Numerical stability of the Halley-iteration for the solution of a system of nonlinear equations

Abstract

Let F : R q → R q F:{{\mathbf {R}}^q} \to {{\mathbf {R}}^q} and x ∗ {x^ \ast } a simple root in R q {{\mathbf {R}}^q} of the system of nonlinear equations F ( x ) = 0 F(x) = 0 . Abstract Padé approximants (APA) and abstract Rational approximants (ARA) for the operator F have been introduced in [2] and [3]. The adjective “abstract” refers to the use of abstract polynomials [5] for the construction of the rational operators. The APA and ARA have been used for the solution of a system of nonlinear equations in [4]. Of particular interest was the following third order iterative procedure: \[ x i + 1 = x i + a i 2 a i + 1 2 F i ′ − 1 F i a i 2 , {x_{i + 1}} = {x_i} + \frac {{a_i^2}}{{{a_i} + \frac {1}{2}F_i^{’- 1}F_i^{}a_i^2}}, \] with F i ′ F_i’ the 1st Fréchet-derivative of F in x 1 , a i = − F i ′ − 1 F i {x_1},{a_i} = - F_i^{’- 1}{F_i} the Newton-correction where F i = F ( x i ) , F i {F_i} = F({x_i}),F_i^{} the 2nd Fréchet-derivative of F in x i {x_i} where F i a i 2 F_i^{}a_i^2 is the bilinear operator F i F_i^{} evaluated in ( a i , a i ) ({a_i},{a_i}) , and componentwise multiplication and division in R q {{\mathbf {R}}^q} . For q = 1 q = 1 this technique is known as the Halley-iteration [6, p. 91]. In this paper the numerical stability [7] of the Halley-iteration for the case q ⩾ 1 q \geqslant 1 is investigated and illustrated by a numerical example.