仿射迭代和包装效果：各种方法
Affine Iterations and Wrapping Effect: Various Approaches
摘要：
如果矩阵A的谱半径小于1，则形式为x（n+1）= Ax（n）+ b的仿射迭代使用实数算术收敛。然而，用区间运算代替实数运算可能导致这些迭代的发散，特别是如果A的绝对值的谱半径大于1。当初始向量x（0）和b的分量是区间时，我们将回顾不同的方法来限制迭代的高估。我们将比较，理论和实验，这些不同的方法计算的迭代的宽度：天真迭代，方法的基础上的QR和SVD分解的A，和Lohner的QR分解方法。基于SVD分解的方法计算量小，并且在矩阵尺度不佳时给出了良好的结果，它被朴素迭代或Lohner方法所取代。
英文摘要：
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the SVD-factorization is computationally less demanding and gives good results when the matrix is poorly scaled, it is superseded either by the naive iteration or by Lohner's method otherwise.
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