Exercises in functional iteration: the function f(x) = ln(2-exp(-x))

Transcription

1 Eiss in funtional itation: th funtion f ln2-p- A slfstudy usin fomal powsis and opato-matis Gottfid Hlms updat Dfinition Th funtion onsidd is an ampl tan fom a pivat onvsation with D.Gisl and W.Jay. H I disuss poptis of fational itation of f ln2 p- f f f 0 f h+ f h f // manin th h'th itat f - ln2-p f // mioin aound th oiin A ouh ida an b ot by th followin plot. Th funtion has al an only fo ln2< ; this is th iht d uv in th followin. Fo -> it appoimats th onstant ln2. Fo <-ln2 th funtion has th imainay omponnt Pi*I n lins and its al pat appoahs th funtion f fo ->- lft d uv Baus th is th fipoint f00 w an onstut fomal powsis fo abitay ontinuous itats and baus f'< fo >0 it is an attatin fipoint and itation to positiv hihts is a vy wll onvin poss. H th offiints h fo th fomal powsis of a tain itation-hiht h an b tan by a st of polynomials in h whih shall b dtmind lat: f h h * + 2h * 2 + 3h * 3 + W shall find that ah offiint is a polynomial in th h-paamt whih an b pliitly b dtmind without nd of usion

2 Itation of fln2-p- S Pod: 2.. Constut th mati-opato fo th funtion fln2 p- Th powsis fo this an b ivn in ational offiints: f /2* 4 + 5/4* 5-54/360* /20* 7 + O 8 Th assoiatd mati-opato M of init siz bins with M W s that th offiints fo f 0 a in th fist olumn olumnind 0 that fo f in th sond olumn that fo f 2 in th thid olumn 2 and so on Th us of th mati-opato fo pssion of th quid powsis With a "Vandmond"-li vto-typ V ow 2 3 w an wit V * M Vf owvto f f 2 f 3 whih immdiatly allows nalization fo itats of f. Lt's wit th h'th itat f h. Thn w hav fist fo int hihts: V * M h V f h wh int pows of M an atly b dtmind by mati-mulipliation of th tianula mati with itslf. This allows still at ational aithmti up to abitay tunation siz. Th qustion is now: an also fational itats b dtmind. Th answ is ys; and this is a nown podu. Eiss in funtional itation Mathmatial Miniatus

3 Itation of fln2-p- S faional pows of a mati-opato via lo/p Sin f is a funtion havin f0 0 f'0 w an dtmin a matiloaithm whih still povids at aithmti: Thn dfin M M ID wh ID is th idntity-mati LoM M M 2 /2 + M 3 /3- + th mato sis fo lo L LoM M h Ep h * L povids th h'th-pow of M aain in ational aithmti. Th loaithm-mati L has an intstin fomat: L Th loaithm of an opato-mati has always th stutu that all olumns a smply shiftd multipls of th sond olumn; w s that th offiints L in som ol and ow a just *L +-.. Thus th mati L is not a opatomati! And th sond olumn ivs th offiints fo a funtion whih w may all th "itativ-loaithm-funtion of f" as a funtion w t: whih is also lf /2! + 4 / 4! + 6 / 6! +. lf -2 osh 2 p + p- A fatoially sald vson of mati L is d F * L * d F - FLf and bins with If w want to omput som nal pow h of th mati M w hav to valuat th ponntial: I'v sn th tm "shliht"-funtion fo this in old litatu man fo "simpl funtion" Eiss in funtional itation Mathmatial Miniatus

