A basic family of iteration functions for polynomial root finding and its characterizations (original) (raw)

1997, Journal of Computational and Applied Mathematics

Let p(x) be a polynomial of degree n 2 2 with coefficients in a subfield K of the complex numbers. For each natural number m > 2, let L,(x) be the m x m lower triangular matrix whose diagonal entries are p(x) and for each j= 1,. , m-1, its jth subdiagonal entries are p"'(x)/j!. For i = 1,2, let L:'(x) be the matrix obtained from Lm(x) by deleting its first i rows and its last i columns. L\')(x)= 1. Then, the function B,(x) =x-p(x) det(Lz',(x))/det(Lg'(x)) defines a fixedpoint iteration function having mth order convergence rate for simple roots of p(x). For m =2 and 3, B,(x) coincides with Newton's and Halley's, respectively. The function B,(x) is a member of S(m, m+n-2), where for any M am, S(m,M) is the set of all rational iteration functions g(x) E K(x) such that for all roots 0 of p(x), then g(x)=B+cE, yl(x)(8-x)', with rl(x) E K(x) and well-defined at any simple root 8. Given g E S(m,M), and a simple root 0 of p(x), g(')(B) = 0, i = 1,.. , m-1, and the asymptotic constant of convergence of the corresponding fixed-point iteration is ~~(0) = (-l)"g(")(0)/m!. For B,(x) we obtain y,(B) = (-1)"'det(L~~,(@)/det(~~)((3)). If all roots of p(x) are simple, B,(x) is the unique member of S(m, m + n-2). By making use of the identity 0 = ~:=,[p(')(x)/i!](B-x)', 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 Schriider, 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(l) in g as well as in r,(e).