Compound Means and Fast Computation of Radicals

Transcription

1 ppled Mathematc Publhed Ole September 4 ScRe Compoud Mea ad Fat Computato of Radcal Ja Šute Departmet of Mathematc Faculty of Scece Uverty of Otrava Otrava Czech Republc Emal: aute@oucz Receved 8 Jue 4; reved ugut 4; accepted 6 ugut 4 Copyrght 4 by author ad Scetfc Reearch Publhg Ic Th wor lceed uder the Creatve Commo ttrbuto Iteratoal Lcee (CC BY btract I lat decade everal algorthm were developed for fat evaluato of ome elemetary fucto wth very large argumet for example for multplcato of mllo-dgt teger The preet paper troduce a ew fat teratve method for computg value x wth hgh accuracy for fxed ad x The method baed o compoud mea ad Padé approx- mato Keyword Compoud Mea Padé pproxmato Computato of Radcal Iterato Itroducto I lat decade everal algorthm were developed for fat evaluato of ome elemetary fucto wth very large argumet for example for multplcato of mllo-dgt teger The preet paper troduce a ew teratve method for computg value x wth hgh accuracy for fxed ad x The bet-ow method ued for computg radcal Newto method ued to olve the equato f t t x Newto method a geeral method for umercal oluto of equato ad for partcular choce of the equato t ca lead to ueful algorthm for example to algorthm for dvo of log umber Th method coverge quadratcally Houeholder [] foud a geeralzato of th method Let d be a parameter of the method Whe olvg the equato f ( t the terato covergg to the oluto are How to cte th paper: Šute J (4 Compoud Mea ad Fat Computato of Radcal ppled Mathematc

2 J Šute ( d ( a f a a ( d ( d ( a f The covergece ha order d The order of covergece ca be made arbtrarly large by the choce of d But for larger value of d t eceary to perform too may operato every tep ad the method get lower The method (34 preeted th paper volve compoud mea It proved that th method perform le operato ad fater tha the former method Defto fucto P : called mea f for every tu mea P called trct f ( tu Ptu ( tu m max t u m tu < Ptu < max tu mea P called cotuou f the fucto P cotuou ow cla of mea the power mea defed for p by for p we defe M p ( tu p p p t u : ; M tu : lm M tu tu p The mot ued power mea are the arthmetc mea M the geometrc mea G M ad the harmoc mea H M ll power mea are cotuou ad trct There a ow equalty betwee power mea α β M M ee eg [] From th oe drectly get the equalty betwee arthmetc mea ad geometrc mea For other clae of mea ee eg [3] Tag two mea oe ca obta aother mea by compog them by the followg procedure Defto Let PQ be two mea Gve two potve umber t u put p α a : t b : u a : P a b b : Q a b If thee two equece coverge to a commo lmt the th lmt deoted by P Qtu : lm a lm b The fucto P Q : called compoud mea The bet ow applcato of compoud mea Gau arthmetc-geometrc mea [4] π β π Gtu dθ t co θ u θ Iterato of the compoud mea the gve a fat umercal algorthm for computato of the ellptc tegral ( Matow [5] proved the followg theorem o extece of compoud mea Theorem Let P Q be cotuou mea uch that at leat oe of them trct The the compoud mea P Q ext ad cotuou ( ( 494

3 J Šute Properte We call a mea P homogeeou f for every ctu cp ( t u P ct cu ll power mea are homogeeou If two mea are homogeeou the ther compoud mea alo homogeeou Homogeeou mea P ca be repreeted by t trace Coverely every fucto p : repreet homogeeou mea p t P t wth property ( t p( t ( t m max ( u P( t u tp t Theorem If the compoud mea P Q ext the t atfe the fuctoal equato ( P Q Ptu Qtu P Qtu ( O the other had there oly oe mea R atfyg the fuctoal equato ( R Ptu Qtu Rtu Eay proof of thee fact ca be foud [5] t u tu Example Tae the arthmetcal mea tu ( ad the harmoc mea H( tu t u The arth- metc-harmoc mea H ext by Theorem ad Theorem mple that H G Hece the terato of the arthmetc-harmoc mea H ( x ca be ued a a fat umercal method of computato of G x x Th lead to a well ow Babyloa method wth a quadratcal covergece to lm a x x a a a a The Babyloa method fact Newto method ued to olve the equato method to olve the equato t x lead to terato wth a quadratcal covergece to lm a 3 Our Method a a ( a a x x t x Ug Newto I the preet paper we wll proceed mlarly a Example For a fxed teger ad a potve real umber x we wll fd a equece of approxmato covergg fat to x We eed the followg lemma Lemma Let fucto p atfy ( p for w ad let ( w p be bouded Let t q( t ume that p ad q atfy ( trctly f t The the fucto q atfe p ( t ( ( ( w q p for w ad q bouded Let P ad Q be the homogeeou mea correpodg to trace p ad q repectvely The the compoud mea P Q ext ad t covergece ha (3 495

