Mansor Monsi

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

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.

Modification on interval symmetric single-step procedure ISS-5δ for bounding polynomial zeros simultaneously

The purpose of this paper is to establish a new modified method. This modified procedure is called the Interval Symmetric Single Step-5 Delta Procedure ISS-5δ. This research start with some disjoints intervals as the initial intervals which contain the polynomial zeros. The procedure of ISS-5δ will generate smaller bounded close intervals. The procedure is run on 5 test polynomials and the results obtained show that this procedure is more efficient than previous procedure.

The interval zoro-symmetric single-step procedure IZSS2-5D for the simultaneous bounding of simple polynomial zeros

This paper describes the Interval Zoro-Symmetric Single-Step Procedure IZSS2-5D which is an extension of the previously proposed procedure IZSS2. It is also an extension of our procedure ISS2-5D. This procedure is developed in order to enhance the rate of convergence. Numerical results are obtained and they indicated that the IZSS2-5D outperformed the ISS2-5D in terms of computational time and number of iterations.

A modified interval symmetric single step procedure ISS-5D for simultaneous inclusion of polynomial zeros

The aim of this paper is to present a new modified interval symmetric single-step procedure ISS-5D which is the extension from the previous procedure, ISS1. The ISS-5D method will produce successively smaller intervals that are guaranteed to still contain the zeros. The efficiency of this method is measured on the CPU times and the number of iteration. The procedure is run on five test polynomials and the results obtained are shown in this paper.

The performance of the IMSS2-5D procedure for simultaneous bounding of polynomial zeros

A new modified interval midpoint symmetric single-step IMSS2-5D procedure which is an extension from the previous ISS2 procedure is formulated in this paper. This procedure is in need of some pre-conditions for the initial interval to converge to the zeros respectively, starting with some disjoint intervals, each of which contains a polynomial zero. The procedure IMSS2-5D will produce a set of intervals of smallest possible width such that each interval includes one or more zeros of the polynomial from a given initial interval. The efficiency of the procedure is measured based on the CPU times, number of iterations and the value of the intervals width after satisfying the convergence criterion. The six test polynomials are used in order to verify the procedure. The numerical results are obtained by using MATLAB. The results indicated that the IMSS2-5D procedure outperformed the existing ISS2 and ISS2-5D procedures. Therefore, this study suggests that it would be practical to use IMSS2-5D procedure for simultaneously bounding the polynomial zeros.

On modified interval repeated zoro symmetric single-step IRZSS1-5D procedure for bounding polynomial zeros simultaneously

We present a new rapidly convergence procedure called the interval repeated zoro symmetric single-step IRZSS1-5D which is an extension of the procedures IZSS1-5D and IRSS1. Theoretical analysis shows that the rate of convergence of IRZSS1-5D is at least 3r + 2 (r ≥ 1), whereas the rates of convergence of IZSS1-5D and IRSS1 are at least five and 2r + 1 (r ≥ 1), respectively. With r = 1, the procedure IRZSS1-5D is identical to IZSS1-5D. The procedures IRSS1 and IZSS1-5D are the basic references for the establishment of IRZSS1-5D, where the forward-backward-forward (FBF) step of IZSS1-5D are repeated r times in IRZSS1-5D, so that some elements of FBF of IZSS1-5D can be saved and reused for the next inner iterations r = 2,3, … in IRZSS1-5D.

The efficiency of convergence rate for IMSS2-5D procedure

A new iterative procedure is formulated in this paper known as the interval midpoint symmetric single-step IMSS2-5D procedure. In this paper, we consider this new procedure in order to describe the rate of convergence of the IMSS2-5D procedure. It is analytically proven that the IMSS2-5D procedure has a higher convergence rate than ISS2 and ISS2-5D, verifying the rate of convergence to be at least 12. Hence, computational time is reduced since this procedure is more efficient for bounding simple zeros simultaneously. Hence, it would be effective to use this procedure in determining the zeros of polynomial simultaneously.

On the performances of IMZSS2 method for bounding polynomial zeros simultaneously

This paper describes the extension of the interval symmetric single-step method IZSS2, namely the interval midpoint symmetric single-step IMZSS2 method which performs a forward-backward-forward step. The algorithm IMZSS2 introduced new reusable correctors where we always update the midpoints of the intervals at every step of the method. We will display the numerical results comparing the CPU times and number of iterations of both methods. The results show that the IMZSS2 method performs better both in CPU times and number of iterations as can be seen in the accompanied figures.

On the convergence rate of interval repeated midpoint zoro symmetric single-step procedure for simultaneous bounding the polynomial zeros

In this paper, we present the analysis of the rate of convergence of the interval repeated midpoint zoro symmetric single-step procedure (IRMZSS) which is the extension of the interval midpoint zoro symmetric single-step procedure (IMZSS). The results show that the procedure IRMZSS has R-order of convergence at least 7r + 1 (r ≥ 1) or 0R (IRMZSS) ≥ 7 r + 1 (r ≥ 1), whereas the procedure IMZSS R-order of convergence at least 8 or 0R (IMZSS) ≥ 8. In fact 0R (IRMZSS) ≥ 0R (IMZSS) and 0R (IRMZSS) = 0R (IRMZSS) = 0R (IMZSS) when r = 1.

Two-step diagonal Newton method for large-scale systems of nonlinear equations

We propose some improvements on a diagonal Newtons method for solving large-scale systems of nonlinear equations. In this approach, we use data from two preceding steps to improve the current approximate Jacobian in diagonal form. Via this approach, we can achieve a higher order of accuracy for Jacobian approximation when compares to other existing diagonal-type Newtons method. The results of our numerical tests, demonstrate a clear enhancement in numerical performance of our proposed method