Convergence of interval symmetric single-step method for simultaneous inclusion of real polynomial zeros

The purpose of this thesis is to find the inclusion of polynomial zeros by using interval analysis approach. We will focus on interval single-step method in order to gain the fastest speed of convergence for bounding simple polynomial zeros simultaneously. Firstly, we will generally describe on some basic mathematical background on interval analysis approach. Then, we will briefly discuss the procedure given in the literature which has been proved by other researchers. We present some information on interval single-step IS method together with the algorithm and the analysis on the rate of convergence. In order to improve IS method, we made several modifications using interval analysis approaches whereby it has been proved that these procedures not only including intervals for roots, but also convergent under a few assumptions. We have new modification namely ISS, IZSS and IZMSS methods which are describe precisely in this thesis. The processing time (CPU) of the algorithm of the modified methods may be done using Matlab 2007a associated with Intlab. Nevertheless, we will also present the theoretical analyses of the convergence rate of the modified procedure. This thesis will cover the algorithms, theoretical analysis and numerical results for each modification. Based on the analysis that has been done, we finally found the rate of convergence for ISS is at least 9, for IZSS is at least 13 and for IMZSS is at least 16 while the rate of convergence of IS is at least 2(1+r)˃3. Finally, we conclude our thesis by comparing all the factors needed in a table and we give some possible extensions for future works.

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