Journal of Applied Mathematics

Volume 2012 (2012), Article ID 709832, 11 pages

doi:10.1155/2012/709832

Research Article

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

Mansor Monsi,¹ Nasruddin Hassan,² and Syaida Fadhilah Rusli¹

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.

Abstract.

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 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 () to converge to the zeros (), respectively, as shown subsequently in the sequel. We also give attractive features of PZSS1 in Section 3.

2. Methods of Estimating polynomial zeros

Let be a polynomial of degree defined by

() and . Let be the distinct zeros of , expressed in the form:

with . Suppose that, for is an estimate of , and let be defined by

Then,

By (2.2), if for , (), then

Now, () so by (2.5),

An iteration procedure PT1 of (2.6) is defined by

which has been studied by Kerner [7]. Furthermore, the following procedure PS1

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 by

The following definitions and theorem (Alefeld and Herzberger [12], Ortega and Rheinboldt [13]) are very useful for evaluation of -order of convergence of an iterative procedure I.

Definition 2.1. If there exists a such that for any null sequence generated from then the R-factor of the sequence is defined to be

where is independent of the norm .

Definition 2.2. We next define the R-order of the procedure in terms of the R-factor as

Suppose that then it follows from Ortega and Rheinboldt [13] that the R-order of satisfies the inequality .

Theorem 2.3. Let be an iterative procedure and let be the set of all sequences generated by which converges to the limit . Suppose that there exists a and a constant such that for any ,

Then, it follows that - of satisfies the inequality .

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 , while the R-order of convergence of (2.8) is greater than 2 or . However, the R-order of convergence of PSS1 is at least 3 or .

3. The Point Zoro Symmetric Single-Step Procedure PZSS1

The value of which is computed from (3.1c) requires multiplications, one division, and subtractions, increasing the lower bound on the R-order by unity compared with the R-order of PS1. Furthermore, the value of which is computed from (3.1d) requires multiplications, one division, and 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 () from

The procedure PZSS1 has the following attractive features.

From (3.1b), (3.1c), and (3.1d), it follows that for ,

(i)the values () which are computed for use in (3.1b) are reused in (3.1c) and (3.1d).(ii) and , so that and need not be computed.(iii)The product

which are computed for use in (3.1b) are reused in (3.1c).(iv)The product

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) is defined by (2.1); (ii) is defined by

(iii) is defined by

where and (); (iv) is defined by

then

Lemma 3.2. If (i)–(iv) of Lemma 3.1 are valid; (v) () are such that (), (), (), and

  , , and (), then

where

Lemma 3.3. If (i)–(v) of Lemma 3.2 are valid; (vi) and (), where and , then ().

Theorem 3.4. If (i) defined by (2.1) has distinct zeros (); (ii) (), where and , and the sequences () are generated from PZSS1 (i.e., from (3.1a)–(3.1e)), then () () and .

Proof. For, let

Then, by (3.5) and (3.6),

where is defined by (3.4).

By Lemmas 3.1 and 3.2 with , , , , (), it follows that, for , ,

where

Similarly, by Lemmas 3.1 and 3.2, with , , , , (), it follows that, for , ,

where

Similarly, by Lemma 3.1 and Lemma 3.2, with , , , , (), it follows that, for , ,

where

It follows from (3.13)-(3.14) and Lemma 3.3 that (), and it follows from (3.15)-(3.16) and Lemma 3.3 that () follows from (3.1e). It follows from (3.17)-(3.18) and Lemma 3.3 that

whence (). It then follows by induction on that, for all ,

whence , (). Let

Then, by (3.13), (3.15), (3.17), and (3.21), for , (recall (3.1b), (3.1c), (3.1d))

for ,

and for ,

Let

For let

Then, by (3.25)–(3.26), for ,

Suppose, without loss of generality, that

Then, by a lengthy inductive argument, it follows from (3.21)–(3.29) that for , for all ,

whence, by (3.28) and (3.1e), for all ,

By (3.21) for ,

then by (3.1e),

So,

Let

Then, by (3.22)–(3.35)

So,

Therefore (Ortega and Rheindboldt [13]),

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.

Acknowledgment

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

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