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

Abstract:

Let $F:{{\mathbf {R}}^q} \to {{\mathbf {R}}^q}$ and ${x^ \ast }$ a simple root in ${{\mathbf {R}}^q}$ of the system of nonlinear equations $F(x) = 0$. 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: \[ {x_{i + 1}} = {x_i} + \frac {{a_i^2}}{{{a_i} + \frac {1}{2}F_i^{’- 1}F_i^{”}a_i^2}},\] with $F_i’$ the 1st Fréchet-derivative of F in ${x_1},{a_i} = - F_i^{’- 1}{F_i}$ the Newton-correction where ${F_i} = F({x_i}),F_i^{”}$ the 2nd Fréchet-derivative of F in ${x_i}$ where $F_i^{”}a_i^2$ is the bilinear operator $F_i^{”}$ evaluated in $({a_i},{a_i})$, and componentwise multiplication and division in ${{\mathbf {R}}^q}$. For $q = 1$ 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 $q \geqslant 1$ is investigated and illustrated by a numerical example.