Research Article The Point Zoro Symmetric Single-Step Procedure for Simultaneous Estimation of Polynomial Zeros

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1 Aled Mathematcs Volume 2012, Artcle ID , 11 ages do: /2012/ Research Artcle The Pont Zoro Symmetrc Sngle-Ste Procedure for Smultaneous Estmaton of Polynomal Zeros Mansor Mons, 1 Nasruddn Hassan, 2 and Syada Fadhlah Rusl 1 1 Deartment of Mathematcs, Faculty of Scence, Unverst Putra Malaysa, UPM Serdang, Selangor, Malaysa 2 School of Mathematcal Scences, Faculty of Scence and Technology, Unverst Kebangsaan Malaysa, Bang, Selangor, Malaysa Corresondence should be addressed to Nasruddn Hassan, nas@ukm.my Receved 15 November 2011; Revsed 29 February 2012; Acceted 20 March 2012 Academc Edtor: Martn Weser Coyrght q 2012 Mansor Mons et al. Ths s an oen access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch ermts unrestrcted use, dstrbuton, and reroducton n any medum, rovded the orgnal work s roerly cted. The ont symmetrc sngle ste rocedure PSS1 has R-order of convergence at least 3. Ths rocedure s modfed by addng another sngle-ste, whch s the thrd ste n PSS1. Ths modfed rocedure s called the ont zoro symmetrc sngle-ste PZSS1. It s roven that the R-order of convergence of PZSS1 s at least 4 whch s hgher than the R-order of convergence of PT1, PS1, and PSS1. Hence, comutatonal tme s reduced snce ths rocedure s more effcent for boundng smle zeros smultaneously. 1. Introducton The teratve rocedures for estmatng smultaneously the zeros of a olynomal of degree n were dscussed, for examle, n Ehrlch 1, Aberth 2, Alefeld and Herzberger 3, Farmer and Lozou 4, Mlovanovć and Petkovć 5 and Petkovć and Stefanovć 6. In ths aer, we refer to the methods establshed by Kerner 7, Alefeld and Herzberger 3, Mons,and Wolfe 8,Mons9 and Rusl et al. 10 to ncrease the rate of convergence of the ont zoro symmetrc sngle-ste method PZSS1. The convergence analyss of ths rocedure s gven n Secton 3. Ths rocedure needs some recondtons for ntal onts x 0 1,...,n to converge to the zeros x 1,...,n, resectvely, as shown subsequently n the sequel. We also gve attractve features of PZSS1 n Secton 3.

2 2 Aled Mathematcs 2. Methods of Estmatng olynomal zeros Let : C C be a olynomal of degree n defned by x a x, a C 1,...,n and a n / 0. Let x x 1,x 2,...,x n T be the dstnct zeros of x 0, exressed n the form: x n x x 0, wth a n 1. Suose that, for 1,...,n, x s an estmate of x,andletq : C C be defned by qx n x x. 2.3 Then, q x n x x, 1,...,n. 2.4 / By 2.2, ffor 1,...,n, x / x 1,...,n; /, then x x / x x x. 2.5 Now, x x 1,...,n so by 2.5, x x / x 1,...,n. 2.6 x x An teraton rocedure PT1 of 2.6 s defned by 1 / 1,...,nk 0, 2.7

3 Aled Mathematcs 3 whch has been studed by Kerner 7. Furthermore, the followng rocedure PS n has been studed by Alefeled and Herzberger 3. The symmetrc sngle-ste dea of Atken 11 and the rocedure PS1 of Alefeld and Herzberger 3 are used to derve the ont symmetrc sngle-ste rocedure PSS1Mons 9. The rocedure PSS1 s defned by 1 1 1,...,n, n 1 n 1 1,...,n, 1,...,n, ,...,n. The followng defntons and theorem Alefeld and Herzberger 12, Ortega and Rhenboldt 13 are very useful for evaluaton of R-order of convergence of an teratve rocedure I. Defnton 2.1. If there exsts a 1 such that for any null sequence {w k } generated from { }, then the R-factor of the sequence {w k } s defned to be R w k lm su w k 1/k, 1, k lm su w k /k, > 1, k where R s ndeendent of the norm. Defnton 2.2. We next defne the R-order of the rocedure I n terms of the R-factor as f R I,x 0, for 1, O R I,x nf { 1,,R I,x 1 }, otherwse Suose that R w k < 1, then t follows from Ortega and Rhenboldt 13 that the R-order of I satsfes the nequalty O R I,x.

