Appled Mathematcs Letters 28 (204) 60 65 Contents lsts avalable at ScenceDrect Appled Mathematcs Letters journal homepage: wwwelsevercom/locate/aml On an effcent smultaneous method for fndng polynomal zeros MS Petkovć a,, LD Petkovć b, J Džunć a a Faculty of Electronc Engneerng, Unversty of Nš, 8000 Nš, Serba b Faculty of Mechancal Engneerng, Unversty of Nš, 8000 Nš, Serba a r t c l e n f o a b s t r a c t Artcle hstory: Receved 25 July 203 Receved n revsed form 28 September 203 Accepted 29 September 203 Keywords: Polynomal zeros Smultaneous methods Kung Traub method Acceleraton of convergence Computatonal effcency A new teratve method for the smultaneous determnaton of smple zeros of algebrac polynomals s stated Ths method s more effcent compared to the all exstng smultaneous methods based on fxed pont relatons A very hgh computatonal effcency s obtaned usng sutable correctons resultng from the Kung Traub three-step method of low computatonal complexty The presented convergence analyss shows that the convergence rate of the basc thrd order method s ncreased from 3 to 0 usng ths specal type of correctons and applyng 2n addtonal polynomal evaluatons per teraton Some computatonal aspects and numercal examples are gven to demonstrate a very fast convergence and hgh computatonal effcency of the proposed zero-fndng method 203 Elsever Ltd All rghts reserved Introducton The am of ths paper s to construct an teratve method for the smultaneous determnaton of smple polynomal roots wth a very hgh computatonal effcency The proposed method s ranked as the most effcent among exstng methods n the class of smultaneous methods for approxmatng polynomal roots based on fxed pont relatons The presented teratve formula reles on the fxed pont relaton of Gargantn Henrc type [] A hgh computatonal effcency s attaned by employng sutable correctons whch enable very fast convergence (equal to ten) wth mnmal computatonal costs In fact, these correctons arse from the Kung Traub three-pont method [2] 2 Accelerated methods Let f (z) = n j= (z ζ j) be a monc polynomal of degree n wth smple real or complex zeros ζ,, ζ n and let u(z) = f (z) d f (z) = n log f (z) = () dz z ζ j j= be Newton s correcton appearng n the quadratcally convergent Newton method To construct an teratve method for the smultaneous ncluson of polynomal zeros, Gargantn and Henrc [] started from () and derved the followng fxed pont relaton ζ = z u(z) j I n \{} z ζ j ( In := {,, n}) (2) Correspondng author E-mal addresses: msp@junsnacrs, msp@eunetrs (MS Petkovć) 0893-9659/$ see front matter 203 Elsever Ltd All rghts reserved http://dxdoorg/006/jaml203090
MS Petkovć et al / Appled Mathematcs Letters 28 (204) 60 65 6 Let z,, z n be dstnct approxmatons to the zeros ζ,, ζ n Settng z = z and substtutng the zeros ζ j by some approxmatons z j n (2), the teratve method ẑ = z u(z ) z z ( In) (3) j j I n \{} for the smultaneous determnaton of all smple zeros of the polynomal f s obtaned The choce z j = z j n (3) gves the well-known cubcally convergent Ehrlch Aberth method [3,4] ẑ = z u(z ) ( In) (4) z j I n \{} z j Comparng (2) and (3) t s evdent that the better approxmatons z j gve the more accurate approxmatons ẑ ; ndeed, f z j ζ j, then ẑ ζ Ths dea was employed by Nouren n [5] for the constructon of the followng fourth-order method by usng the Newton approxmatons z j = z j u(z j ) n (3): ẑ = z u(z ) ( In) (5) z j I n \{} z j + u(z j ) In ths paper we wll prove that further ncrease of computatonal effcency can be acheved by combnng a sutable three-pont method More detals about multpont methods may be found n [6,7] In fact, we construct a tenth-order smultaneous method of the form (3) usng 2n addtonal polynomal evaluatons These addtonal evaluatons provde a huge ncrease of the order of convergence from 3 (method (4)) to the ncredble 0 Let f be a functon wth an solated zero ζ and let x m be ts approxmaton obtaned at the mth teratve step To acheve a very fast convergence of the method (3), we wll apply a specal case of