DKA method for sgle varable holomorphc fuctos TOSHIAKI ITOH Itegrated Arts ad Natural Sceces The Uversty of Toushma -, Mamhosama, Toushma, 770-8502 JAPAN Abstract: - Durad-Kerer-Aberth (DKA method for sgle varable holomorphc fuctos are treated. Fte seres appromato of the sgle varable holomorphc fucto s used to fd local roots of sgle varable holomorphc fuctos stead of algebrac equato. We ca fd roots of sgle varable holomorphc fuctos wde rego by cotug local solutos. Key-Words: - Durad-Kerer-Aberth (DKA method, root fdg Itroducto Advatage of Newto s terato method for o-lear equato or algebrac equato s well-ow ad detaled eplaatos ad ts applcatos are foud may publcatos []. However, brdge for the gap betwee algebrac equato ad aalytc fucto wth ths method was ot gvg much. To cover ths gap [2] s wored towards costructg mathematcal base. Here we wll attempt to cover ths gap actual problem. I ths artcle, root fdg method DKA [3, 4] ad 3rd order methods are treated from both algebrac ad aalytc pots. From ths pot, DKA method resembles to Euclda elmato (algebra ad Weerstrass preparato theorem (aalyss. Numercal algorthms are ot well vestgated from above med pot up to ths pot. We epect to obta ew sght from ths pot.. DKA method ad 3rd order method Durad -Kerer -Aberth method s umercal root fdg algorthm for algebrac equato. Cosder -th degree algebrac equato wth real umber coeffcet, ( P( + a + + a 0, a 0, C. The umber of roots ca be foud umercally by followg Newto s method, (2 ( ( f(,, f(, f (, J ( ( f ( /, ( f(,, f( J( f (, (3 : terato umber ad ϕ (,,...,, m, ϕ, m 2 2 m 0 < < < m f(, 2,..., ( ϕ (, 2,..., a 0,,2,...,. The (3 ca be wrtte as (4 ( ( ( P( / (,,2,...,. We ca assume that (4 s holomorphc mappg, f ( 0 at every terato step wth proper tal codtos. It s a acceptable assumpto, because (0 tal codtos for are gve o the crcle usually. Therefore (4 maps tal crcle where tal values are gve to the closed curve o that eact roots. It s ow that such uque holomorphc fucto that maps crcle to closed curve C ests. From algebrac geometrcal pot, (4 gves umbers of geerators for deals, because each ( + equato (4 s depedet. We ca troduce ope coverg whch gves restrcto mappg, ad etc., D D, D D φ, ( + ρ : D D (, ϕ : +, see Fg.. ( + (0 ( ( D ( D Fg. DKA method ad rego of the mappg. ( D + ( Z ( + D ( D ( D
Ths property may gve superor coverget property. DKA method (4 ca be deformed as, (5 ( + P ( ( (,,2,...,. It resembles to followg Euclda elmato, (6 P ( ( h( + r(,,2,...,, here h( (. I addto r( 0,,...,, the (6 becomes DKA method. DKA method seems umercal appromato of Euclda elmato. Sce t s ow that polyomal fuctos C are UFD, estece of umercal soluto of the method s guarateed. The followg method s modfcato of DKA method, ad t s called 3rd order method, (7 ( ( P ( /{ P'( P ( },,2,...,. 2 DKA method for sgle varable holomorphc fucto We ca modfy DKA method for holomorphc ( fucto f( w, wth several varables, (8 u ( w, f ( w, / (,,2,...,. 2 + P( w, u ( w, f ( w,, w {,,...,,,..., } P w (, ( s Weerstrass polyomal by, ad u ( w, are uts. Estece of such dvso Puf s guarateed by Weerstrass preparato theorem [5]. Fgure 2 llustrates mage of ut U(. Holomorphc fuctos coverge the covergece doma of radus r, (uformly ad absolutely coverge wth approprate radus s<r, see Apped also, (9 f( a, a C, 0 U( here Cauchy-Hadamard theorem for r, 3 2.5 2.5 0.5 Z - 2-0.5 Fg. 2 uw (,, Pw (, fw (, ( lm sup a, r ( ( a. r We troduce varable trasform, ad appromate f( by fte seres, rw (, rw (, :t <, (2 f ( f ( frw ( ( ar ( w bw, b ar (. 0 0 The r of fw ( respect to coeffcets b ad depedet varable w roughly equals to, ad r( s polyradus that depeds o f ad. For eample, appromato by fte seres epaso of Bessel fucto, ths case r(209.05746, behaves as Fg.3. - - 20 40 60 80 00 BesselJ 0( BesselJ 0( r(20 2 4 6 8 0 20 order seres appromato Fg. 3 Comparso betwee orgal fucto ad appromato fucto by fte seres epaso We wll use polyomal fucto by fte seres P ( f ( stead of f( wth DKA ad 3rd order method to resolve local roots of f(. 2. Uform property of holomorphc (aalytc fucto Frstly, we should cofrm uform property of the appromato fuctos by fte seres (polyomals. Fortuately, uform property of the appromato fucto s preserved the approprate treatmet of the radus of the polydsc of orgal holomorphc fucto. If we use r( whch s suffcetly smaller tha radus of covergece, appromato fucto closes orgal fucto, because both fuctos are holomorphc fuctos whch uformly coverge the polydsc of r(. Fgure 4 shows behavor of r( respect to for ep(. The fgure shows that covergece radus r( of the appromato fucto becomes large by
