Abstract
In this work, two broad classes of iteration functions in n-dimensional vector spaces are introduced. They are called iteration functions of the first and second kind at a fixed point of the corresponding iteration function. Two general local convergence theorems are presented for Picard-type iterative methods with high Q-order of convergence. In particular, it is shown that if an iterative method is generated by an iteration function of first or second kind, then it is Q-convergent under each initial approximation that is sufficiently close to the fixed point. As an application, a detailed local convergence analysis of two fourth-order iterative methods is provided for finding all zeros of a polynomial simultaneously. The new results improve the previous ones for these methods in several directions.
Highlights
This paper is devoted to the convergence of iterative methods for the simultaneous approximation of all zeros of an algebraic polynomial of degree n ≥ 2
It is shown that if an iterative method is generated by an iteration function of first or second kind, it is Q-convergent under each initial approximation that is sufficiently close to the fixed point
Two general local convergence theorems were established for Picard-type iterative methods with high Q-order of convergence
Summary
This paper is devoted to the convergence of iterative methods for the simultaneous approximation of all zeros of an algebraic polynomial of degree n ≥ 2. The first method for simultaneously finding polynomial zeros was introduced by Weierstrass [1] in 1891. Classical Iterative Methods for Simultaneous Approximation of Polynomial Zeros. Ξ n ) in the space Kn. Every iterative method for simultaneously finding all the zeros of a polynomial f ∈ K[z] is given by a fixed point iteration distributed under the terms and conditions of the Creative Commons x ( k +1) = T ( x ( k ) ), k = 0, 1, 2, . Let us recall two well-known iteration functions for simultaneous approximation of polynomial zeros: Definition 1 (Weierstrass [1]). The second formula in (4) is due to Börsch–Supan [6]
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