Fast Toeplitz orthogonalization

Abstract

Summary. Algorithms are presented which compute the Q R factorization of an order-n Toeplitz matrix in O(n 2) operations. The first algorithm computes only R explicitly, and the second computes both Q and R. The algorithms are derived from a well-known procedure for performing the rank-1 update of QR factors, using the shift-invariance property of the Toeplitz matrix. The algorithms can be used to solve the Toeplitz leastsquares problem, and can be modified to solve Toeplitz systems in O(n) space.

Efficient minimum-phase prefilter computation using fast QL-factorization

“…Different methods have been proposed for performing the fast QL-factorization 3 [8], [9], [10], each of which has different numerical properties and slightly different complexity as well. They do however all use the shift-invariance property of Toeplitz matrices to partition it in two ways, and it is this partitioning that leads to the low complexity schemes.…”

“…The methods described in [8], [9], and [10] all deal with real-valued matrices, but the results can be extended to be valid over the complex field, [10]. Furthermore, the methods can be extended to handle block Toeplitz matrices for the general MIMO case as well, [11].…”

An Efficient Parallel Algorithm to Solve Block?Toeplitz Systems

“…Block Schur-type algorithms can be found in [25,30,34]. More recently, Thirumalai [32] developed several block algorithms that use BLAS3 operations in order to increase their performance.…”

“…The computational cost of computing R is lower than the communication cost of broadcasting R, especially in networks with a high latency. For the computation ofŜ (30), each computed row of S is placed in the correct memory location of the processor according to the permutation matrix P, avoiding the interchange of rows in memory after the computation of S.…”

Stability analysis of a general toeplitz systems solver

“…The first such O(n 2 ) algorithm was introduced by Sweet [73,74]. Unfortunately, Sweet's algorithm depends on the condition of a certain submatrices of A, so is unstable [8,56].…”