creasg. Sce ar ( O( as to appromato fucto, therefore polyomal fucto for appromato dverges where > r (. I geeral we eed careful treatmet for r( respect to the meag of lm sup (. 2.2 Covergece radus ad ts behavor by fte seres appromato Wth appromate radus r(, we troduce varable trasform r(w. Dscardg the term of + ad hghers the seres epaso of f(, the appromato fucto s, (3 r( f ( w f ( w ar ( w, 0 the, Trucato error Oa ( r + + (. Sce the covergece radus r by ew coeffcets b a r( roughly equals to oe, we oly tae up resolved roots by the DKA whch are sde ut crcle wth followg error correcto. We defe, ( ε f ( ( ( ( 0, δ f(, ad γ f( f (, Corrected radus (trustable, (4 r r ( r, correct error ma ε, δ, γ, γ rerror,,..., m ( ( ( ( f '( f '( f '( f '( ( ( here rw (, lower de s the umber of roots ad m. r 35 30 25 20 5 0 5 f f(e 20 40 60 80 00 Fg. 4 Plot of r( ε f ( '( ε ( Fg. 5 Error estmato ad correcto of radus γ δ f ( f( Easly we ca do error estmato by whether δ < Tolerace Error s satsfed or ot for the resolved roots. If there are ay resolved roots that do ot satsfy ths codto, they are reected. Moreover, resolved roots outsde of the polydsc radus of whch s defed as dstace betwee the ceter of covergece ad reected resolved root wth (4 are also reected. Ths frst process decreases comple error estmato wth comple error treatmet. 2.3 Normalg requremet for actual eecuto by fte seres appromato We defed polyomal fucto for appromato as (2. Applg DKA method to f ( w (2, we must ormale t as P( (. The we devde f ( w by b for ormalato. It s mportat to avod dvergece of coeffcets of polyomal fucto for appromato. We foud that varable trasform (2 removes ths dvergece. If we scamp ths treatmet, we eed large dgt for computato. Because a s very small for large, we eed large dgt for / a for ormalato. 3 Eamples 3. Low order polyomal As a eample, we appled the method to BesselJ 0 ( wth 20 order appromato for DKA method. Fgure 6(a shows plots of BesselJ 0 ( ad polyomal fucto for appromato. Fgure 6(b shows resolved roots. Horotal le the ceter of fgure correspods to real as. Meshes ths fgure show covergece traectores of resolved roots each terato step. We fd that 4 roots are sde ut crcle at least by umercal resoluto. Appromato Bessel -0-5 5 0 - Ut crcle r(209. Fg. 6 (a Plots of BesselJ 0 ( ad 20 order polyomal fucto for appromato, (b Numercal result I ths umercal treatmet, we used Mathematca, therefore we have doe symbolc treatmet for the computato wheever t s possble. Traectory
0-0 0 20-0 -0 0 20-0 -0 0 20-3.2 Hgher order appromato BesselJ 0 ( fucto ad ts 40 order appromato are show Fg. 7(a ad resoluto by DKA s show Fg. 7(b. We ca obta more roots at the same tme by the hgher order appromato method tha by the lower order oe. Ut crcle 0-0 0 20 coverg of the some rego usg polydscs. Each local polydsc overlap compactly wthout gaps the rego. Resoluto of the holomorphc fucto o each polydsc s possble wth proposed method the prevous secto. Small crcles ad tragles Fg. 9 correspod to roots by the method o each polydsc ad Fg.0 shows mage of cotuato of resoluto. Cosdet Error?, Neglect - r(406.7 Fg. 7 (a Plots of BesselJ0( ad 40 order appromato fucto, (b Numercal result 3.2 Cos( by 3rd order method wth 80 order appromatos Numercal resoluto of perodc fucto cos( ad ts appromato by 80 order polyomal fucto are show Fg. 8(a, ad ts roots by umercal resoluto wth 3rd order method are show Fg. 8(b. We ca fd may uecessary roots outsde ut crcle. It seems