4 J Šute order w Proof The aumpto mple that The q t ( p t t O t t p w w ( t t O( t t w t O t t O t w ( w hece q p for w ad q bouded The compoud mea P Q ext by Theorem Let a ad b a order of covergece put δ : The a w ad b a O( δ O( b a t b be the terato of P Q To fd the ( p a p a a δ a δ a δ O δ w w ( ( ( ( q b p a a δ a δ a δ O δ w w ( ( ( w R tu : t u Th mea trct cotuou ad homogeeou ad t ha the property R x x We wll cotruct two mea P Q uch that R P Q p t be the Padé approxmato of the fucto aroud t Tae the mea ( Let ad let et ft p t t O t t of order [ ] The exact formula for e f wll be derved Lemma 5 I Lemma we wll prove that p atfe ( t p( t ( t for every t Hece t a trace of a trct homogeeou mea Relato R ad t trace P Q ad Theorem mply that (3 m < < max (3 u e u t P( t u tp t t u f t ( Qtu ( tu t u (33 P t q( t Q( t p Iequalte (3 mply that Q a trct homogeeou mea too Defto deote the equece gve by the compoud mea P Q by a b tartg wth ( t 496

5 J Šute a ad b x From (33 we obta that hece a b a b b Q a b ( P ( a b a a b a b a b x ad x b So the terato of the compoud mea P Q are a x e ex ( a x a ( a x f fx ( a a a P a b P a a a Note that we do t have to compute the equece b ccordg to (3 ad Lemma the covergece of the equece (34 to t lmt x ha order 4 Complexty Let M N deote the tme complexty of multplcato of two N-dgt umber The clacal algorthm of multplcato ha aymptotc complexty M ( N O( N But there are alo algorthm wth aymptotc complexty or M N O N O N log ( log log log M N O N N N log N M N O Nlog N ee Karatuba [6] Schöhage ad Strae [7] or Fürer [8] repectvely The fatet algorthm have large aymptotc cotat hece t better to ue the former algorthm f the umber N ot very large The complexty of dvo of two N-dgt umber dffer from the complexty of multplcato oly by ome multplcatve cotat D Hece the complexty of dvo DM ( N aly [9] how that th 7 cotat ca be a mall a D We wll deote by σ ( the mmal umber of multplcato eceary to compute the power t t See [] for a urvey o ow reult about the fucto σ Before the ma computato of complexty we eed th auxlary lemma Lemma ume that r ad that M ( N a fucto uch that for ome ω [ ] the fucto M ( N f N odecreag wth N ω N ω δ o( For every δ > put gδ ( N ad aume that M ( N for every T the mage et gδ ([ T] bouded ad there N ( δ wth gδ ( gδ ( m for every m N ( δ The (34 f N N (4 logr N N M r lm N M N r ω 497

6 J Šute Proof From the mootocty of f we have for every N Let logr N logr N δ > The (4 mple lm g ( N ext a umber every N N ( δ ω N N N M f r r r ω logr N ω ω ω M N N f N r r N δ N δ uch that g ( g ( m for every N ( δ Iequalte (4 ad (43 yeld r δ δ logr N (4 From th ad properte ad we deduce that there N ( N ad every m Th mple for logr N N ( ω δ N M r gδ log r N log r N r ω δ r r ω δ ( ω δ ω δ M N N r ω δ g δ ω δ logr N logr N N N M M r r lmf lm up N ω M N r N M ( N (43 r Pag to the lmt δ mple the reult Note that all the above metoed fucto M ( N atfy all aumpto of Lemma wth ω ω log 3 ω ad ω repectvely Now we compute the complexty V ( N of algorthm computg x to wth N dgt The fucto M ( N ad V ( N have aymptotcally the ame order a N ee for tace Theorem 63 [3] Hece all fat algorthm for computg x dffer oly the aymptotc cotat Let the algorthm for computg x perform Z multplcato of two N-dgt umber before the terato perform multplcato ad B dvo of two log umber every tep ha order of covergece r The accuracy to wth N dgt eceary oly the lat tep I the prevou tep we eed accuracy N oly to wth dgt ad o o Hece r N V ( N V ( BD M ( N O( N r The error term O( N correpod to addto of N-dgt umber Th ad Lemma mply that logr N N V N BD M ZM N O N r ( o BD Z M N O N ω r ( BD Z o( M ( N ω r 4 Complexty of Newto Method Newto method (3 ha order ad every tep t perform ( σ multplcato (evaluato of a I the lat le we aume the hypothe that all multplcato algorthm atfy N o ee [] M N 498

