Быстродействующие алгоритмы диагонализации трехдиагональных симметричных матриц на основе элементарных плоских вращений

Abstract

This paper discusses two algorithms for diagonal-ization of real symmetric tridiagonal matrices. The proposed algorithms preserve the invariant (unchanged) tridiagonal form and provide high performance comparable with the most high-performance QR-algorithms and similar techniques. Contrary to the transformations of reflections, the presented algorithms of matrices di-agonalization using elementary planar rotations have absolute stability of calculations (Wilkinson stability). In this paper, the absolutely stable convergence of the proposed algorithms is proved. Convergence criteria are defined inherently by “integral” properties in contrast with “differential” properties of neighborhood convergence of QR-algorithms (in practice, QR-algorithms are prone to instability and loss of accuracy for eigenvalues computation under certain conditions). A priori estimates of maximum values of diagonalization error for proposed algorithms are provided. Also, the paper discusses the results of numerical experiments for a wide class of symmetric matrices conducnted to elaborate matrices di-agonalization time dependencies for QR-techniques and proposed algorithms.