Modification of interval symmetric single-step procedure for simultaneous bounding polynomial zeros

The focus of this research is on the bounding of simple and real polynomial zeros simultaneously, focusing on the interval analysis approaches. This procedure started with some disjoint intervals XXXX each of which contains a zero of the polynomial and finally produced successively smaller closed bounded intervals, which always converge XX to the zeros for XXXX respectively. In relation to that, the previous work on Interval Symmetric Single-step (ISS2) procedure is investigated to ensure this procedure is useful for solving polynomials. Thus, this procedure is extended to some modifications in order to improve the efficiency of the procedure. Starting from the authentic ISS2 procedure, four modified procedures are developed. The procedures are Interval Symmetric Single-Step (ISS2-5D) procedure, Interval Zoro-Symmetric Single-Step (IZSS2-5D) procedure,Interval Midpoint Symmetric Single-Step (IMSS2-5D) procedure and Interval Midpoint Zoro-Symmetric Single-Step (IMZSS2-5D) procedure. The programming language Intlab toolbox for Matlab is used to record the numerical results, whereby the stopping criterion used is XXXX 10−10. The results are numerically compared to the original ISS2 procedure to supervise the improvements and efficiencies of the modified procedures. In order to assure that the outcomes of the procedures are promising,convergence rate for each modified procedures is analyzed for comparing purposes. Other than that, the analysis of inclusion to certify the convergence of the modified procedures is included. All the modifications are proven to have better rate of convergences and these are well-supported on the reduction of CPU times, number of iterations and the value of the interval width of the procedures. In a nutshell, this study reveals that the new modified procedures are capable and efficient for bounding the simple and real polynomial zeros simultaneously.

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