7 J Šute ad dvo So the complexty where N ( N V N S o M N S ( ( σ ( D N ω For the choce of multplcato ad dvo algorthm wth ω ad 4 Complexty of Houeholder Method ( σ ( N V N o M N 7 7 D we have Coder Houeholder method ( appled to the equato f ( t t x calculato (ee [] for tace lead to terato where λ ( d ad ( d a ( a d d λ x a d d µ x ( a Let d eay µ are utable cotat The method ha order d before the terato t perform d multplcato of N-dgt umber (evaluato of d x x ad every tep t perform σ ( d multplcato (evaluato of a the evaluato of umerator ad deomator by Horer method ad the the fal multplcato ad dvo So the complexty of Houeholder algorthm where H d ( d H ( V N S o M N H Sd ( ( σ ( d D d ω ( d For the choce of multplcato ad dvo algorthm wth ω ad ( H 7 σ Vd ( N 3d σ ( o( M( N d The optmal value of d whch mmze the complexty th cae 43 Complexty of Our Method ( σ ( σ d D we have Gve our method (34 ha order before the terato t perform multplcato of N-dgt umber (evaluato of x x ad every tep t perform σ ( multplcato (evaluato of a the evaluato of umerator ad deomator by Horer method ad the the fal multplcato ad dvo So the complexty of our algorthm where S ( S ( V N S o M N Not alway Horer method optmal ee [3] I thoe cae Houeholder ad our algorthm are fater 499

8 J Šute S S ( ( σ ( D ω ( For the choce of multplcato ad dvo algorthm wth ω ad ( 7 D we have S 5 σ 5 V ( N 3 σ ( o( M( N 4 The optmal value of whch mmze the complexty th cae ( σ 5 σ ( 5 (44 5 Example Example Compare the algorthm for computato of 4 x For 4 we have σ ( 4 ad accordg to (44 the optmal value of for our algorthm Padé approxmato of the fucto 4 t aroud t 4 5t 3 t O( t 3 3t 5 Hece the terato of our algorthm are 4 5x 3a a : a : a (5 4 3x 5a wth covergece of order 3 For computato of N dgt of 4 x we eed to perform operato Newto method S 57 V N o M N 4 ( 4 a : a : 3a 3 4 a ha order ad for computato of N dgt t eed ( σ ( 3 5 V N ( 4 N ( 7 o( M ( N operato Hece our method ave 6% of tme compared to Newto method For Houeholder method the optmal value d ad t lead to the ame terato (5 a our method Example 3 Compare the algorthm for computato of 79 x For 79 we have σ ( ad accordg to (44 the optmal value of for our algorthm Padé approxmato of the fucto 79 t aroud t 79 Hece the terato of our algorthm are t t 5 t O t t 47t 77 x 77x 47xa 59a a a a : : x 47xa 77a wth covergece of order 5 For computato of N dgt of 79 x we eed to perform S 73 V N o M N 8 ( 79 5

9 J Šute operato Newto method a : a : 78a ha order ad for computato of N dgt t eed a ( V N 79 N 7 o M N operato For Houeholder method the optmal value d ad t lead to terato 8x 78a a : a : a x 8a Th method ha order 3 ad for computato of N dgt t eed H 87 V ( 79 N o M ( N 4 operato Hece our method ave % of tme compared to Newto method ad ave 6% of tme compared to Houeholder method Example 4 Compare the algorthm for computato of x For the exact value of σ ( ot ow We aume that σ ( 59 ccordg to (44 the optmal value of for our algorthm 3 The terato of our algorthm are a : x xa xa a a : a x xa xa a wth covergece of order 7 For computato of N dgt of x we eed to perform S 33 V3 ( N o M ( N 4 operato For Houeholder method the optmal value d 3 ad t lead to terato a : x xa xa a a : a x xa xa a x 5

