Interval arithmetic multistep methods for nonlinear systems of equations

Abstract

In the present paper we introduce interval arithmetic multistep methods for nonlinear systems of equations. These methods are derived from several types of singlestep methods discussed in previous papers. An interval arithmetic particularity consists in the fact that both the function defining the system and its Jacobi matrix can be kept constant in interval multistep methods for a well defined number of steps while it is still possible to prove superlinear convergence. The solution of nonlinear systems of equations by interval arithmetic methods has been discussed by many authors. See [1, 7] or some papers cited in [11], for example. For interval methods convergence results can be obtained which go beyond those for “point”, i.e. noninterval methods. There are, for example, methods converging to the solution without requiring any convexity condition. The inclusion of the solution by all iterates can be proved and the convergence to the solution requires the existence of a computable initial inclusion as main condition, i.e. the convergence can be said to be almost global. We report numerical results for applications of some of our methods on a vector and a sequential computer.