Nov Sad J. Math. Vol. 36, No. 2, 2006, 0-09 IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUARE-ROOT INTERVAL METHOD FOR MULTIPLE ZEROS Modrag S. Petkovć 2, Dušan M. Mloševć 3 Abstract. A new theorem concerned wth the convergence of the Ostrowsk-lke method for the smultaneous ncluson of multple complex zeros n crcular complex arthmetc s establshed. Computatonally verfable ntal condton that guarantees the convergence of ths parallel ncluson method s sgnfcantly relaxed compared wth the classcal theorem stated n [Z. Angew. Math. Mech. 62 982), 627 630]. AMS Mathematcs Subject Classfcaton 2000): 65H05, 65G20, 30C5 Key words and phrases: Polynomal zeros, convergence, smultaneous methods, ncluson methods, crcular nterval arthmetc. Introducton Iteratve methods for the smultaneous ncluson of polynomal zeros, realzed n nterval arthmetc, produce resultng real or complex ntervals dsks or rectangles) that contan the sought zeros. In ths manner, nformaton about the upper error bounds of approxmatons to the zeros s provded. Besdes, there exsts the ablty to ncorporate roundng errors wthout alterng the fundamental structure of the nterval method. An extensve study and hstory of nterval methods for solvng algebrac equatons may be found n the books [9] and [0]. The purpose of ths paper s to present the mproved ntal convergence condton of the square-root ncluson method proposed n [8]. A smlar problem was consdered n the recent paper [], where the ntal condton for the convergence of the thrd-order Newton-lke ncluson method, presented n [4], s relaxed. The presentaton of the paper s organzed as follows. Some basc defntons and operatons of crcular complex nterval arthmetc, necessary for the convergence analyss and the constructon of ncluson methods, are gven n secton. The dervaton of the square-root ncluson method and the crteron for the choce of a proper square root of a dsk are presented n secton 2. The convergence analyss of the consdered method under the relaxed ntal condton s gven n secton 3. Ths research was supported by the Serban Mnstry of Scence and Envronmental Protecton under grant number 44024 2 Faculty of Electronc Engneerng, Unversty of Nš, A. Medvedeva 4, 8000 Nš, Serba, msp@juns.n.ac.yu 3 Faculty of Electronc Engneerng, Unversty of Nš, A. Medvedeva 4, 8000 Nš, Serba, dmlosev@elfak.n.ac.yu
02 M. S. Petkovć, D. M. Mloševć The constructon of the ncluson method and ts convergence analyss, presented n ths paper, need the basc propertes of the so-called crcular complex arthmetc ntroduced by Gargantn and Henrc [4]. A crcular closed regon dsk) Z : {z : z c r} wth center c : md Z and radus r : rad Z we denote by parametrc notaton Z : {c; r}. The followng s vald: α{c; r} {αc; α r} α C), {c ; r } ± {c 2 ; r 2 } {c ± c 2 ; r + r 2 }. The nverson of a non-zero dsk Z s defned by the Möbus transformaton, { ) Z c c 2 r 2 ; r } c 2 r 2 c > r,.e., 0 / Z). The addton, subtracton and nverson Z are exact operatons. The set {z z 2 : z Z, z 2 Z 2 }, n general, s not a dsk. In order to reman wthn the realm of dsks, Gargantn and Henrc [4] ntroduced the multplcaton by Z Z 2 : {c c 2 ; c r 2 + c 2 r + r r 2 } {z z 2 : z Z, z 2 Z 2 }. Then the dvson s defned by Z : Z 2 Z Z 2. The square root of a dsk {c; r} that does not contan the orgn, where c c e θ and c > r, s defned as the unon of two dsjont dsks see [2]): 2) {c; r} /2 : { c e θ/2 ; r c + c r } { c e θ/2 ; In ths paper we wll use the followng obvous propertes: r c + c r }. 3) 4) z {c; r} z c r, {c ; r } {c 2 ; r 2 } c c 2 > r + r 2. More detals about crcular arthmetc can be found n the books [], [9] and [0]. 2. Ostrowsk-lke method Let P be a monc polynomal of degree N 3 n 5) P z) z ζ j ) µj wth n N) dstnct real or complex zeros ζ,..., ζ n of respectve multplctes µ,..., µ n, where µ + + µ n N and let δ 2 z) P z P z)p z) P z.
