ON AN EFFICIENT METHOD FOR THE SIMULTANEOUS APPROXIMATION OF POLYNOMIAL MULTIPLE ROOTS

Applicable Analysis and Discrete Mathematics

article

Miodrag Petković¹  Ljiljana Petković¹  Jovana Džunić² 

[1] Department of Mathematics, Faculty of Electronic Engineering, University of Niˇs;Department of Mathematics, Faculty of Mechanical Engineering, University of Niˇs

关键词: Polynomial roots;    multiple roots;    simultaneous methods;    acceleration of convergence;   

DOI  :  10.2298/AADM140310005P

学科分类：社会科学、人文和艺术（综合）

来源: Univerzitet u Beogradu * Elektrotehnicki Fakultet / University of Belgrade, Faculty of Electrical Engineering

【 摘 要 】

An iterative method in parallel mode for the simultaneous determinationof multiple roots of algebraic polynomials is stated together with its singlestep variant. These methods are more efficient compared to all simultaneousmethods based on fixed point relations. To attain very high computationalefficiency, a suitable correction resulting from Li-Liao-Cheng’s two-pointfourth-order method of low computational complexity and Gauss-Seidel’s approach are applied. Considerable increase of the convergence rate is obtainedapplying only ν additional polynomial evaluations per iteration, where ν isthe number of distinct roots. A special emphasis is given to the convergence analysis and computational efficiency of the proposed methods. Thepresented convergence analysis shows that the R-order of convergence of theproposed single-step method is at least 2 +τν, where τν ∈ (4, 6) is the uniquepositive root of the polynomial gν(τ ) = τ ν −4ν−1τ −22ν−1. The convergenceorder of the corresponding total-step method is six. Computational aspectsand some numerical examples are given to demonstrate high computationalefficiency and very fast convergence of the proposed methods.

【 授权许可】

Unknown