High Performance Algorithms for Toeplitz and block Toeplitz matrices

Abstract

this paper we discuss several high performance variants of the classical Schur algorithm algorithms to factor symmetric block Toeplitz matrices. Specifically we discuss routines to factor symmetric positive definite, positive semidefinite and indefinite matrices. Algorithms to obtain the QR factorization of exactly and nearly rank deficient Toeplitz matrices are also discussed. In this paper the classical Schur algorithm for obtaining the Cholesky factorization of symmetric positive definite block Toeplitz matrices [9, 8] is generalized to the block Toeplitz matrix case using a block generalization of the hyperbolic Householder reflectors. The block generalization of the Schur algorithm and various blocking schemes differing in the amount of storage and computational primitives used are described in Section 2. Blocking the hyperbolic Householder transformations allows us to apply these transformations using BLAS 3 primitives rather than the BLAS 2 primitives which are required for plain hyperbolic Householder transformations. On machines with a memory hierarchy this provides us with a faster algorithm. For symmetric indefinite block Toeplitz matrices the Schur algorithm breaks down if the matrix has singular principal minors. A scheme to modify the block Schur algorithm by per