Interval iterative methods on simultaneous inclusion of polynomial zeros

The main aim of the thesis is to modified procedures of bounding real zeros of polynomials simultaneously. For this purpose the interval approach is used in order to obtain faster and more accurate results. The research is based on the existing procedures: the interval symmetric single-step ISS1 and the interval repeated single-step IRSS1. To begin with, the basic concepts of interval computations and some brief introductions on Newton’s method are provided. The modifications done in this thesis can be grouped into two types. The first is the repeated procedures and the second type is the Newton’s modified procedures. The modified procedures proposed consist of four repeated procedures and two Newton’s modified procedures. The algorithms of these modified procedures are elaborated to show the significance of each procedure. Theoretically, the analyses of inclusions for all procedures are presented to ensure the inclusions property of the procedures. In order to find the rate of convergence of the procedures, the analyses of R-order of convergence are discussed in detail. To obtain the numerical results, coding for the procedures are developed and implemented using the MATLAB R2007a combined with the Intlab toolbox. Numerical results are presented in terms of CPU times, number of iterations and the widths of final intervals to indicate the accuracies of the procedure. For the conclusion, faster computational time and good accuracies are achieved from the new modified procedures. Furthermore, they attained higher rate of convergences than the existing procedures.

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