4 Itation of fln2-p- S. -4- M h Ep h * Lo M 2.4. Th half-pow of M usin h/2 iv th half-itat of f W an do this fo som atual valu of h say h/2 to t th powsis fo th half-itat. Th top lft smnt of th mati is M /2 Ep /2 * L and in th sond olumn w find th offiints of th fomal powsis fo f 0.5 : suh that f /2* 2 + /4* 3 - /6* 4 + /8* 5-37/440* 6 + 7/960* 7 + O 8 V * M 0.5 V f 0.5 o with a bat-[owol]-notation fo th tation of th sond olumn of mati M 0.5 w wit f 0.5 V * M 0.5 [ ] //omittin th ow-ind mans th whol olumn 2.5. Th nal pow of M and th nal itat of f; symbolially If w want this in mo nality pin th itation-hiht paamt h as vaiabl w an omput th mati-ponntial symbolially ttin th followin polynomials in h as ntis of th h'th pow M h : M h wh th sond olumn M h [] only is ndd to povid th lvant polynomials fo th omputation of f h. If w inst fo instan h w t th oiinal powsis fo f if w inst h/2 w t th offiints of th powsis fo th half-itat f 0.5 and so foth. Eiss in funtional itation Mathmatial Miniatus

5 Itation of fln2-p- S Th bivaiat offiints-mati POLY If th offiints of h in that sond olumn a aain psntd as a mati w an wit this as mati POLY of offiints fo th bivaiat funtion f h V * POLY * Vh~ POLY So POLY * Vh~ ivs th sond olumn in M h and as a bivaiat pssion in th mati-notation w hav: f h V * POLY * Vh~ * + 2 * *h + 3 * *h * /2*h *h * /4*h 2 + *h * + Eiss in funtional itation Mathmatial Miniatus

6 Itation of fln2-p- S Epliit dsiptions of ntis in POLY It miht b of intst that POLY an b sald to povid int ntis only huistially no poof yt. This is possibl usin a fatoial similaity salin; h is th top-lft smnt assum d F as diaonal mati of fatoials dia[0!!2! ] d F *POLY * d F - FPf I sudd in findin a nal pssion fo th ntis in FPf and thus fo that in POLY. Assumin th mati-indis ow and ol binnin at zo w hav fo th lmnts in POLY: p! o with th binomial-offiint mo onvnintly adaptd: δ p! wh d is th Kon-symbol if if 0 > 0 W s that th p a finitly omposd polynomials whos numb of tms is just qual to th olumn-ind. So to dsib th powsis fo th h'th itat of f w wit: f h + p 2 *h 2 + p 32 * h p 4 * h + p 43 * h and baus w do no mo nd th mati-loaithm/mati-ponntial w an dtmin that offiints to abitaily many tms in at ational aithmti and thus th whol funtion with optimal pision. Eiss in funtional itation Mathmatial Miniatus

7 Itation of fln2-p- S. -7- Eiss in funtional itation Mathmatial Miniatus 2.8. Th funtion w h with fid as powsis in h It miht b of intst to fomulat th funtion to a fid paamt wh only th itation h is a vaiabl aumnt. This mans that w must intodu a family of funtions whih dpnd on and us th offiints of on olumn fo its taylosis. W shall thn valuat th funtions fist ivin th offiints fo th powsis of w h in tms of h whih is atually only a witin as w h f h wh w h 0 * h Th funtions a fist powsis in usin th ntis p alon th olumns of POLY: p Fist w an han od of summation baus ah p is a finitly omposd sum of tms: 0 0!!! Th inn sums an b fomulatd into losd foms as ponntials: This ivs at vn indd olumns 2j >0 : E sinh 2 and at odd indd olumns 2j+ : + O osh 2

8 Epssd as funtion in Pai/GP this is Itation of fln2-p- S. -8- \\ us-funtion POLY_olumn_sum if0tun; if % 2 _odd_vn \\ intnal funtions _vn 2/*sum-^*binomial*sinh* _odd 2/*sum-^*binomial*osh*- With this th funtion w hf h is w h 0 * h 2.9. Numial ampls At this ivs numially: w h *h *h *h *h *h *h *h *h 8 + Oh 9 Th offiints sm to inas but altnat in sin. Bfo futh analysis if I apply Eul-summation I find sults fo fational hihts in th unit-intval 0<h< hiht h w hf h Gottfid Hlms Eiss in funtional itation Mathmatial Miniatus