The Newton’s method in the interval symmetric single step procedure for polynomial zeros

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.

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The efficiency of convergence rate for IMSS2-5D procedure

Jamaludin

Monsi

Hassan

et al. 2015

AIP Conference Proceedings

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

Jamaludin

Monsi

Hassan³

et al. 2016

AIP Conference Proceedings

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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 preconditions 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.

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On the efficiencies of the IMSS1 method for bounding polynomial zeros simultaneously

Rusli

Monsi

Hassan

et al. 2015

2015 International Conference on Research and Education in Mathematics (ICREM7)

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This paper presents an improvement of the existing interval symmetric single-step method ISS1 which will be called the interval midpoint symmetric single-step method IMSS1. The term 'midpoint' is referred to the updated midpoints used in every step in the method. The idea of midpoint will potentially reduce the time and improve the effectiveness of the method. This method is tested numerically in terms of CPU times and number of iterations of which comparison for both methods will be presented. This procedure is verified on five test polynomials and the results were obtained using MATLAB in association with Intlab toolbox. Based on the numerical results, the IMSS1 method shows a better performance than does the ISS1 method.

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The Newton’s method in the interval symmetric single step procedure for polynomial zeros

Cited by 3 publications

References 10 publications

The efficiency of convergence rate for IMSS2-5D procedure

The efficiency of convergence rate for IMSS2-5D procedure

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

On the efficiencies of the IMSS1 method for bounding polynomial zeros simultaneously

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