  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scaled, it is superseded either by the naive iteration or by Lohner's method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.
