## On Newton-like methods to enclose solutions of nonlinear equations.(English)Zbl 0669.65039

The solution of nonlinear equation systems $$F(x)=0$$, where $$x\in {\mathbb{R}}^ n$$, is commonly based on a local linearization of F(x). G. Alefeld and J. Herzberger [Introduction to Interval Computations (1983; Zbl 0552.65041)] have shown an interval arithmetic solution may be affected by splitting the interval Jacobian matrix [A] as [M]-[N], with suitable choice of [M]. Rather than compute the Jacobian and hence perform a complete interval Gaussian algorithm each iterate, various Newton-like methods are described, based on several steps with [M] and [N] being fixed.
Theorems concerning the feasibility of the method and its global convergence, as well as the speed of convergence and the quality of the enclosure attained, are derived. Two numerical examples, one based on a simple nonlinear Dirichlet problem, the other on a nonlinear integral equation, are used to illustrate the method, each using both Jacobi and Gauss-Seidel splitting.
Reviewer: A.Swift

### MSC:

 65H10 Numerical computation of solutions to systems of equations 65G30 Interval and finite arithmetic

Zbl 0552.65041
Full Text:

### References:

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