@misc{rev-22-aa-iterate, title = {Affine Iterations and Wrapping Effect: Various Approaches}, author = {Nathalie Revol}, howpublished = {ArXiv article, number 2201.00513}, doi = {10.48550/arXiv.2201.00513}, year = 2022, month = jan, comment = {Spurious google hit. But may be a possible application of AA}, 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the SVD-factorization is computationally less demanding and gives good results when the matrix is poorly scaled, it is superseded either by the naive iteration or by Lohner's method otherwise.} }