求解多项式和多项式特征值问题的多精度算法

Dario Bini, L. Robol
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摘要

现实世界中的许多应用都是由矩阵多项式[EQUATION]建模的,其中Ai是m × m个矩阵,例如参见[2],[6]。在这个框架中遇到的一个计算任务是计算P(x)的特征值,即多项式方程det P(x) = 0的解。这个任务通常是通过将P(x)简化为适合大小为mn的矩阵K, L的线性铅笔xL—K,并通过标准数值算法解决特征值问题(λL—K)v = 0来完成的。关于这种方法存在大量文献,我们建议读者参考[7]作为一个例子。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A multiprecision algorithm for the solution of polynomials and polynomial eigenvalue problems
Many applications of the real world are modelled by matrix polynomials [EQUATION] where Ai are m x m matrices, see for instance [2], [6]. A computational task encountered in this framework is computing the eigenvalues of P(x), that is, the solutions of the polynomial equation det P(x) = 0. This task is generally accomplished by reducing P(x) to a linear pencil of the kind xL -- K for suitable matrices K, L of size mn, and to solving the eigenvalue problem (λL -- K)v = 0 by means of standard numerical algorithms. A wide literature exists on this approach, we refer the reader to [7] for an example.
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