An Extension of Divide-and-Conquer for Real Symmetric Tridiagonal Eigenproblem(<Special Issue>Algorithms for Matrix・Eigenvalue Problems and their Applications)

Abstract

Divide-and-conquer (DC) is one of the fastest algorithms for eigenproblem of a large-size symmetric tridiagonal matrix (STM). In the original DC, a STM is supposed to be divided in half. In this paper, we propose an extended DC (EDC) where a STM is divided into k parts (k>2). Compared to DC, EDC requires only 3k/(2(k^2-1)) floating operation counts if k is much smaller than the matrix size. In implementation of EDC, the orthogonality among eigenvectors with nearly multiple eigenvalues is ensured by an appropriate usage of quadruple-precision floating-point number processing. We give a formula for the floating operation counts of the present implementation, whose validity is confirmed by numerical experiment.