4 4 Aled Mathematcs Theorem 2.3. Let I be an teratve rocedure and let ΩI,x be the set of all sequences { } generated by I whch converges to the lmt x. Suose that there exsts a 1 and a constant γ such that for any { } ΩI,x, 1 x γ x, k k0 k 0 {} Then, t follows that R-order of I satsfes the nequalty O R I,x. We wll use ths result n order to calculate the R-order of convergence of PZSS1 n the subsequent secton. For comarson, the rocedure 2.7 has R-order of convergence at least 2 or O R PT1,x 2, whle the R-order of convergence of 2.8 s greater than 2 or O R PS1,x > 2. However, the R-order of convergence of PSS1 s at least 3 or O R PSS1,x The Pont Zoro Symmetrc Sngle-Ste Procedure PZSS1 The value of whch s comuted from 3.1c requres n multlcatons, one dvson, and n1 subtractons, ncreasng the lower bound on the R-order by unty comared wth the R-order of PS1. Furthermore, the value of x k,3 whch s comuted from 3.1d requres n multlcatons, one dvson, and n 1 subtractons, ncreasng the lower bound on the R-order by unty comared wth the R-order of PSS1. Ths observaton gves rse to the dea that t mght be advantageous to add another ste n PSS1. Ths leads to what s so called the ont zoro symmetrc sngle-ste rocedure PZSS1 whch conssts of generatng the sequences { } 1,...,n from x k, ,...,n, 3.1a n 1 n 1 x k,3 n 1 1,...,n, 3.1b 1,...,n, 3.1c 1,...,n, 3.1d x k,3 1,...,nk e The rocedure PZSS1 has the followng attractve features. From 3.1b, 3.1c,and3.1d, t follows that for k 0, the values 1,...,n whch are comuted for use n 3.1b are reused n 3.1c and 3.1d. n n and x k,3 1 1,sothat n and x k,3 1 need not be comuted.

5 Aled Mathematcs 5 The roduct 1 2,...,n, 3.2 whch are comuted for use n 3.1b are reused n 3.1c. v The roduct n 1 n 1,...,1 3.3 whch are comuted for use n 3.1c are reused n 3.1d. The followng lemmas Mons 9 are requred n the roof of Theorem 3.4. Lemma 3.1. If : C C s defned by 2.1; : C C s defned by 1 n x x xm x xm 1,...,n; 3.4 m1 m1 q : C C s defned by 1 n q x x x m x x m 1,...,n, 3.5 m1 m1 where x / x m and x / x m v φ : C C s defned by, m 1,...,n; / m; 1 φ x q x q x q x x x x 1q x q x 1,...,n, 3.6 x x x then φ x x x C 1,...,n. 3.7 Lemma 3.2. If v of Lemma 3.1 are vald; v ˇx 1,...,n are such that ˇx / 0 1,...,n, ˇx / x m m 1,..., 1, ˇx / x m m 1,...,n,and ˇx x ˇx 1 m1 ˇx x m n m1 ˇx x m 1,...,n; 3.8