Improvement of convergence condton of the square-root... 03 From the factorzaton of 5) we fnd δ 2 z) d2 dz 2 log P z) ) n n z ζ j µ z ζ + z ζ j. Solvng the last equaton n ζ we obtan the followng fxed-pont relaton 6) ζ z [ δ 2 z) µ z ζ j ] /2 It s assumed that only one complex value of two) of the square root has to be taken n the last formula, whch s ndcated by the symbol. Ths value s chosen n such a way that the rght-hand sde reduces to ζ. Let I n : {,..., n} be the ndex set and suppose that n dsjont dsks Z,..., Z n such that ζ j Z j j I n ) have been found. Let us put z z md Z n 6). Snce ζ j Z j j I n ), accordng to the ncluson sotoncty property we obtan. 7) ζ z [ δ 2 z ) µ z Z j ] /2 I n ). Remark. We wrte rad 0 / Z), see [6]. Z2 rather than z Z j z Z j snce rad Z Let Z 0),..., Z0) n be ntal dsjont dsks contanng the zeros ζ,..., ζ n, that s, ζ Z 0) for all I n. The relaton 7) suggests the followng method for the smultaneous ncluson of all multple zeros of P : 8) Z m+) z m) µ [ δ 2 z m) ) z m) Z m) j ] /2, I n ; m 0,,...). Assumng that the denomnator does not contan the number 0, accordng to 3) there follows that the square root of a dsk gves two dsks. Snce these dsks are dsjont, only one of them gves a crcular outer approxmaton that contans the exact zero ζ. The choce of ths proper dsk s ndcated by the symbol. The crteron for the choce of a proper dsk s smlar to that consdered n [2] and reads:
04 M. S. Petkovć, D. M. Mloševć Let [ δ 2 z m) ) z m) Z m) j ] /2 D m) m), D 2,, where D m), and D m) 2, are determned accordng to 2). Among the dsks D m), and D m) 2, one has to choose that dsk whose center mnmzes P z m) )/µ P z m) )) md D m) k, k, 2). For the teraton ndex m let us ntroduce the abbrevatons r m) max n rm), µ mn j n ρ m) mn { z m),j n j z m) j r m) j }, z m) md Z m), r m) rad Z m). For smplcty, we wll omt sometmes the teraton ndex. Remark 2. The teratve method 8) was proposed by M. Petkovć n [8]. It was proved that the order of convergence s equal four under the ntal condton 9) ρ 0) > 3N µ)r 0). The man goal of ths paper s to mprove ths condton, that s, to fnd a multpler sgnfcantly smaller than 3N µ). Remark 3. Omttng the sum n the teratve formula 8) we obtan the Ostrowsk teratve formula z m+) z m) µ [ ] /2 δ 2 z m) ) wth the cubc convergence, extensvely studed by Ostrowsk [5]. For ths reason, the ncluson method 8) s referred to as Ostrowsk-lke method. 3. Convergence analyss In ths secton we gve the convergence analyss of the nterval method 8). In the sequel we wll always assume that N 3. Lemma. Let the nequalty 0) ρ > 2 N µ r hold. Then
