Affine Iterations and Wrapping Effect

Abstract

Affine iterations of the form xn+1=Axn+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the SVD-factorization is computationally less demanding and gives good results when the matrix is poorly scaled, it is superseded either by the naive iteration or by Lohner's method otherwise.