10 J Šute Th method ha order 5 ad for computato of N dgt t eed operato Newto method a : a H 69 V3 N o M N 8 ( : a a ha order ad for computato of N dgt t eed (aumg that ( V N ( N 5 o( M N x σ 59 operato Hece our method ave 35% of tme compared to Newto method ad ave 7% of tme compared to Houeholder method 6 Proof I th ecto we wll prove that fucto p defed by (3 atfe equalte (3 For the ae of brevty wll ue the ymbol u 6 Combatoral Idette for the et [ u Frt we eed to prove everal combatoral dette Our otato wll be chaged th ubecto Here wll be a varable ued mathematcal ducto wll be a ummato dex ad B wll be addtoal parameter The chage of otato made becaue of eay applcato of the followg method ϕ we wll deote t dfferece by baed o [4] For a fucto ϕ( : ϕ( ϕ( ϕ( : ϕ( ϕ( Gve a fucto F( there ome fucto G( atfyg ome relato betwee F ( ad ( Th ew fucto the ued for eaer evaluato of um cotag ( G for egatve teger For ad B wth max ( B put ( ( B B ( B ( B ( B ( ( B ( B ( ( B F B : R ( B ( B B B B : G B : F B R B S( B : F( B Lemma 3 For every S ( ( B ( B B atfyg max ( B we have ( ( B B ( B ( B ( B ( B ( ( B ( B ( ( B Proof From the polyomal detty F Recall that 5

11 J Šute we mmedately obta ad (( ( B( B( B ( B ( B( B( ( B ( B (( ( B ( B ( ( B ( B (( B( B( ( B ( B (( B ( B ( ( ( B 6B B 6B B B ( B 4B B B ( F B ( ( B( B( B ( B ( B( B( ( B ( B F B ( ( B( B( B ( B ( B( B( ( B ( B ( B( B( ( B ( B ( F B ( B B B 6B B 6B ( B 6B B 4B ( ( B ( B F B ( ( B ( B ( ( B ( B ( B ( B ( ( ( B ( B ( ( ( ( G( B F B R B F B R B ( R B R B F B ( ( ( (6 For fxed > we have max ( B Smlarly for ad we have The (6 (6 (63 ad (64 mply < ad hece ( ( B Thu F B (6 G B (63 G B (64 53

12 J Šute If ad B S ( B the ( ummad S( B ( ( F B F B ( ( ( ( ( F B G B (65 F B G B G B B the oe for B ad for < B ad ( B for ( B B ( B ( B ( B ( ( B ( B ( ( B > B Hece the oly ozero S B (66 B Smlarly for < ad B we obta From th ad (66 we obta that S < B ( S max B B Equato (65 mple that S wll ot chage for greater For wth ad put F ( ( ( ( ( ( ( ( : ( ( ( ( T : S : F Lemma 4 For every wth S ( Proof From the polyomal detty we obta for that ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( F T F F Th ad the fact that T( mply ( ( ( ( ( S F F T ( ( ( ( F T F T ( ( ( ( ( ( ( ( ( ( ( 54

13 J Šute For B wth B < ad put F G 3 3 ( B ( ( B ( ( B ( ( : ( B ( ( B ( ( B ( ( ( : S B : F B 3 3 Lemma 5 For every B wth B < S 3 ( B Proof From the polyomal detty we obta for B < ad that For ( B ( ( B ( ( B ( ( ( B ( B ( ( ( ( ( B ( B ( ( ( ( ( B ( ( ( B ( ( B ( ( ( B ( ( ( ( ( ( B ( B F( B G3 ( ( ( ( B ( B F3 ( B B B F B F ( B 3 3 we have G3 ( B Th ad (67 mply ( B ( B S ( B ( ( S ( B 3 3 (( B ( B F3( B ( ( F3( B ( ( F( B 3 ( ( G B F B 3 3 G B G B F B B B B B Lemma 4 for hece ad B mple B B B ( ( B ( B ( B ( ( B ( B B B B B B B (67 (68 55