Improvement of convergence condton of the square-root... 05 ) ) δ 2 z ) z ζ j ) 2 z z j ) 2 2N µ)r ρ 3 ; µ j 4µ > z z j 5r 2. Proof. Of ): Snce z ζ j z z j z j ζ j z z j r j ρ, we conclude that ) z ζ j ρ and z z j ρ. Usng 0) and ) we estmate z ζ j ) 2 z z j ) 2 n r ζ j z j z z j + z ζ j ) z ζ j 2 z z j 2 r z z j z ζ j z ζ j 2 z z j 2 ) z ζ j 2 z z j + z ζ j z z j 2 z z j + z ζ j ) z ζ j 2 z z j 2 2N µ)r ρ 3. Of ): Usng the nequalty 0) and the asserton ) of Lemma we obtan δ n 2z ) µ z z j z ζ 2 z ζ j ) 2 z z j ) 2 > 4µ 5r 2. µ 2N µ)r r2 ρ 3 > µ r 2 ) 4µ N µ Now we state the convergence theorem of the Ostrowsk-lke method 8) under the relaxed ntal condton of the form 0). Theorem. Let be gven the ntal dsjont dsks Z 0),..., Z0) n contanng the zeros ζ,..., ζ n of the polynomal P and let the nterval sequences { Z m) } I n ) be defned by the teratve formula 8). Then, under the condton 2) ρ 0) > 2 N µ r 0), for each I n and m 0,,... we have
06 M. S. Petkovć, D. M. Mloševć ζ Z m) ; 2 r m+) < 8N µ) r m)) 4 5µ ρ 0) 5 3 r0)) 3. Proof. Of ): We wll prove the asserton by nducton. Suppose that ζ Z m) for I n and m. Then z m) [ δ 2 z m) ) µ ] /2 m) z ζ j ζ, where the symbol denotes the complex) value of the square root equal to µ / z m) ) ζ. Snce z m) ζ j z m) Z m) j from 8) one obtans ζ Z m+). Snce ζ Z 0), the asserton follows by mathematcal nducton. Let us prove now that the nterval method 8) has the order of convergence equal to four asserton 2 ). We use nducton and start wth m 0. For smplcty, all ndces are omtted and all quanttes n the frst teraton are denoted by. We use the followng ncluson derved n [7] 3) ) k { z Z j z z j ) k ; kr } ρ k+ Accordng to the ncluson 3) for k 2) we obtan 4) ) { 2 n z Z j, k, 2,... ). } 2N µ)r ; z z j ) 2 ρ 3 : {c ; η}. Let u δ 2 z ) c. Then, usng ), 2) and 4), from 8) we obtan 5) µ ˆr rad Ẑ rad { µ rad {u ; η} /2 ū ) µ rad {u ; η} } /2 u 2 η 2 ; η u 2 η 2 µ η u 2 η 2 ) /2 u /2 + u η) /2). ) /2
Improvement of convergence condton of the square-root... 07 Here we have used the nequalty rad ) /2 Z rad 0 / Z) proved n [6]. /2 Z By vrtue of Lemma and the nequalty 2) we estmate η 2N µ)r ρ 3 < 4 N µ r 2 < µ 5r 2 and u η > 4µ 5r 2 µ 5r 2 3µ 5r 2. Usng the last two nequaltes and Lemma ), we obtan from 5) ˆr < and, usng 2), µ 3/2 4 5 2 µn µ)r 4 5 6) ˆr < 7 r. ρ 3 ) /2 4 5 + 3 5) < 8 5 N µ µ Accordng to a geometrc constructon and the fact that the dsks Z m) and must have at least one common pont the zero ζ ), the followng relaton Z m+) can be derved see [3]): 7) ρ m+) ρ m) r m) 3r m+). Usng the nequaltes 6) and 7) for m 0), we fnd r 4 ρ 3 ρ ) ρ 0) r 0) 3r ) > 2 N µ r 0) r 0) 3 7 r0) > 7r N µ 3 ), 7 wherefrom t follows 8) ρ ) > 2 N µ r ). Ths s the condton 0) for the ndex m, whch means that all assertons of Lemma are vald for m. Usng the defnton of ρ and 8), for arbtrary par of ndces, j I n j) we have 9) z ) z ) j ρ ) > 2 N µ r ) 2r ) r ) + r ) j. Therefore, n regard to 4), the dsks Z ),..., Z) n, produced by 8), are dsjont. Applyng mathematcal nducton wth the argumentaton used for the dervaton of 6), 8) and 9) whch makes the part of the proof wth respect to
