Rigorous numerics for nonlinear operators with tridiagonal dominant linear parts
Maxime Breden, Jean‐Philippe Lessard, Laurent Desvillettes · Corpus Université Laval (Université Laval) · 2015
We present a method designed for computing solutions of infinite dimensional nonlinear operators f(x) = 0 with a tridiagonal dominant linear part. 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 present rigorous computer-assisted calculations showing that T is a contraction near ¯x, thus yielding the 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 ideas for computing A, and proposes a new rigorous method for proving existence of solutions of nonlinear operators with tridiagonal dominant linear part.