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Journal of Applied Mathematics
Volume 2012, Article ID 709832, 11 pages
Research Article

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

1Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia
2School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Received 15 November 2011; Revised 29 February 2012; Accepted 20 March 2012

Academic Editor: Martin Weiser

Copyright © 2012 Mansor Monsi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The point symmetric single step procedure PSS1 has R-order of convergence at least 3. This procedure is modified by adding another single-step, which is the third step in PSS1. This modified procedure is called the point zoro symmetric single-step PZSS1. It is proven that the R-order of convergence of PZSS1 is at least 4 which is higher than the R-order of convergence of PT1, PS1, and PSS1. Hence, computational time is reduced since this procedure is more efficient for bounding simple zeros simultaneously.

1. Introduction

The iterative procedures for estimating simultaneously the zeros of a polynomial of degree 𝑛 were discussed, for example, in Ehrlich [1], Aberth [2], Alefeld and Herzberger [3], Farmer and Loizou [4], Milovanović and Petković [5] and Petković and Stefanović [6]. In this paper, we refer to the methods established by Kerner [7], Alefeld and Herzberger [3], Monsi, and Wolfe [8], Monsi [9] and Rusli et al. [10] to increase the rate of convergence of the point zoro symmetric single-step method PZSS1. The convergence analysis of this procedure is given in Section 3. This procedure needs some preconditions for initial points 𝑥𝑖(0) (𝑖=1,,𝑛) to converge to the zeros 𝑥𝑖 (𝑖=1,,𝑛), respectively, as shown subsequently in the sequel. We also give attractive features of PZSS1 in Section 3.

2. Methods of Estimating polynomial zeros

Let 𝑝𝐶𝐶 be a polynomial of degree 𝑛 defined by𝑝(𝑥)=𝑛𝑖=0𝑎𝑖𝑥𝑖,(2.1)𝑎𝑖𝐶 (𝑖=1,,𝑛) and 𝑎𝑛0. Let 𝑥=(𝑥1,𝑥2,,𝑥𝑛)𝑇 be the distinct zeros of 𝑝(𝑥)=0, expressed in the form:𝑝(𝑥)=𝑛𝑖=1𝑥𝑥𝑖=0,(2.2) with 𝑎𝑛=1. Suppose that, for 𝑗=1,,𝑛,𝑥𝑗 is an estimate of 𝑥𝑗, and let 𝑞𝐶𝐶 be defined by𝑞(𝑥)=𝑛𝑗=1𝑥𝑥𝑗.(2.3) Then, 𝑞𝑥𝑖=𝑛𝑗𝑖𝑥𝑖𝑥𝑗,(𝑖=1,,𝑛).(2.4) By (2.2), if for 𝑖=1,,𝑛, 𝑥𝑖𝑥𝑗 (𝑗=1,,𝑛;𝑗𝑖), then𝑥𝑖=𝑥𝑖𝑝𝑥𝑖𝑗𝑖𝑥𝑖𝑥𝑗.(2.5) Now, 𝑥𝑗𝑥𝑗 (𝑗=1,,𝑛) so by (2.5),𝑥𝑖𝑥𝑖𝑝𝑥𝑖𝑗𝑖𝑥𝑖𝑥𝑗(𝑖=1,,𝑛).(2.6) An iteration procedure PT1 of (2.6) is defined by 𝑥𝑖(𝑘+1)=𝑥𝑖(𝑘)𝑝𝑥𝑖(𝑘)𝑗𝑖𝑥𝑖(𝑘)𝑥𝑗(𝑘)(𝑖=1,,𝑛)(𝑘0),(2.7) which has been studied by Kerner [7]. Furthermore, the following procedure PS1𝑥𝑖(𝑘+1)=𝑥𝑖(𝑘)𝑝𝑥𝑖(𝑘)𝑖1𝑗=1𝑥𝑖(𝑘)𝑥𝑗(𝑘+1)𝑛𝑗=𝑖+1𝑥𝑖(𝑘)𝑥𝑗(𝑘)(2.8) has been studied by Alefeled and Herzberger [3].

