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

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 Lm(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 pj(x)j!. For i=1,2, let Lmi)(x) be the matrix obtained from Lm(x) by deleting its first i rows and its last i columns. L1(1)(x)≡1. Then, the function Bm(x)=x−p(x)det(Lm−1(1)(x))det(Lm(1)(x)) defines a fixed-point iteration function having mth order convergence rate for simple roots of p(x). For m=2 and 3, Bm(x) coincides with Newtonʹs and Halleyʹs, respectively. The function Bm(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=mMγi(x)(θ−x)i, with γi(x) ∈ K(x) and well-defined at any simple root θ. Given g ∈ S(m,M), and a simple root θ of p(x), gi(θ)=0, i=1, …, m−1 and the asymptotic constant of convergence of the corresponding fixed-point iteration is γm(θ) = (−1)gm(θ)m!. For Bm(x) we obtain γm(θ)=(−1)mdet(Lm+1(2)(θ))det(Lm(1)(θ)). If all roots of p(x) are simple, Bm(x) is the unique member of S(m,m + n − 2). By making use of the identity 0 = ∑i=0n[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 Bm(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 γm(θ).