Iterative Solution of Indefinite Symmetric Systems by Methods Using Orthogonal Polynomials over Two Disjoint Intervals.

摘要

It is shown in this paper that certain orthogonal polynomials over two disjoint intervals can be particularly useful for solving large symmetric indefinite linear systems or for finding a few interior eigenvalues of a large symmetric matrix. There are several advantages of the proposed approach over the techniques which are based upon the polynomials having the least uniform norm in two intervals. While a theoretical comparison will show that the norms of the minimal polynomial of degree n in the least squares sense differs from the minimax polynomial of the same degree by a factor not exceeding 2(n+1)to the 0.5 power, the least squares polynomials are by far easier to compute and to use thanks to their three term recurrence relation. A number of suggestions will be made for the problem of estimating the optimal parameters and several numerical experiments will be reported. (Author)