Interlacing Properties of Tridiagonal Symmetric Matrices with Applications to Parallel Computing

Abstract

In this paper we present new interlacing properties for the eigenvalues of an unreduced tridiagonal symmetric matrix in terms of its leading and trailing submatrices. The results stated in Hill and Parlett [SIAM J. Matrix Anal. Appl., 13 (1992), pp. 239–247] are hereby improved. We further extend our results to reduced symmetric tridiagonal matrices and to specially structured full symmetric matrices. We then present new fast and efficient parallel algorithms for computing a few eigenvalues of symmetric tridiagonal matrices of very large order.