The symmetric single-step idea of Aitken [11] and the procedure PS1 of Alefeld and Herzberger [3] are used to derive the point symmetric single-step procedure PSS1(Monsi [9]). The procedure PSS1 is defined by𝑥𝑖(𝑘,0)=𝑥𝑖(𝑘)𝑥(𝑖=1,,𝑛),𝑖(𝑘,1)=𝑥𝑖(𝑘)𝑝𝑥𝑖(𝑘)𝑖1𝑗=1𝑥𝑖(𝑘)𝑥𝑗(𝑘,1)𝑛𝑗=𝑖+1𝑥𝑖(𝑘)𝑥𝑗(𝑘,0)𝑥(𝑖=1,,𝑛),𝑖(𝑘,2)=𝑥𝑖(𝑘)𝑝𝑥𝑖(𝑘)𝑖1𝑗=1𝑥𝑖(𝑘)𝑥𝑗(𝑘,1)𝑛𝑗=𝑖+1𝑥𝑖(𝑘)𝑥𝑗(𝑘,2)𝑥(𝑖=1,,𝑛),𝑖(𝑘+1)=𝑥𝑖(𝑘,2)(𝑖=1,,𝑛).(2.9) The following definitions and theorem (Alefeld and Herzberger [12], Ortega and Rheinboldt [13]) are very useful for evaluation of 𝑅-order of convergence of an iterative procedure I.

Definition 2.1. If there exists a 𝑝1 such that for any null sequence {𝑤(𝑘)}generated from {𝑥(𝑘)},then the R-factor of the sequence {𝑤(𝑘)}is defined to be 𝑅𝑝𝑤(𝑘)=lim𝑘𝑤sup(𝑘)1/𝑘,𝑝=1,lim𝑘𝑤sup(𝑘)1/𝑝𝑘,𝑝>1,(2.10) where 𝑅𝑝is independent of the norm .

Definition 2.2. We next define the R-order of the procedure 𝐼 in terms of the R-factor as 𝑂𝑅𝐼,𝑥=+if𝑅𝑝𝐼,𝑥[=0,for𝑝1,inf𝑝𝑝1,),𝑅𝑝𝐼,𝑥=1,otherwise.(2.11) Suppose that 𝑅𝑝(𝑤(𝑘))<1,then it follows from Ortega and Rheinboldt [13] that the R-order of 𝐼 satisfies the inequality 𝑂𝑅(𝐼,𝑥)𝑝.

Theorem 2.3. Let 𝐼 be an iterative procedure and let Ω(𝐼,𝑥) be the set of all sequences {𝑥(𝑘)}generated by 𝐼 which converges to the limit 𝑥. Suppose that there exists a 𝑝1 and a constant 𝛾 such that for any {𝑥(𝑘)}Ω(𝐼,𝑥), 𝑥(𝑘+1)𝑥𝑥𝛾(𝑘)𝑥𝑝,𝑘𝑘0=𝑘0𝑥(𝑘).(2.12) Then, it follows that 𝑅-order of 𝐼 satisfies the inequality 𝑂𝑅(𝐼,𝑥)𝑝.

We will use this result in order to calculate the R-order of convergence of PZSS1 in the subsequent section.

For comparison, the procedure (2.7) has R-order of convergence at least 2 or 𝑂𝑅(PT1,𝑥)2, while the R-order of convergence of (2.8) is greater than 2 or 𝑂𝑅(PS1,𝑥)>2. However, the R-order of convergence of PSS1 is at least 3 or 𝑂𝑅(PSS1,𝑥)3.

