Affine Iterations and Wrapping Effect: Various Approaches

TL;DR

This paper reviews various interval arithmetic methods to improve the convergence and accuracy of affine iterations, comparing their effectiveness in controlling overestimation and divergence.

Contribution

It provides a comparative analysis of naive, QR, SVD, and Lohner's methods for affine iterations with interval data, highlighting their advantages and limitations.

Findings

SVD-based method is efficient for poorly scaled matrices.

Lohner's method reduces overestimation better in some cases.

Naive iteration performs well with well-scaled matrices.

Abstract

Affine iterations of the form x(n+1) = Ax(n) + b converge, using real arithmetic, if the spectral radius of the matrix A is less than 1. However, substituting interval arithmetic to real arithmetic may lead to divergence of these iterations, in particular if the spectral radius of the absolute value of A is greater than 1. We will review different approaches to limit the overestimation of the iterates, when the components of the initial vector x(0) and b are intervals. We will compare, both theoretically and experimentally, the widths of the iterates computed by these different methods: the naive iteration, methods based on the QR-and SVD-factorization of A, and Lohner's QR-factorization method. The method based on the SVD-factorization is computationally less demanding and gives good results when the matrix is poorly scaled, it is superseded either by the naive iteration or by Lohner's method otherwise.