On approximate zeros and rootfinding algorithms for a complex polynomial

Abstract

In this paper we give criteria for a complex number to be an approximate zero of a polynomial <italic>f</italic> for Newton’s method or for the <italic>k</italic> th-order Euler method. An approximate zero for the <italic>k</italic> th-order Euler method is an initial point from which the method converges with an order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis k plus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(k + 1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Also, we construct families of Newton (and Euler) type algorithms which are surely convergent.