3. The Point Zoro Symmetric Single-Step Procedure PZSS1

The value of 𝑥𝑖(𝑘,2) which is computed from (3.1c) requires (𝑛𝑖) multiplications, one division, and (𝑛𝑖+1) subtractions, increasing the lower bound on the R-order by unity compared with the R-order of PS1. Furthermore, the value of 𝑥𝑖(𝑘,3) which is computed from (3.1d) requires (𝑛𝑖) multiplications, one division, and (𝑛𝑖+1) subtractions, increasing the lower bound on the R-order by unity compared with the R-order of PSS1. This observation gives rise to the idea that it might be advantageous to add another step in PSS1. This leads to what is so called the point zoro symmetric single-step procedure PZSS1 which consists of generating the sequences {𝑥𝑖(𝑘)} (𝑖=1,,𝑛) from𝑥𝑖(𝑘,0)=𝑥𝑖(𝑘)𝑥(𝑖=1,,𝑛),(3.1a)𝑖(𝑘,1)=𝑥𝑖(𝑘)𝑝𝑥𝑖(𝑘)𝑖1𝑗=1𝑥𝑖(𝑘)𝑥𝑗(𝑘,1)𝑛𝑗=𝑖+1𝑥𝑖(𝑘)𝑥𝑗(𝑘,0)𝑥(𝑖=1,,𝑛),(3.1b)𝑖(𝑘,2)=𝑥𝑖(𝑘)𝑝𝑥𝑖(𝑘)𝑖1𝑗=1𝑥𝑖(𝑘)𝑥𝑗(𝑘,1)𝑛𝑗=𝑖+1𝑥𝑖(𝑘)𝑥𝑗(𝑘,2)𝑥(𝑖=1,,𝑛),(3.1c)𝑖(𝑘,3)=𝑥𝑖(𝑘)𝑝𝑥𝑖(𝑘)𝑖1𝑗=1𝑥𝑖(𝑘)𝑥𝑗(𝑘,3)𝑛𝑗=𝑖+1𝑥𝑖(𝑘)𝑥𝑗(𝑘,2)(𝑥𝑖=1,,𝑛),(3.1d)𝑖(𝑘+1)=𝑥𝑖(𝑘,3)(𝑖=1,,𝑛)(𝑘0).(3.1e)

The procedure PZSS1 has the following attractive features.

From (3.1b), (3.1c), and (3.1d), it follows that for 𝑘0, (i)the values 𝑝(𝑥𝑖(𝑘)) (𝑖=1,,𝑛) which are computed for use in (3.1b) are reused in (3.1c) and (3.1d).(ii)𝑥𝑛(𝑘,2)=𝑥𝑛(𝑘,1) and 𝑥1(𝑘,3)=𝑥1(𝑘,2), so that 𝑥𝑛(𝑘,2) and 𝑥1(𝑘,3) need not be computed.(iii)The product 𝑖1𝑗=1𝑥𝑖(𝑘)𝑥𝑗(𝑘,1)(𝑖=2,,𝑛),(3.2) which are computed for use in (3.1b) are reused in (3.1c).(iv)The product 𝑛𝑗=𝑖+1𝑥𝑖(𝑘)𝑥𝑗(𝑘,2)(𝑖=𝑛1,,1)(3.3) which are computed for use in (3.1c) are reused in (3.1d).

The following lemmas (Monsi [9]) are required in the proof of Theorem 3.4.

Lemma 3.1. If (i)𝑝𝐶𝐶 is defined by (2.1); (ii)𝑝𝑖𝐶𝐶 is defined by 𝑝𝑖(𝑥)=𝑖1𝑚=1𝑥𝑥𝑚𝑛𝑚=𝑖+1𝑥𝑥𝑚(𝑖=1,,𝑛);(3.4)(iii)𝑞𝑖𝐶𝐶 is defined by 𝑞𝑖(𝑥)=𝑖1𝑚=1𝑥𝑥𝑚𝑛𝑚=𝑖+1𝑥̂𝑥𝑚(𝑖=1,,𝑛),(3.5) where 𝑥𝑗𝑥𝑚 and ̂𝑥𝑗̂𝑥𝑚 (𝑗,𝑚=1,,𝑛;𝑗𝑚); (iv)𝜙𝑖𝐶𝐶 is defined by 𝜙𝑖(𝑥)=𝑞𝑖(𝑥)+𝑖1𝑗=1𝑝𝑖𝑥𝑗𝑞𝑖(𝑥)𝑞𝑖𝑥𝑗𝑥𝑥𝑗+𝑛𝑗=𝑖+1𝑝𝑖̂𝑥𝑗𝑞𝑖(𝑥)𝑞𝑖̂𝑥𝑗𝑥̂𝑥𝑗(𝑖=1,,𝑛),(3.6) then 𝜙𝑖(𝑥)=𝑝𝑖(𝑥)(𝑥𝐶)(𝑖=1,,𝑛).(3.7)