possble that the umber of these fcttous roots become smaller by ecellet seres epaso. By ths method, we obta the same umber of resolved roots to the umber of order of polyomal fucto for the appromato, whether we desre t or ot. Fg. 9 Cotuato or coecto wth polydscs for global resoluto of roots. r(5020 Appromato 4 3 2 0 0-0 0 20 30 - Ut crcle r(8030 Fg. 8 (a Plots of cos( by 80 order appromato fucto, (b resoluto by the method 4 Cotuato of local soluto We cosdered local resoluto of roots for the holomorphc fucto up to ths pot. I the followg secto, we cosder global dstrbuto of the roots by coecto ad cotuato. Fgure 9 llustrates global dstrbuto of roots by coecto wth may local resolutos by DKA. We cosder Fg. 0 Image of cotuato by proposed method. I ths case, we ca defe ad use covergece radus o each polydsc. Therefore, we wll obta relable roots ad fcttous roots by the method the each polydsc. Whe a root by the method a polydsc s ot foud aother polydsc where the rego of the both polydscs s overlapped, we must cosder such root s fcttous oe. Moreover, we must detfy or dstgush two close roots. The oe s obtaed a polydsc ad the other s obtaed aother polydsc. It s dffcult to detfy or dstgush such two roots overlapped rego, because both are very close ad we have o formato of the dstrbuto about the roots ths rego. As for the prescrpto of ths dffculty, we use square regos. The se of square regos are defed by the smallest radus of all polydscs the
rego. Sdes of each square rego cosst of two ds of boudares, whch are defed as Fg.. Closed r( Fg. Global cotuato by polydsc for the method. We oly allow resolved roots each square rego whch are covered by a polydsc that cludes the square wholly. Usg ths coecto, cotuato ad selecto, we ca obta all resolved roots wthout redudacy the whole rego. 5 Applcato of DKA method wth ths procedure We ca use above method to appromate u( (8. To costruct ut fucto u(, we treat as followg, (5 m m ( ( P r w ( ( ( (, Ope (,2,..., m, here w,,..,m are resolved roots by DKA or 3rd method sde polydsc. m (m s umber of roots sde polydsc of whch radus s (4. The we put, (6 / u ( f (/ P (, u ( u ( for the appromato of u(. Rough estmato of u( gves effectve formato for the eact treatmet of Weerstrass preparato theorem for f(, ad P ( each polydsc gves algebrac obect to represet orgal holomorphc fucto by dfferece (algebrac formula. 7 Summares ad Coclusos Eteded treatmet of DKA a d 3rd order method for holomorphc (aalytc fuctos are proposed. Weerstrass preparato theorem s treated from umercal pot of vew. Local dscrete represetatos of holomorphc fuctos are possble usg proposed eteded method. New umercal treatmet for evaluatg resdues of holomorphc fuctos wll be developed by ths approach. Refereces: [] W. C. Rheboldt, Methods for solvg Systems of Nolear Equatos, SIAM, 987. [2] T. Itoh, Algebrac Geometrcal Treatmet of Numercal Algorthms, WSEAS Trasactos o Mathematcs, Vol. 2, No., 2003, pp.30. [3] E. Durad, Soluto umérques des équatos algébrques, Masso, 960. [4] T. Yamamoto ad T. Ktagawa, Suchaseesyu, Scece-sha, Japa,99. [5] R. C. Gug ad H. Ross, Aalytc Fuctos of Several Varables, Pretce-Hal, Ic., Eglewood Clffs, N. J.,l965. Apped Holomorphc fucto C Defto : A comple-valued fucto f defed o C s called holomorphc fucto C f each pot w C has a ope eghborhood U, w U C, such that the fucto f has a power seres epresso v (A. v f( a ( w ( w v v 0 v v whch coverges for all U. Here C C C whch s Cartesa product of copes of comple plae, ad y are real umbers ad + y C, (,,. Defto 2: A ope polydsc C s a subset ( wr ; C of the form (A.2 ( wr ; ( w,, w; r,, r ; { C ; w < r, } the pot w C s called the ceter of the polydsc, ad r ( r,, r R,( r > 0 s called the polyradus. Note that polyomals the fuctos,..., are holomorphc all C. It s famlar result from elemetary aalyss that a power seres epaso of the form (A. s absolutely uformly coverget all sutable small ope polydscs ( wr ; cetered at the pot w. Moreover, the fucto f s holomorphc each varable separately throughout the doma whch t s aalytc.