6 6 Aled Mathematcs ˇw ˇx x, ŵ x x,andw x x 1,...,n, then 1 w ˇw γ w γ ŵ 1 1,...,n, 3.9 where m /, x xm γ q x x ˇx m /, x xm γ x x ˇx q 1,..., 1, 1,...,n Lemma 3.3. If v of Lemma 3.2 are vald; v ˇx x θd/2n 1 and x x θd/2n 1 1,...,n, where d mn{ x x, 1,...n; / } and 0 <θ<1, then w θ ˇw 1,...,n. Theorem 3.4. If : C C defned by 2.1 has n dstnct zeros x 1,...,n; x 0 x θd/2n 1 1,...,n, where 0 <θ<1andd mn{ x x, 1,..., n : / }, and the sequences { } 1,...,n are generated from PZSS1.e., from 3.1a 3.1e, then x k 1,...,n and O R PZSS1,x 4. Proof. For 1,...,n,let Then, by 3.5 and 3.6, 1 φ 1, x q 1, x 1 n q 1, x x m m1 m1 1 n q 2, x x m m1 m1 1 n q 3, x x x k,3 m q 1, 1 φ 2, x q 2, x q 2, 1 φ 3, x q 3, x q 3, m1 q 1, x x q 2, x x x k,3 x k,3 q 3, x x x k,3 m1 x m x m x m 1 q 1, 1 q 2, 1q 3,,,. q 1, x x q 2, x x q 3, x x,,, where x s defned by 3.4.

7 Aled Mathematcs 7 By Lemmas 3.1 and 3.2 wth q q 1,,ˇx, x, x, φ φ 1, 1,..., n, t follows that, for 1,...,n, k 0, w k,1 w k 1 α k,1 w k,1 α k,0 w k,0 1, 3.13 where α k,1 α k,0 w k,s q 1, q 1, x k,s x s 0,...,3, xm 1,..., 1, m /, m /, x m 1,...,n Smlarly, by Lemmas 3.1 and 3.2, wthq q 2,, ˇx, x, x, φ φ 2, 1,...,n, t follows that, for 1,...,n, k 0, w k,2 w k 1 β k,1 w k,1 β k,2 w k,2 1, 3.15 where β k,1 β k,2 q 2, q 2, m /, m /, xm 1,..., 1, xm 1,...,n Smlarly, by Lemma 3.1 and Lemma 3.2,wthq q 3,,ˇx, x, x x k,3, φ φ 3, 1,...,n, t follows that, for 1,...,n, k 0, w k,3 w k 1 γ k,3 w k,3 γ k,2 w k,2 1, 3.17

8 8 Aled Mathematcs where γ k,3 γ k,2 q 3, q 3, m /, x k,3 m /, x k,3 x k,3 xm 1,..., 1, x m 1,...,n It follows from and Lemma 3.3 that w 0,1 from and Lemma 3.3 that w 0,2 θ 2 w 0,0 follows from and Lemma 3.3 that w 0,3 θ w 0,0 1,...,n, and t follows 1,...,n follows from 3.1e. It θ 3 0,0 w 1,...,n, 3.19 whence w 1,0 θ 3 w 0,0 1,...,n. It then follows by nducton on k that, for all k 0, w k,0 θ 4k 1 0,0 w 1,...,n, 3.20 whence x k, 1,...,n. Let h k,m 2n 1 d w k,m 1,...,nm 0,..., Then, by 3.13, 3.15, 3.17, and3.21, for 1,...,n, recall 3.1b, 3.1c, 3.1d h k,1 1 1 n 1 hk,0 h k,1 h k,0 1, 3.22 for n,...,1, h k,2 1 1 n 1 hk,0 h k,1 h k,2 1, 3.23 and for 1,...,n, h k,3 1 1 n 1 hk,0 h k,3 h k,

9 Aled Mathematcs 9 Let u 1,1 u 1,2 u 1,3 2, 1,...,n 1, 3, n, 4, 1, 3, 2,...,n, 4, 1,...,n 1, 5, n For r 1, 2, 3, let u k1,r 4u k,r, 1,...,n 1, 4u k,r 1, n Then, by , for all k 1, u k,1 u k,2 u k,3 Suose, wthout loss of generalty, that 2 4 k1, 1,...,n 1, 10 4 k1 1, 3 3 n, 4 4 k1, 1, 3 4 k1, 2,...,n k1 1, n, k1, 1,...,n 1, 16 4 k , n. h 0,0 h<1 1,...,n Then, by a lengthy nductve argument, t follows from that for 1,...,n,for all k 1, h k,1 h uk1,1, h k,2 h uk1,2, 3.30 h k,3 h uk1,3,