Lemma 3.2. If (i)–(iv) of Lemma 3.1 are valid; (v) 𝑥𝑖 (𝑖=1,,𝑛) are such that 𝑝(𝑥𝑖)0 (𝑖=1,,𝑛), 𝑥𝑖𝑥𝑚 (𝑚=1,,𝑖1), 𝑥𝑖̂𝑥𝑚 (𝑚=𝑖+1,,𝑛), and 𝑥𝑖=𝑥𝑖𝑝𝑥𝑖𝑖1𝑚=1𝑥𝑖𝑥𝑚𝑛𝑚=𝑖+1𝑥𝑖̂𝑥𝑚(𝑖=1,,𝑛);(3.8)   𝑤𝑖=𝑥𝑖𝑥𝑖, 𝑤𝑖=̂𝑥𝑖𝑥𝑖, and 𝑤𝑖=𝑥𝑖𝑥𝑖 (𝑖=1,,𝑛), then 𝑤𝑖=𝑤𝑖𝑖1𝑗=1𝛾𝑖𝑗𝑤𝑗+𝑛𝑗=𝑖+1̂𝛾𝑖𝑗𝑤𝑗(𝑖=1,,𝑛),(3.9) where 𝛾𝑖𝑗=𝑚𝑖,𝑗𝑥𝑗𝑥𝑚𝑞𝑖𝑥𝑗𝑥𝑗𝑥𝑖(𝑗=1,,𝑖1),̂𝛾𝑖𝑗=𝑚𝑖,𝑗̂𝑥𝑗𝑥𝑚𝑞𝑖̂𝑥𝑗̂𝑥𝑗𝑥𝑖(𝑗=𝑖+1,,𝑛).(3.10)

Lemma 3.3. If (i)–(v) of Lemma 3.2 are valid; (vi) |𝑥𝑖𝑥𝑖|𝜃𝑑/(2𝑛1) and |̂𝑥𝑖𝑥𝑖|𝜃𝑑/(2𝑛1) (𝑖=1,,𝑛), where 𝑑=min{|𝑥𝑖𝑥𝑗|𝑖,𝑗=1,𝑛;𝑗𝑖} and 0<𝜃<1, then |𝑤𝑖𝑤|𝜃|𝑖| (𝑖=1,,𝑛).

Theorem 3.4. If (i) 𝑝𝐶𝐶 defined by (2.1) has 𝑛 distinct zeros 𝑥𝑖 (𝑖=1,,𝑛); (ii) |𝑥𝑖(0)𝑥𝑖|𝜃𝑑/(2𝑛1) (𝑖=1,,𝑛), where 0<𝜃<1 and 𝑑=min{|𝑥𝑖𝑥𝑗|𝑖,𝑗=1,,𝑛𝑗𝑖}, and the sequences {𝑥𝑖(𝑘)} (𝑖=1,,𝑛) are generated from PZSS1 (i.e., from (3.1a)–(3.1e)), then 𝑥𝑖(𝑘)𝑥𝑖 (𝑘) (𝑖=1,,𝑛) and 𝑂𝑅(PZSS1,𝑥)4.