14 J Šute B For B we have > ad (68 yeld ( ( B ( ( S3 B B (69 B ( B ( B ( ( 3 3 S B S B Th wth (69 mple the reult for every > B For wth put F G 4 4 ( ( ( ( ( ( ( : ( ( ( ( ( ( ( ( : 3 S ( : F( 4 4 Lemma 6 For every wth S 4 ( Proof From the polyomal detty ( ( ( ( ( ( ( ( ( ( ( ( ( ( 3 ( ( ( ( ( ( we obta for ad that F ( 4 ( ( ( ( ( ( ( ( 3 ( ( ( ( 3 ( ( ( ( ( ( F 4 ( ( ( ( ( 3 ( ( ( ( ( ( ( ( ( 3 ( ( ( ( 3 ( ( ( ( ( ( ( ( ( 3 ( ( ( ( ( ( ( ( 3 G ( 4 F 4 For we have G4 ( From th (6 ad the polyomal detty F 4 ( (6 56

15 J Šute we obta For the um ( ( ( S F F ( ( F G 4 4 F4( G4( G4( ( ( ( ( ( ( ( 3 ( ( ( ( ( ( ( ( ( 3 ( ( ( ( ( ( 3 ( 3 S4 cota oly oe ozero ummad for Equato (6 mple that S 4 wll ot chage for greater For B wth B < put F G 5 5 ( B ( B ( B ( B ( ( ( B B ( B ( B : ( B ( B ( B ( B ( ( ( B B ( B ( B : ad we have S ( (6 4 B S ( B : F( B 5 5 Lemma 7 For every B wth B < S 5 ( B Proof From the polyomal detty B ( B ( B ( B ( ( ( B B ( B ( B ( B ( ( B ( B ( B ( ( B ( B we obta for B < ad F 5 ( B ( B ( ( B ( B F 5 ( B ( B ( B ( B ( B ( B ( F5 ( B ( B ( B ( B ( B ( B ( B ( B ( ( ( B B ( B ( B ( B ( B ( B ( ( ( B B ( B ( B G( B 5 (6 57

16 J Šute ad For ad B we have From th ad (6 we obta Lemma 6 for B yeld G5 B G5 B B S5 ( B B B F5( B G5( B (63 G B B G B 5 5 ( B ( B ( B ( B ( ( B B ( B S B B Equato (63 mple that 5 S B 4 S5 B ha the ame value for greater For B ad put F ( B ( ( ( ( ( ( ( B ( B ( 3 B B B B : ( : T B B B B B B B B B G ( B 5B 69 B 6 B 5 B 7 B 3B B B B B B B B B B B B B B B 5 4 B B ( ( ( B ( B ( B T6 ( B ( ( ( B ( B ( 3 B 3 : 6 S ( B : F( B 6 6 Lemma 8 For every B S ( B Proof From the polyomal detty we obta ( ( ( ( ( ( ( B ( B ( 3 B B B B 6 ( ( B ( B ( B ( B ( ( ( B ( B ( 3 B ( 3 B ( 3 B 3 ( T( B( ( B T( B ( B

17 J Šute For we obta F 6 ( B ( ( B ( B ( B ( B ( ( ( B ( B ( 3 B ( 3 B ( 3 B 3 F ( ( B ( B ( 3 B ( 3 B ( 3 B 3 ( T6( B( ( B T6( B ( B ( B ( ( B ( 3 B ( 3 B ( 3 B 3 ( T6 ( B ( B ( B ( 3 B ( 3 B ( 3 B 3 ( B T6 ( B ( ( B ( 3 B ( 3 B ( 3 B 3 F ( B ( ( ( B ( B ( B T6 ( B ( ( B ( B ( 3 B 3 B ( B ( B T 6 B B B 3 B 3 G 6 ( B we have G( B 6 From th (64 ad the polyomal detty ( ( ( ( ( T6 B B B B ( ( ( S B F B F B ( ( F B G B 6 6 ( ( ( F B G B G B ( ( ( B ( B ( B ( ( ( B ( B ( 3 B 3 ( ( ( B ( B ( B T6 ( B ( B ( B ( 3 B 3 ( ( ( B ( B ( B ( B ( B ( 3 B 3 (( ( B ( B ( B T6 ( B 6 6 ( B F 6 ( B (64 (65 For we have 6 ( B ( B ( B ( B ( B ( B B ( B 3 B( B 3 B( B 3 S B Equato (65 mple that S6 ( B ha the ame value for greater 59

