Noraini Jamaludin

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.

Numerical solution of second order ODE directly by two point block backward differentiation formula

Direct Two Point Block Backward Differentiation Formula, (BBDF2) for solving second order ordinary differential equations (ODEs) will be presented throughout this paper. The method is derived by differentiating the interpolating polynomial using three back values. In BBDF2, two approximate solutions are produced simultaneously at each step of integration. The method derived is implemented by using fixed step size and the numerical results that follow demonstrate the advantage of the direct method as compared to the reduction method.

On the convergence of two point block backward differentiation formula for second order ODEs

The Two Point Block Backward Differentiation Formula (BBDF2) is a direct solver for second order Ordinary Differential Equations (ODEs). It had shown its efficiency by having less total number of steps and less computational time over the first order ODEs solver. In this paper, the convergence of the BBDF2 is justified by its consistency and zero-stability properties.

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.

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.