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Journal Articles Acta Cybernetica Year : 2023

Affine Iterations and Wrapping Effect: Various Approaches

Itérations affines et effet enveloppant: différentes approches.


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the SVD-factorization is computationally less demanding and gives good results when the matrix is poorly scaled, it is superseded either by the naive iteration or by Lohner's method otherwise.
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hal-03505854 , version 1 (31-12-2021)





Nathalie Revol. Affine Iterations and Wrapping Effect: Various Approaches. Acta Cybernetica, 2023, 26 (1), pp.129-147. ⟨10.14232/actacyb.295251⟩. ⟨hal-03505854⟩
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