18 J Šute For B ad wth B put F G 7 7 ( B ( ( ( B ( B ( ( B ( ( B ( B ( ( B ( : ( B ( ( ( B ( B ( ( B ( ( ( B ( B ( ( B ( : S ( B : F( B 7 7 Lemma 9 For every B wth B S 7 ( B ( ( ( B ( B ( ( B ( ( B ( B ( ( B ( Proof From the polyomal detty ( ( B ( ( ( B ( ( ( ( B ( ( B we obta for B ad wth B F( B 7 ( ( B ( ( ( B ( F7 ( B ( ( B ( ( ( ( B ( ( B F 7 ( B ( ( B ( ( ( ( ( B F7 ( B ( ( ( ( B ( ( ( ( B ( B ( ( B ( ( B ( B ( ( B ( ( ( ( B ( B ( ( B ( ( ( B ( B ( ( B ( G ( B 7 (66 For ad we have ad G7 B G7 B From th ad (66 we obta For B Lemma 8 mple ( ( ( S B F B G B G B G B 7 7 (67 5

19 J Šute 7 ( S BB ( ( ( B ( B ( B ( B ( ( B ( B ( B ( ( B S ( B 6 3 Equato (67 mple that S7 ( B ha the ame value for greater 6 Formula Now the ymbol aga ha t orgal meag For ad put ( g : ad g : The umber g are the coeffcet of Taylor polyomal of the fucto Lemma For ad t ( t g t Proof See for tace Theorem 59 [5] Now we prove a techcal lemma that we eed later Lemma For ad ( ( ( ( ( h h h t ( h ( ( ( h ( h ( ( h h (68 Proof O both de of (68 there are polyomal of degree varable Therefore t uffce to prove the equalty for ad for value wth m m For we mmedately have For LHS ( h h RHS wth m we obta o the left-had de of (68 m For uch umber m we have h m LHS h m m m h Therefore we obta o the rght-had de of (68 h m ( ( ( ( ( m h ( h m ( ( m h h h h h RHS h m h m h m 5

20 J Šute The frt product o the lat le equal to zero for m the ecod product o the lat le equal to zero for m For other value of both product cota oly potve term Hece Th mple that For m ( ( m ( ( m h ( h m ( ( m m LHS m h h ( m ( m ( m ( ( m ( m ( m m m ( m ( m ( m ( ( m ( ( ( m ( m ( ( m m RHS LHS m < we have ( m The Lemma 3 mple for > we have ( S ( mb ( m ( m ( ( m ( ( ( m ( m ( ( m ( m ( m ( m ( ( m ( ( ( m ( m ( ( m m m RHS LHS ad for > m we have Hece equalty (68 follow for wth m m For ad put For ad put ( ( h c : h ( ( h d : h p ( t c ( t : d ( t (69 We wll prove that (69 Padé approxmato of t Lemma For the umber c ad d atfy the ytem of equato Proof Lemma mple d g c 5

21 J Šute ( ( c h h ( h ( h h h ( h h ( h h ( h ( h h h d g d g Lemma 3 For the umber c ad d atfy the ytem of equato d g Proof For we obta o the left-had de ( d g ( h ( h ( ( h h LHS Th expreo multpled by a polyomal of degree varable Therefore t uffce to prove the equalty for value wth m For m we have m h h ad hece the whole expreo equal to zero For m we obta ( ( h ( ( h LHS h m h m The ecod product equal to zero for > m therefore the ummato ed for m The Lemma 5 mple Lemma 4 Fucto p m ( ( ( ( LHS ( h m m h h h m ( ( ( m ( ( ( m S3 m B t the Padé approxmato of the fucto Proof For t Lemma Lemma ad Lemma 3 mply The reult follow d ( t t d ( t g ( t ( t d g ( t d g O( t c t O t t of order [ ] aroud t 53

22 J Šute Now we fd the coeffcet e ad f of the Padé approxmato p ( t Lemma 5 For every ad e t f t : e h h Proof Frt we prove (6 From (69 we obta Bomal theorem the mple that Thu equalty (6 equvalet to ( (6 h h f h h ( (6 h h f t d t ( d f ( ( h ( h ( ( h h h (6 h O both de of (6 there are polyomal of degree varable Therefore t uffce to prove the equalty for every wth r r r Lemma 7 mple S5 ( r B ( ( r ( ( r ( ( ( r ( r ( ( r ( ( r ( r ( r ( r ( r ( ( ( h r ( h ( h r ( h r h h ( ( h ( h ( h h h h hece (6 ad (6 follow Puttg : to (6 ad applyg bomal theorem the ame way we obta (6 54