Proof. For𝑖=1,,𝑛, let 𝑞1,𝑖(𝑥)=𝑖1𝑚=1𝑥𝑥𝑚(𝑘,1)𝑛𝑚=𝑖+1𝑥𝑥𝑚(𝑘,0),𝑞2,𝑖(𝑥)=𝑖1𝑚=1𝑥𝑥𝑚(𝑘,1)𝑛𝑚=𝑖+1𝑥𝑥𝑚(𝑘,2),𝑞3,𝑖(𝑥)=𝑖1𝑚=1𝑥𝑥𝑚(𝑘,3)𝑛𝑚=𝑖+1𝑥𝑥𝑚(𝑘,2).(3.11)
Then, by (3.5) and (3.6), 𝜙1,𝑖(𝑥)=𝑞1,𝑖(𝑥)+𝑖1𝑗=1𝑝𝑖𝑥𝑗(𝑘,1)𝑞1,𝑖(𝑥)𝑞1,𝑖𝑥𝑗(𝑘,1)𝑥𝑥𝑗(𝑘,1)+𝑛𝑗=𝑖+1𝑝𝑖𝑥𝑗(𝑘,0)𝑞1,𝑖(𝑥)𝑞1,𝑖𝑥𝑗(𝑘,0)𝑥𝑥𝑗(𝑘,0),𝜙2,𝑖(𝑥)=𝑞2,𝑖(𝑥)+𝑖1𝑗=1𝑝𝑖𝑥𝑗(𝑘,1)𝑞2,𝑖(𝑥)𝑞2,𝑖𝑥𝑗(𝑘,1)𝑥𝑥𝑗(𝑘,1)+𝑛𝑗=𝑖+1𝑝𝑖𝑥𝑗(𝑘,2)𝑞2,𝑖(𝑥)𝑞2,𝑖𝑥𝑗(𝑘,2)𝑥𝑥𝑗(𝑘,2),𝜙3,𝑖(𝑥)=𝑞3,𝑖(𝑥)+𝑖1𝑗=1𝑝𝑖𝑥𝑗(𝑘,3)𝑞3,𝑖(𝑥)𝑞3,𝑖𝑥𝑗(𝑘,3)𝑥𝑥𝑗(𝑘,3)+𝑛𝑗=𝑖+1𝑝𝑖𝑥𝑗(𝑘,2)𝑞3,𝑖(𝑥)𝑞3,𝑖𝑥𝑗(𝑘,2)𝑥𝑥𝑗(𝑘,2),(3.12) where 𝑝𝑖(𝑥) is defined by (3.4).
By Lemmas 3.1 and 3.2 with 𝑞𝑖=𝑞1,𝑖, 𝑥𝑖=𝑥𝑖(𝑘), ̂𝑥𝑖=𝑥𝑖(𝑘,0), 𝑥𝑖=𝑥𝑖(𝑘,1), 𝜙𝑖=𝜙1,𝑖 (𝑖=1,,𝑛), it follows that, for 𝑖=1,,𝑛, 𝑘0, 𝑤𝑖(𝑘,1)=𝑤𝑖(𝑘)𝑖1𝑗=1𝛼(𝑘,1)𝑖𝑗𝑤𝑗(𝑘,1)+𝑛𝑗=𝑖+1𝛼(𝑘,0)𝑖𝑗𝑤𝑗(𝑘,0),(3.13) where 𝑤𝑖(𝑘,𝑠)=𝑥𝑖(𝑘,𝑠)𝑥𝑖𝛼(𝑠=0,,3),(𝑘,1)𝑖𝑗=𝑚𝑖,𝑗𝑥𝑗(𝑘,1)𝑥𝑚𝑞1,𝑖𝑥𝑗(𝑘,1)𝑥𝑗(𝑘,1)𝑥𝑖(𝑘)𝛼(𝑗=1,,𝑖1),(𝑘,0)𝑖𝑗=𝑚𝑖,𝑗𝑥𝑗(𝑘,0)𝑥𝑚𝑞1,𝑖𝑥𝑗(𝑘,0)𝑥𝑗(𝑘,0)𝑥𝑖(𝑘)(𝑗=𝑖+1,,𝑛).(3.14) Similarly, by Lemmas 3.1 and 3.2, with 𝑞𝑖=𝑞2,𝑖, 𝑥𝑖=𝑥𝑖(𝑘), ̂𝑥𝑖=𝑥𝑖(𝑘,2), 𝑥𝑖=𝑥𝑖(𝑘,1), 𝜙𝑖=𝜙2,𝑖 (𝑖=1,,𝑛), it follows that, for 𝑖=1,,𝑛, 𝑘0, 𝑤𝑖(𝑘,2)=𝑤𝑖(𝑘)𝑖1𝑗=1𝛽(𝑘,1)𝑖𝑗𝑤𝑗(𝑘,1)+𝑛𝑗=𝑖+1𝛽(𝑘,2)𝑖𝑗𝑤𝑗(𝑘,2),(3.15) where 𝛽(𝑘,1)𝑖𝑗=𝑚𝑖,𝑗𝑥𝑗(𝑘,1)𝑥𝑚𝑞2,𝑖𝑥𝑗(𝑘,1)𝑥𝑗(𝑘,1)𝑥𝑖(𝑘)𝛽(𝑗=1,,𝑖1),(𝑘,2)𝑖𝑗=𝑚𝑖,𝑗𝑥𝑗(𝑘,2)𝑥𝑚𝑞2,𝑖𝑥𝑗(𝑘,2)𝑥𝑗(𝑘,2)𝑥𝑖(𝑘)(𝑗=𝑖+1,,𝑛).