A basic family of iteration functions for polynomial root finding and its characterizations

Abstract

Let p( x) be a polynomial of degree n⩾2 with coefficients in a subfield K of the complex numbers. For each natural number m⩾2, let L m ( x) be the m× m lower triangular matrix whose diagonal entries are p( x) and for each j=1,…, m−1, its jth subdiagonal entries are p j(x) j! . For i=1,2, let L m i)(x) be the matrix obtained from L m ( x) by deleting its first i rows and its last i columns. L 1 (1)( x)≡1. Then, the function B m ( x)= x− p( x) det(L m−1 (1)(x)) det(L m (1)(x)) defines a fixed-point iteration function having mth order convergence rate for simple roots of p( x). For m=2 and 3, B m ( x) coincides with Newton's and Halley's, respectively. The function B m ( x) is a member of S( m, m+ n−2), where for any M⩾ m, S( m, M) is the set of all rational iteration functions g( x) ∈ K( x) such that for all roots θ of p( x), then g( x)= θ+∑ i= m Mγ i(x)(θ−x) i, with γ i ( x) ∈ K( x) and well-defined at any simple root θ. Given g ∈ S( m, M), and a simple root θ of p( x), g i ( θ)=0, i=1, …, m−1 and the asymptotic constant of convergence of the corresponding fixed-point iteration is γ m(θ) = (−1)g m(θ) m! . For B m ( x) we obtain γ m(θ)= (−1) m det(L m+1 (2)(θ)) det(L m (1)(θ)) . If all roots of p( x) are simple, B m ( x) is the unique member of S( m, m + n − 2). By making use of the identity 0 = ∑ i=0 n[p (i)(x) i!] (θ − x) i , we arrive at two recursive formulas for constructing iteration functions within the S( m, M) family. In particular, the family of B m ( x) can be generated using one of these formulas. Moreover, the other formula gives a simple scheme for constructing a family of iteration functions credited to Euler as well as Schröder, whose mth order member belongs to S( m, mn), m>2. The iteration functions within S( m, M) can be extended to any arbitrary smooth function f, with the uniform replacement of p ( j) with f ( j) in g as well as in γ m ( θ).