Infinite-Dimensional Newton-Type Method

Abstract

This chapter presents two numerical verification methods which are based on some infinite-dimensional fixed-point theorems. The first approach is a technique using sequential iteration. Although this method is simple and can be applied to general nonlinear functional equations in Banach spaces, the relevant compact map has to be retractive in some neighborhood of the fixed point to be verified. The second verification approach is Newton-like iteration. We consider a linearized operator (denoted by $\text{{$\mathcal {L}$}}$) of the problem and verify the invertibility of $\text{{$\mathcal {L}$}}$ and compute guaranteed norm bounds for $\text{{$\mathcal {L}$}}^{-1}$ by applying the same principle as in Chaps. 1 and 2. After that, we confirm the existence of solutions by proving the contractility of the infinite-dimensional Newton-like operator with a residual form. Note that a projection into a finite-dimensional subspace and constructive error estimates of the projection play important and essential roles.