the Kung Traub famly of multpont methods of arbtrary order of convergence [2], gven through the followng three steps: y m = x m f (x m) f (x m ) = x m u(x m ), v m = y m f (x m)f (y m )u(x m ) f (xm ) f (y m ), 2 x m+ = K(x m ) := v m (y m v m )f (v m )u(x m ) f (xm ) f (v m ) f (x m ) 2 (6) f (y 2 m ) + f (y m ) f (v m ) For smplcty, the three-pont Kung Traub teraton (6) s denoted as x m+ = K(x m ) Now we can construct a new smultaneous method takng the Kung Traub approxmatons z j = K(z j ) (gven by (6)) n (3) If,, z(0) n are ntal approxmatons to the polynomal zeros ζ,, ζ n, then the new smultaneous method s defned by the teratve formula z (m+) = z (m) u(z (m) ) j I n \{} z (m) K z (m) j, ( I n, m = 0,, ) (7) Remark To decrease the total computatonal cost, before executng an teraton step t s frst necessary to calculate all entres K z (m) j 3 Convergence analyss The followng theorem deals wth the order of convergence of the smultaneous method (7) Theorem Assume that ntal approxmatons,, z(0) n are suffcently close to the dstnct zeros ζ,, ζ n of the polynomal f Then the order of convergence of the smultaneous method (7) s 0 Proof For smplcty, we omt the teraton ndex m and denote all quanttes at the (m + )th teraton wth the symbol Let us ntroduce the errors ε j = z j ζ j, ˆε j = ẑ j ζ j, and let z j = K(z j ), λ j = z K(z j ), θ = j I n \{} K(z j ) ζ j (z ζ j )λ j Then, startng from (7) and usng () we obtan ẑ = z + = z ε, ε z ζ j ε θ j I n \{} j I n λ \{} j
62 MS Petkovć et al / Appled Mathematcs Letters 28 (204) 60 65 and hence ˆε = ẑ ζ = ε ε ε θ = ε2 θ ε θ Accordng to the condtons of Theorem, we can assume that ε = O M (ε j ) for any par, j; let ε {ε,, ε n } be the error of the maxmal modulus Here O M s the symbol whch ponts to the fact that two complex numbers w and w 2 have modul of the same order (that s, w = O( w 2 ), O s the Landau symbol), wrtten as w = O M (w 2 ) The order of convergence of the three-pont method (6) s eght, that s, the followng relaton s vald (8) K(z j ) ζ j = O(ε 8 j ) (9) See [2] for the proof Accordng to (9), we have θ = O M (ε 8 ) and from (8) we fnd ˆε = O M (ε 0 ), whch means that the order of convergence of the method (7) s 0 Remark 2 Exceptonal acceleraton of the order of convergence from 3 (the Ehrlch Aberth method (4)) to 0 (the new method (7)) s attaned usng 2n addtonal evaluatons of the polynomal f per teraton As a consequence, the computatonal effcency of the method (7) s ncreased, as shown n Secton 4 Let ζ = ζ,, ζ n and z (0) =,, z(0) n be the vectors of smple polynomal zeros and dstnct ntal approxmatons to these zeros, respectvely One of the most nterestng and challengng problems n studyng teratve root-fndng methods s to get a ball of convergence B(ζ, R) wth center ζ and radus R such that the mplemented method converges startng from any ntal pont belongng to B(ζ, R) A ball of convergence has been found for some relatvely smple one-pont methods for solvng scalar equatons of the form f (x) = 0 as well as systems of equatons of Newton s and secant type, see [8] and references cted theren In the case of n-pont methods for scalar equatons a ball of convergence has not been consdered n the lterature for n 3 due to ther very complcated structure For ths reason, the determnaton of a ball of convergence for the proposed method (7), nvolvng the three-pont method (6), also appears as a very dffcult task Let {c; r} := {z : z c r} denote a dsk n the complex plane wth center c and radus r We consder a ball of convergence for an teratve method (IM) for the smultaneous determnaton of all n smple zeros ζ,, ζ n of a polynomal f of degree n In general, the major problem n fndng a ball of convergence of the form B(ζ, R) = ({ζ, R },, {ζ n ; R n }) for the smultaneous methods for polynomal zeros s to state computatonally verfable condtons whch guarantee the convergence startng wth B(ζ, R) Desgnng such a procedure s a very dffcult task even for smple teratve methods Ths fact has forced numercal analysts, begnnng from the 