Rigorous numerics for nonlinear operators with tridiagonal dominant linear part

TL;DR

This paper introduces a rigorous computer-assisted method for proving the existence of solutions to infinite-dimensional nonlinear operators with a tridiagonal dominant linear part, using a Newton-like approach and contraction mapping principles.

Contribution

It develops a novel rigorous method for solving nonlinear operators with tridiagonal dominant linear parts, including a new way to compute an approximate inverse for the derivative.

Findings

Rigorous computer-assisted proofs of solution existence.

New method for computing approximate inverses of non-diagonally dominant derivatives.

Application to infinite-dimensional nonlinear operators with tridiagonal structure.

Abstract

We present a method designed for computing solutions of infinite dimensional non linear 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 $Df(\overline x)$ at an approximate solution $\overline x$. We present rigorous computer-assisted calculations showing that $T$ is a contraction near $\overline x$, thus yielding the existence of a solution. Since $Df(\overline x)$ 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.