, estmate ca erefore, ca be sow as foows do t we coeffcets α of τ, are very bg [ [yx = e above sows tat f [ ad [ -1 satsfy te crtero 9, [ τ ad [ τ -1 are t cuded Ad, uder fte dgt -1 artmetc, [ - fte dgts [ I te formua 2, we crtero 9, t ca dcate [ = [-1 s ot cuded fuf te erefore, we ca get te foowg tree formuae [ [yx =, [-1 = [yx -1 vaue comparso wt te assumptos we te formua 13 does't aceve t I suc case, we regard te probem 1 as avg umerca sguarty As for cuded te foowg formua by te recursve formua 5 ad te expaso s deotes tat t < ε [ = [ -1 erefore, t was sow tat te sceme 2 satsfed te crtero 9 e approxmato fuf te stoppg crtero 8 wc we

5 proposed, does t ecompass a trucato error, ad t ecompasses oy a roud off error e accuracy of ts approxmato ca be estmated by te formua 7 6 Cocuso e cocusos obtaed are summarzed as foows: e proposed stoppg crtero does ot deped o toerace, ece te error cotaed te approxmato s equa to te estmate of te roud off error It s possbe to fd sguarty propertes of te probem 1, based o te ew stoppg crtero ad cotrog terva I As te error of approxmato s equvaet to accumuated roud off error obtaed by te ew stoppg crtero, we cofrm tat te reducto of te error of approxmato s due to Mφ er's agortm wc reduces roud off Order ad Stepsze Cotro Extrapoato Metods,Numer, Mat, Vo41, 1983, pp [5 MMUOFUSHI ad HNAGASAKA e reatosp betwee te roud-off error ad Mφ er's agortm extrapoato metod, Aas of Numerca Matematcs, Vo1,1994, error[5 eferece: [1 Bursc ad JStoer, Numerca reatmet of Ordary ffereta Equato by Extrapoato Metods, NumerMat,Vo8, 1966pp1-13 [2 CBEZINSKI, EXAPLOAION MEHOS teory ad practce,noh-hollan,1991 [3 MMUOFUSHI ad HNAGASAKA, O te ta stepsze a extrapoato agortm for IVP OE, Numerca gortms, Vo3,1992,pp [4 Peufard,