(3.16) Similarly, by Lemma 3.1 and Lemma 3.2, with 𝑞𝑖=𝑞3,𝑖, 𝑥𝑖=𝑥𝑖(𝑘), ̂𝑥𝑖=𝑥𝑖(𝑘,2), 𝑥𝑖=𝑥𝑖(𝑘,3), 𝜙𝑖=𝜙3,𝑖 (𝑖=1,,𝑛), it follows that, for 𝑖=1,,𝑛, 𝑘0, 𝑤𝑖(𝑘,3)=𝑤𝑖(𝑘)𝑖1𝑗=1𝛾(𝑘,3)𝑖𝑗𝑤𝑗(𝑘,3)+𝑛𝑗=𝑖+1𝛾(𝑘,2)𝑖𝑗𝑤𝑗(𝑘,2),(3.17) where 𝛾(𝑘,3)𝑖𝑗=𝑚𝑖,𝑗𝑥𝑗(𝑘,3)𝑥𝑚𝑞3,𝑖𝑥𝑗(𝑘,3)𝑥𝑗(𝑘,3)𝑥𝑖(𝑘)𝛾(𝑗=1,,𝑖1),(𝑘,2)𝑖𝑗=𝑚𝑖,𝑗𝑥𝑗(𝑘,2)𝑥𝑚𝑞3,𝑖𝑥𝑗(𝑘,2)𝑥𝑗(𝑘,2)𝑥𝑖(𝑘)(𝑗=𝑖+1,,𝑛).(3.18) It follows from (3.13)-(3.14) and Lemma 3.3 that |𝑤𝑖(0,1)|𝜃|𝑤𝑖(0,0)| (𝑖=1,,𝑛), and it follows from (3.15)-(3.16) and Lemma 3.3 that |𝑤𝑖(0,2)|𝜃2|𝑤𝑖(0,0)| (𝑖=1,,𝑛) follows from (3.1e). It follows from (3.17)-(3.18) and Lemma 3.3 that ||𝑤𝑖(0,3)||𝜃3||𝑤𝑖(0,0)||(𝑖=1,,𝑛),(3.19) whence |𝑤𝑖(1,0)|𝜃3|𝑤𝑖(0,0)| (𝑖=1,,𝑛). It then follows by induction on 𝑘 that, for all 𝑘0, ||𝑤𝑖(𝑘,0)||𝜃4𝑘1||𝑤𝑖(0,0)||(𝑖=1,,𝑛),(3.20) whence 𝑥𝑖(𝑘)𝑥𝑖(𝑘), (𝑖=1,,𝑛). Let 𝑖(𝑘,𝑚)=(2𝑛1)𝑑||𝑤𝑖(𝑘,𝑚)||(𝑖=1,,𝑛)(𝑚=0,,3).(3.21) Then, by (3.13), (3.15), (3.17), and (3.21), for 𝑖=1,,𝑛, (recall (3.1b), (3.1c), (3.1d)) 𝑖(𝑘,1)1(𝑛1)𝑖(𝑘,0)𝑖1𝑗=1𝑗(𝑘,1)+𝑛𝑗=𝑖+1𝑗(𝑘,0),(3.22) for 𝑖=𝑛,,1, 𝑖(𝑘,2)1(𝑛1)𝑖(𝑘,0)𝑖1𝑗=1𝑗(𝑘,1)+𝑛𝑗=𝑖+1𝑗(𝑘,2),(3.23) and for 𝑖=1,,𝑛, 𝑖(𝑘,3)1(𝑛1)𝑖(𝑘,0)𝑖1𝑗=1𝑗(𝑘,3)+𝑛𝑗=𝑖+1𝑗(𝑘,2).(3.24) Let 𝑢𝑖(1,1)=𝑢2,(𝑖=1,,𝑛1),3,(𝑖=𝑛),𝑖(1,2)=𝑢4,(𝑖=1),3,(𝑖=2,,𝑛),𝑖(1,3)=4,(𝑖=1,,𝑛1),5,(𝑖=𝑛).(3.25) For 𝑟=1,2,3, let 𝑢𝑖(𝑘+1,𝑟)=4𝑢𝑖(𝑘,𝑟),(𝑖=1,,𝑛1),4𝑢𝑖(𝑘,𝑟)+1,(𝑖=𝑛).(3.26) Then, by (3.25)–(3.26), for all𝑘1, 𝑢𝑖(𝑘,1)=24(𝑘1),(𝑖=1,,𝑛1),1034(𝑘1)13𝑢,(𝑖=𝑛),𝑖(𝑘,2)=44(𝑘1)34,(𝑖=1),(𝑘1),(𝑖=2,,𝑛1)1034(𝑘1)13𝑢,(𝑖=𝑛),(3.27)𝑖(𝑘,3)=44(𝑘1),(𝑖=1,,𝑛1),1634(𝑘1)13,(𝑖=𝑛).