10 10 Aled Mathematcs whence, by 3.28 and 3.1e, for all k 1, By 3.21 for m 3, then by 3.1e, So, w k,3 w k1 w k h k h 4k 1,...,n d 2n 1 hk,3 1,...,n, 3.32 d 2n 1 hk1 1,...,n d 2n 1 hk 1,...,nk Let { } w k w k max 1 n, { } h k k max h. 1 n 3.35 Then, by w k d 2n 1 h4k k So, R 4 w k { lm su w k } { 1/4 k d 1/4 k } lm su h h<1. k k 2n Therefore Ortega and Rhendboldt 13, O R PZSS1,x 4 1,...,n Concluson The result above shows that the rocedure PZSS1 has R-order of convergence at least 4 that s hgher than does PT1, PS1, and PSS1. The attractve features gven n Secton 3 of ths

11 Aled Mathematcs 11 rocedure wll gve less comutatonal tme. Our exerences n the mlementaton of the nterval verson of PZSS1, that s, the rocedure IZSS1Rusl et al. 10 showed that ths rocedure s more effcent for boundng the zeros smultaneously. Acknowledgment The authors are ndebted to Unverst Kebangsaan Malaysa for fundng ths research under the grant UKM-GUP References 1 L. W. Ehrlch, A modfed Newton method for olynomals, Communcatons of the ACM, vol. 10, , O. Aberth, Iteraton methods for fndng all zeros of a olynomal smultaneously, Mathematcs of Comutaton, vol. 27, , G. Alefeld and J. Herzberger, On the convergence seed of some algorthms for the smultaneous aroxmaton of olynomal roots, SIAM Journal on Numercal Analyss, vol. 11, , M. R. Farmer and G. Lozou, A class of teraton functons for mrovng, smultaneously, aroxmatons to the zeros of a olynomal, vol. 15, no. 3, , G. V. Mlovanovć and M. S. Petkovć, On the convergence order of a modfed method for smultaneous fndng olynomal zeros, Comutng, vol. 30, no. 2, , M. S. Petkovć and L. V. Stefanovć, On a second order method for the smultaneous ncluson of olynomal comlex zeros n rectangular arthmetc, Comutng. Archves for Scentfc Comutng, vol. 36, no. 3, , O. Kerner, Total ste rocedure for the calculaton of the zeros of olynomal, Numersche Mathematk, vol. 8, , M. Mons and M. A. Wolfe, Interval versons of some rocedures for the smultaneous estmaton of comlex olynomal zeros, Aled Mathematcs and Comutaton, vol. 28, no. 3, , M. Mons, The ont symmetrc sngle-ste PSS1 rocedure for smultaneous aroxmaton of olynomal zeros, Malaysan Mathematcal Scences, vol. 6, no. 1, , S. F. Rusl, M. Mons, M. A. Hassan, and W. J. Leong, On the nterval zoro symmetrc sngle-ste rocedure for smultaneous fndng of real olynomal zeros, Aled Mathematcal Scences, vol. 5, no. 75, , A. C. Atken, Studes n ractcal mathematcs. V. On the teratve soluton of a system of lnear equatons, Proceedngs of the Royal Socety of Ednburgh A, vol. 63, , G. Alefeld and J. Herzberger, Introducton to Interval Comutatons, Comuter Scence and Aled Mathematcs, Academc Press, New York, NY, USA, J. M. Ortega and W. C. Rhenboldt, Iteratve Soluton of Nonlnear Equatons n Several Varables, Academc Press, New York, NY, USA, 1970.

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