TY - JOUR
AU - Nathalie Revol
PY - 2023/06/02
Y2 - 2024/10/08
TI - Affine Iterations and Wrapping Effect: Various Approaches
JF - Acta Cybernetica
JA - Acta Cybern
VL - 26
IS - 1
SE - Special Issue of SCAN 2020
DO - 10.14232/actacyb.295251
UR - https://cyber.bibl.u-szeged.hu/index.php/actcybern/article/view/4279
AB - 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the SVD-factorization is computationally less demanding and gives good results when the matrix is poorly scaled, it is superseded either by the naive iteration or by Lohner's method otherwise.
ER -