23 J Šute 63 Boud By Lemma 4 the fucto p Padé approxmato hece we ow t properte the eghbourhood of Here we fd global boud for p that are eceary for fuctoalty of our algorthm We eed aother two techcal lemma Lemma 6 For ad m ( ( m ( m h h h m h h m ( m ( m h Proof O both de there are polyomal of degree varable Therefore t uffce to prove the equalty for every wth r r m r Lemma 9 mple 7 S r B m ( ( ( m ( m ( r ( r m r ( ( m ( m ( r ( r m ( r ( ( ( m ( m m ( r h ( r h ( h r ( ( m ( m h h m h ( ( ( m ( m ( ( m ( m Th mple the reult Lemma 7 For ad wth Proof Put m: The Lemma 6 mple LHS d d g m ( h ( h ( h h h m h > ( h ( h h ( h ( ( h ( h ( h h ( ( h h ( ( m ( h ( h ( h h ( ( m h h m m m g ( m m ( h ( h ( m ( m h h LHS > m Now we fd lower ad upper boud for the fucto p Lemma 8 For ad ( Proof Put p t t t we have > α ( t : d ( t t d ( t g ( t (63 55

24 J Šute Lemma ad Lemma 3 mply From (63 we obta for g We have g α ( t c ( t (64 α d g Lemma 7 mple α ( t ( d g ( t < (65 From (63 (64 ad (65 we obta Lemma 5 mple that hece d ( t t α ( t α ( t < c ( t d ( t c > ( t p t > t d ( t Lemma 9 For every ad p t > Proof Drectly from the defto of c ad d we obta p ( t x > we have ( c ( t h ( ( h ( h ( t > d t ( h ( t (66 The equalty trct ce for the ma bracet the umerator greater tha the ma bracet the deomator Now we prove that fucto p ( t atfe equalte (3 Lemma For every ad t { } ( t < p ( t < ( t m max Proof The proof plt to four cae For < t < Lemma 8 mple that t < p ( t Lemma 5 mple that e f for every Hece p t p t Ug th ad the frt cae we obta that for 3 For t > Lemma 9 mple that p ( t > 4 Ug th ad (67 we obta for t cowledgemet p t t t > we have < < that < (67 p t < The author would le to tha to profeor drze Schzel for recommedato of the paper [4] ad alo to 56

25 J Šute author colleague Kaml Breza Luáš Novotý ad Ja Štĕpča for checg the reult Publcato of th paper wa upported by grat P//35 of the Czech Scece Foudato by grat 798//RRC of the Morava-Slea rego ad by grat SGS8/PrF/4 of the Uverty of Otrava Referece [] Houeholder S (97 The Numercal Treatmet of a Sgle Nolear Equato McGraw-Hll New Yor [] Hardy GH Lttlewood JE ad Pólya G (95 Iequalte Cambrdge Uverty Pre Cambrdge [3] Borwe JM ad Borwe PB (987 P ad the GM Joh Wley & So Hoboe [4] Gau CF (866 Were Göttge [5] Matow J (999 Iterato of Mea-Type Mappg ad Ivarat Mea ale Mathematcae Sleaae -6 [6] Karatuba ad Ofma Yu (96 Umozhee mogozachyh chel a avtomatah Dolady adem au SSSR [7] Schöhage ad Strae V (97 Schelle Multplato Großer Zahle Computg [8] Fürer M (7 Fater Iteger Multplcato Proceedg of the 39th ual CM Sympoum o Theory of Computg Sa Dego Calfora -3 Jue [9] Bret RP (975 Multple-Preco Zero-Fdg Method ad the Complexty of Elemetary Fucto Evaluato I: Traub JF Ed alytc Computatoal Complexty cademc Pre New Yor 5-76 [] Kuth DE (998 The rt of Computer Programmg Volume : Semumercal lgorthm ddo-weley Boto [] Breza K ( Smíšeé Průměry Mater The Uverty of Otrava Otrava [] Karatuba (995 The Complexty of Computato Proceedg of the Stelov Ittute of Mathematc [3] Pa VYa (96 Neotorye hemy dlya vychleya zache polomov vehchetveym oefftetam Problemy Kberet [4] Wlf H ad Zelberger D (99 Ratoal Fucto Certfy Combatoral Idette Joural of the merca Mathematcal Socety [5] Jarí V (984 Dferecálí počet cadema Praha 57

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