970s, to search for ncluson dsks of the form {z ; R }, centered at sutable approxmatons to the zeros, nstead of {ζ ; R } Here we present the followng useful result: Theorem 2 For n 3 let W = f (z )/ j I n \{} (z z j ) and η = z W If the nequalty max n holds, then the dsks W < 2n mn z z j,j n j D := {η ; W },, D n := {η n ; W n } are mutually dsjont and each of them contans one and only one zero of f The proof of ths asserton follows accordng to the study gven n [9, pp 28 3] It remans to state convergence condtons for the guaranteed convergence whch deal wth the dsks D,, D n For ths purpose we use Smale s pont estmaton theory based on estmates n one pont Ths approach, ntroduced n [0,], deals wth the computatonally verfable doman of convergence Followng Smale s dea, pont estmaton theory for teratve methods for the smultaneous determnaton of smple polynomal zeros was developed n the book [9] We brefly present ths estmaton procedure at an ntal pont Let,, z(0) n be components of the vector of ntal approxmatons and let W (0) = f ( ) (0) z j ( In ) j I n \{} Followng the result for the Ehrlch Aberth method (5) wth Newton s correctons, gven n [9, pp 02 ], the new method wth Kung Traub correctons (6) wll converge under the followng ntal condton: max W (0) < c n mn (0) z j, () n,j n j (0)
MS Petkovć et al / Appled Mathematcs Letters 28 (204) 60 65 63 where c n = /(αn) < /(2n) (α > 2) s a constant that depends only on the polynomal degree n Condton () s actually condton (0) wth the constant c n nstead of /(2n) Snce c n < /(2n), then the convergence condton () also provdes the constructon of non-overlappng dsks D,, D n defned n Theorem 2 Let us emphasze that the ntal condton () s computatonally verfable snce t depends only on avalable data (ntal approxmatons and coeffcents of a gven polynomal), whch s of practcal mportance Accordng to the values of c n for the Ehrlch Aberth method (4) (see [9, pp 99 05]) and the Ehrlch Aberth method wth Newton s correctons (5) (see [9, pp 05 ]), we may expect that α (appearng n the bound of c n ) belongs to the nterval [25, 4] However, the convergence analyss of the new method (7) s consderably more complcated than that of the methods (4) and (5) so that the determnaton of a sharp bound of c n (that s, as small as possble parameter α) requres a laborous and very lengthy study For ths reason, ths subject wll be consdered n a future work 4 Computatonal aspects In ths secton we compare the convergence behavor and computatonal effcency of the Ehrlch Aberth method (4), the Nouren method (5), the new smultaneous method (7), and two combned methods of order 0 The knowledge of the computatonal effcency s of partcular nterest n desgnng a package of root-solvers Ths comparson procedure s entrely justfed snce the analyss of effcency gven n [2, Chapter 6] for several computng machnes showed that the Nouren method (5) has the hghest computatonal effcency n the class of smultaneous methods based on fxed pont relatons As presented n [3, Chapter ], [2, Chapter 6] and [6, Chapter ], the effcency of an teratve method (IM) can be successfully estmated usng the effcency ndex gven by E(IM) = log r d, where r s the R-order of convergence of the teratve method (IM), and d s the computatonal cost The rank lst of methods obtaned by ths formula manly matches well a real CPU (central processor unt) tme In order to evaluate the computaton cost d t s preferable to use arthmetc operatons per teraton taken wth certan weghts dependng on the executon tmes of operatons Denote these weghts wth w as, w m and w d for addton/subtracton, multplcaton, and dvson, respectvely Let AS n, M n and D n be the number of addtons+subtractons, multplcatons and dvsons per teraton for all n zeros of a gven polynomal of degree n Then the computatonal cost d can be (approxmately) expressed as d = d(n) = w as AS(n) + w m M n + w d D n and from (2) and (3) we obtan (2) (3) E(IM, n) = Introduce the abbrevatons log r w as AS(n) + w m M n + w d D n A 2, = f (z ) 2f (z ), u = f (z ) f (z ), S k, = f (z z j + u j ) (k =, 2), h (z ) k = f (z ) f (z ) 2f (z ) j I n \{} (4) In our numercal experments we tested the smultaneous methods (4), (5), (7) and two combned methods of order 0 Namely, to the authors knowledge, there are no other smultaneous methods of order 0 apart from the new method (7) For ths reason, n an artfcal way we have constructed two methods of order 0 by combnng Newton s method of order two and two smultaneous methods of order fve gven below Wang Wu method [4]: ẑ = z u S 2, h 2 + S 2, ( I n ) (5) Farmer Lozou-lke method [5]: u ( u A 2, ) ẑ = z 2u A 2, + (u 2 /2)(A2 2, S 2,) ( I n ) (6) We construct two combned methods, referred to as N W W and N F L, and execute one teraton through two steps: Startng wth approxmatons z,, z n, apply Newton method to obtan approxmatons y,, y n 2 Contnue the teratve process employng ether method (5) or (6) dealng wth y,, y n The order of convergence of the combned methods N W W and N F L s 2 5 = 0
64 MS Petkovć et al / Appled Mathematcs Letters 28 (204) 60 65 Table The number of basc operatons Methods A n + S n M n D n The Ehrlch Aberth method (4) 4n 2 + O(n) 0n 2 + O(n) 2n 2 + O(n) The Nouren method (5) 4n 2 + O(n) 0n 2 + O(n) 2n 2 + O(n) The new method (7) 22n 2 + O(n) 8n 2 + O(n) 2n 2 + O(n) Combned N W W method 29n 2 + O(n) 26n 2 + O(n) 2n 2 + O(n) Combned N F L method 27n 2 + O(n) 26n 2 + O(n) 2n 2 + O(n) Table 2 The (percent) domnance of computatonal effcency of the new method (7) (X) = (4) (X) = (5) (X) = (N W W) (X) = (N F L) ρ((7), (X)) 46% 22% 32% 3% We consder complex polynomals wth real or complex zeros The numbers of basc complex operatons, reduced to operatons of real arthmetc, are gven n Table as functons of the polynomal degree n takng the domnant terms of order O(n 2 ) To compare the smultaneous methods (4), (5), (7), (N W W) and (N F L), we have used the weghts (appearng n (4)) determned accordng to the estmaton of complexty of basc operatons n multple-precson arthmetc Wthout loss of generalty, we assume that floatng-pont number representaton s used, wth a bnary fracton of b bts In other words, we deal wth precson b numbers, gvng results wth a relatve error of approxmately 2 b Followng results gven n [6], the executon tme t b (A) and t b (S) of addton and subtracton s O(b) Usng Schönhage Strassen multplcaton (see [6]), we have t b (M) = O b log b log log b and t b (D) = 35t b (M) We chose the weghts w as, w m and w d proportonal to t b (A), t b (M) and t b (D), respectvely, for a 28-bt archtecture Applyng (4) and data gven n Table, we calculated the percent ratos E((7)) ρ((7), (X)) = E((X)) 00 (n %), where (X) s one of the methods (4), (5), (N W W), (N F L) The entres of ρ are gven n Table 2 Note that very smlar values are obtaned usng the weghts proportonal to the processor executon tmes of basc operatons for octuple precson (machne epslon 0 67 ) for a Pentum M 28 GHz runnng Fedora core 3 and an Opteron 64-bt processor (data taken from [7]) It s evdent from Table 2 that the new method (7) s more effcent than the tested methods (4), (5), (N W W) and (N F L) The domnant effcency (about 30%) of method (7) n regard to (N W W) and (N F L) s expected snce the latter methods requre addtonal n polynomal evaluatons Havng n mnd the mentoned fact on the domnant effcency of the Nouren method, t follows that the proposed smultaneous method (7) s the most effcent method for the smultaneous determnaton of polynomal zeros n the class of methods based on fxed pont relatons To demonstrate the convergence behavor of the methods (4), (5), (7), (N W W) and (N F L), we have tested a number of polynomal equatons mplementng the computatonal software package Mathematca For llustraton, among a number of tested algebrac polynomals we have selected one numercal example As a measure of accuracy of the obtaned approxmatons, we have calculated Eucld s norm /2 n e (m) := z (m) ζ 2 = (m) z ζ 2 (m = 0,, ) (7) = Example We have appled the teratve methods (4), (5), (7), (N W W) and (N F L) for the smultaneous approxmaton of the zeros of the polynomal of the 2st degree f 2 (z) = (z 4)(z 2 )(z 4 6)(z 2 + 9)(z 2 + 6)(z 2 + 2z + 5)(z 2 + 2z + 2) (z 2 2z + 2)(z 2 4z + 5)(z 2 2z + 0) The followng ntal approxmatons were used, yeldng e (0) 025: = 42 + 0, 2 = 2 + 0, 3 = 22 + 0, 4 = 22 0, 5 = 02 + 2, 6 = 02 2, 7 = 02 + 3, 8 = 02 3, 9 = 2 + 2, 0 = 2 2, = 2 +, z(0) 2 = 2,
