homoclinics

Rigorous numerics for nonlinear operators with tridiagonal dominant linear parts

Maxime Breden, Laurent Desvillettes and Jean-Philippe Lessard, submitted, 2014.

We propose a method to compute solutions of infinite dimensional nonlinear operators f(x) = 0 with tridiagonal dominant linear parts. We recast the operator equation into an equivalent Newton-like equation x = T(x) = x - Af(x), where A is an approximate inverse of the derivative Df(x*) at an approximate solution x*. We develop rigorous computer-assisted bounds to show that T is a contraction near x* which yields existence of a solution. Since Df(x*) does not have an asymptotically diagonal dominant structure, the computation of A is not straightforward. This paper provides a method to obtain A and proposes a new rigorous computational method to prove existence of solutions of nonlinear operators with tridiagonal dominant linear parts.

The following links provide the programs sufficient to carry out the main result of the above mentioned work.


The paper in its pdf form can be found here.

Here are the MATLAB codes. Some of them use the interval arithmetic package INTLAB.