(3.28) Suppose, without loss of generality, that 𝑖(0,0)<1(𝑖=1,,𝑛).(3.29) Then, by a lengthy inductive argument, it follows from (3.21)–(3.29) that for 𝑖=1,,𝑛, for all 𝑘1, 𝑖(𝑘,1)𝑢𝑖(𝑘+1,1),𝑖(𝑘,2)𝑢𝑖(𝑘+1,2),𝑖(𝑘,3)𝑢𝑖(𝑘+1,3),(3.30) whence, by (3.28) and (3.1e), for all 𝑘1, 𝑖(𝑘)4𝑘(𝑖=1,,𝑛).(3.31) By (3.21) for 𝑚=3, ||𝑤𝑖(𝑘,3)||=𝑑(2𝑛1)𝑖(𝑘,3)(𝑖=1,,𝑛),(3.32) then by (3.1e), ||𝑤𝑖(𝑘+1)||=𝑑(2𝑛1)𝑖(𝑘+1)(𝑖=1,,𝑛).(3.33) So, ||𝑤𝑖(𝑘)||=𝑑(2𝑛1)𝑖(𝑘)(𝑖=1,,𝑛)(𝑘0).(3.34) Let 𝑤(𝑘)=max1𝑖𝑛||𝑤𝑖(𝑘)||,(𝑘)=max1𝑖𝑛𝑖(𝑘).(3.35) Then, by (3.22)–(3.35) 𝑤(𝑘)𝑑(2𝑛1)4𝑘(𝑘0).(3.36) So, 𝑅4𝑤(𝑘)=lim𝑘𝑤sup(𝑘)1/4𝑘lim𝑘𝑑sup2𝑛11/4𝑘=<1.(3.37) Therefore (Ortega and Rheindboldt [13]), 𝑂𝑅PZSS1,𝑥𝑖4(𝑖=1,,𝑛).(3.38)

4. Conclusion

The result above shows that the procedure PZSS1 has R-order of convergence at least 4 that is higher than does PT1, PS1, and PSS1. The attractive features given in Section 3 of this procedure will give less computational time. Our experiences in the implementation of the interval version of PZSS1, that is, the procedure IZSS1(Rusli et al. [10]) showed that this procedure is more efficient for bounding the zeros simultaneously.


The authors are indebted to Universiti Kebangsaan Malaysia for funding this research under the grant UKM-GUP-2011-159.


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