08 M. S. Petkovć, D. M. Mloševć m ), we prove that the dsks Z m),..., Z m) n are dsjont for each m 0,,..., and the followng relatons are true: 20) 2) 22) r m+) < 8N µ) r m)) 4 5µ ρ m)) 3, r m+) < 7 rm), ρ m) > 2 N µ r m). In addton, we note that the last nequalty 22) means that the assertons of Lemma hold for each m 0,, 2,.... Fnally, from 2) we conclude that the sequence {r m) } monotoncally converges to 0, n other words, the Ostrowsk-lke ncluson method 8) s convergent under the ntal condton 2). Settng ω /7 we fnd 23) + 4ω + ω 2 + ω m ) ω m < + 4ω ω 5 3. By the successve applcaton of 7) and 2) we obtan ρ m) > ρ m ) r m ) 3ωr m ) ρ m ) r m ) + 3ω) > ρ m 2) r m 2) 3ωr m 2) ωr m 2) + 3ω) ρ m 2) r m 2) + 4ω + 4ω 2 ω 2). > ρ 0) r 0) + 4ω + 4ω 2 + + 4ω m ω m ) > ρ 0) 5 3 r0), where we used 23). Accordng to the last nequalty and 20) we fnd r m+) < 8N µ) r m)) 4 5µ ρ 0) 5 3 r0)) 3. Therefore, the asserton 2 of Theorem holds. The last relaton shows that the order of convergence of the ncluson method 8) s four. We conclude ths paper wth the remark that the ntal condton 2) s sgnfcantly weakened compared to 9), see Remark 2. The rato RN, µ) 3N µ)/2 N µ ).5 N µ of multplers appearng n 9) and 2), for µ, 2, 3 and N 4, s gven n Fg..
Improvement of convergence condton of the square-root... 09 References Fg. Rato of multplers [] Alefeld, G., Herzberger, J., Introducton to Interval Computatons. New York: Academc Press 983. [2] Gargantn, I., Parallel Laguerre teratons: The complex case. Numer. Math. 26 976), 37 323. [3] Gargantn, I., Further applcaton of crcular arthmetc: Schröder-lke algorthms wth error bound for fndng zeros of polynomals. SIAM J. Numer. Anal. 5 978), 497 50. [4] Gargantn, I., Henrc, P., Crcular arthmetc and the determnaton of polynomal zeros. Numer. Math. 8 972), 305 320. [5] Ostrowsk, A., Soluton of Equatons and Systems of Equatons. New York: Academc Press 966. [6] Petkovć, Lj. D., A note on the evaluaton n crcular arthmetc. Z. Angew. Math. Mech. 66 986), 37 373. [7] Petkovć, M. S., On a generalzaton of the root teratons for polynomal complex zeros n crcular nterval arthmetc. Computng 27 98), 37 55. [8] Petkovć, M. S., Generalzed root teratons for smultaneous determnaton of multple complex zeros. Z. Angew. Math. Mech. 62 982), 627 630. [9] Petkovć, M. S., Iteratve Methods for Smultaneous Incluson of Polynomal Zeros. Berln-Hedelberg-New York: Sprnger-Verlag 989. [0] Petkovć, M. S., Petkovć, Lj. D., Complex Interval Arthmetc and ts Applcatons. Berln-Wenhem-New York: Wley-VCH 998. [] Zhu, L., On the convergent condton of Newton-lke method n parallel crcular teraton for smultaneously fndng all multple zeros of a polynomal. Appl. Math. Comp. 68 2005), 677 685. Receved by the edtors May 7, 2006