Abstract
The problem of obtaining the smallest possible region containing all the zeros of a polynomial has been attracting more and more attention recently, and in this paper, we obtain several results providing the annular regions that contain all the zeros of a complex polynomial. Using MATLAB, we construct specific examples of polynomials and show that for these polynomials our results give sharper regions than those obtainable from some of the known results.
Highlights
IntroductionOf degree n with complex coefficients, has exactly n complex zeros
The Fundamental Theorem of Algebra states that every nonzero, single-variable, polynomial p (z) = a0 + a1z + a2z2 + a3z3 + ⋅ ⋅ ⋅ + anzn, (1)of degree n with complex coefficients, has exactly n complex zeros
The above theorem tells about the number of zeros, it does not mention anything about the location of these zeros
Summary
Of degree n with complex coefficients, has exactly n complex zeros. For polynomials of degree five or higher with arbitrary coefficients, the AbelRuffini theorem states that there is no algebraic solution. For example, Aberth-Ehrlich method [5, 6], have been proposed for the simultaneous determination of zeros of algebraic polynomials and there are studies [7, 8] to accelerate convergence and increase computational efficiency of these methods. Approximations to the zeros of a polynomial can be drawn by these methods, and these methods can become more efficient when an annulus containing all the zeros of the polynomial is provided. This paper is focused on finding new theorems that can provide smaller annuli containing all the zeros of a polynomial
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