High performance algorithms for Toeplitz and block Toeplitz matrices Original Research Article

In this paper, we present several high performance variants of the classical Schur algorithm to factor various Toeplitz matrices. For positive definite block Toeplitz matrices, we show how hyperbolic Householder transformations may be blocked to yield a block Schur algorithm. This algorithm uses BLAS3 primitives and makes efficient use of a memory hierarchy. We present three algorithms for indefinite Toeplitz matrices. Two of these are based on look-ahead strategies and produce an exact factorization of the Toeplitz matrix. The third produces an inexact factorization via perturbations of singular principal minors. We also present an analysis of the numerical behavior of the third algorithm and derive a bound for the number of iterations to improve the accuracy of the solution. For rank-deficient Toeplitz least-squares problems, we present a variant of the generalized Schur algorithm that avoids breakdown due to an exact rank-deficiency. In the presence of a near rank-deficiency, an approximate rank factorization of the Toeplitz matrix is produced. Finally, we suggest an algorithm to solve the normal equations resulting from a real Toeplitz least-squares problem based on transforming to Cauchy-like matrices. This algorithm exploits both realness and symmetry in the normal equations.