MS Petkovć et al / Appled Mathematcs Letters 28 (204) 60 65 65 Table 3 Norm of approxmaton errors Methods e () e (2) e (3) The Ehrlch Aberth method (4) 876( 2) 03( 4) 26( 3) The Nouren method (5) 46( 2) 574( 7) 26( 26) The new method (7) 33( 2) 75( 7) 709( 66) Combned N W W method 324( 3) 05( 23) 7( 228) Combned N F L method 2( 2) 68( 6) 257( 48) 3 7 9 = 2 +, z(0) 4 = 2 + 3, z(0) 8 = 02 + 4, z(0) 20 = 2, z(0) 5 = 2 3, = 02 4, z(0) 2 = 22 +, z(0) 6 = + 02 = 22, The errors e (m) calculated by (7) are gven n Table 3, where the notaton A( h) means A 0 h From Table 3 and a number of tested polynomal equatons we can conclude that the proposed method (7) produces approxmatons of consderable accuracy; two teratve steps are usually suffcent n solvng most practcal problems when ntal approxmatons are reasonably good and polynomals are well condtoned The thrd teraton s gven only to demonstrate very fast convergence and, most frequently, t s not needed for real-lfe problems In ths concrete example the new method (7) gves more accurate approxmatons than the N F L method but s less accurate compared to the N W W method However, a lot of tested numercal examples showed that the mentoned methods of order 0 produce approxmatons of approxmately the same qualty and that none of these methods s the best for all the examples On the other hand, from Table 2 t s evdent that the proposed method (7) s consderably more effcent (about 30%) than the combned methods N W W and N F L, whch s ts man advantage among the methods of the same order of convergence Acknowledgment Ths work was supported by the Serban Mnstry of Scence under the grant 74022 References [] I Gargantn, P Henrc, Crcular aruthmetc and the determnaton of polynomal zeros, Numer Math 8 (972) 305 320 [2] HT Kung, JF Traub, Optmal order of one-pont and multpont teraton, J ACM 2 (974) 643 65 [3] O Aberth, Iteraton methods for fndng all zeros of a polynomal smultaneously, Math Comp 27 (973) 339 344 [4] LW Ehrlch, A modfed Newton method for polynomals, Commun ACM 0 (967) 07 08 [5] AWM Nouren, An mprovement on two teraton methods for smultaneously determnaton of the zeros of a polynomal, Int J Comput Math 6 (977) 24 252 [6] MS Petkovć, B Neta, LD Petkovć, J Džunć, Multpont Methods for Solvng Nonlnear Equatons, Elsever, Amsterdam, Boston, Hedelberg, 203 [7] MS Petkovć, LD Petkovć, Famles of optmal multpont methods for solvng nonlnear equatons: a survey, Appl Anal Dscrete Math 4 (200) 22 [8] WH B, QB Wu, HM Ren, Convergence ball and error analyss of Ostrowsk Traub s method, Appl Math J Chnese Unv 25 (200) 374 378 [9] MS Petkovć, Pont Estmaton of Root Fndng Methods, Sprnger, Berln, Hedelberg, 2008 [0] S Smale, The fundamental theorem of algebra and complexty theory, Bull Amer Math Soc 4 (98) 35 [] S Smale, Newton s method estmates from data at one pont, n: RE Ewng, KI Gross, CF Martn (Eds), The Mergng Dscplnes: New Drectons n Pure, Appled and Computatonal Mathematcs, Sprnger-Verlag, New York, 986, pp 85 96 [2] MS Petkovć, Iteratve Methods for Smultaneous Incluson of Polynomal Zeros, Sprnger-Verlag, Berln, Hedelberg, New York, 989 [3] JM McNamee, Numercal Methods for Roots of Polynomals, Part I, Elsever, Amsterdam, 2007 [4] D Wang, Y Wu, Some modfcatons of the parallel teraton method and ther convergence, Computng 38 (987) 75 87 [5] MS Petkovć, L Rančć, M Mloševć, The mproved Farmer Lozou method for fndng polynomal zeros, Int J Comput Math 89 (202) 499 509 [6] R Brent, P Zmmermann, Modern Computer Arthmetc, Cambrdge Unversty Press, Cambrdge, 20 [7] J Fujmoto, T Ishkawa, D Perret-Gallx, Hgh Precson Numercal Computatons, Techncal Report, ACCP-N-, 2005