Displacement Structure Approach to Discrete-Trigonometric-Transform Based Preconditioners of G.Strang Type and of T.Chan Type

Abstract

In this paper we use a displacement structure approach to design a class of new preconditioners for the conjugate gradient method applied to the solution of large Toeplitz linear equations. Explicit formulas are suggested for the Strang-type and for the T.Chan-type preconditioners belonging to any of eight classes of matrices diagonalized by the corresponding discrete cosine or sine transforms. Under the standard Wiener class assumption the clustering property is established for all of these preconditioners, guaranteeing rapid convergence of the preconditioned conjugate gradient method. All the computations related to the new preconditioners can be done in real arithmetic and to fully exploit this advantageous property one has to suggest a fast real-arithmetic algorithm for multiplication of a Toeplitz matrix by a vector. It turns out that the obtained formulas for the Strang-type preconditioners allow a number of representations for Toeplitz matrices leading to a wide variety of real-arithmetic multiplication algorithms based on any of eight discrete cosine or sine transforms. Recently, transformations of Toeplitz matrices to Vandermonde-like or Cauchy-like matrices have been found to be useful in developing accurate direct methods for Toeplitz linear equations. In this paper we suggest to further extend the range of the transformation approach by exploring it for iterative methods; this technique allowed us to reduce the complexity of each iteration of the preconditioned conjugate gradient method. We conclude the paper with a suggestion on how to exploit the displacement structure to efficiently organize numerical experiments in a numerically reliable way.

Clustering/Distribution Analysis and Preconditioned Krylov Solvers for the Approximated Helmholtz Equation and Fractional Laplacian in the Case of Complex-Valued, Unbounded Variable Coefficient Wave Number μ

“…it is known that every τ matrix is diagonalized as τ(T n ) = S n Λ n S n , where Λ n is a diagonal matrix constituted by all eigenvalues of τ(T n ), and S n = ([S n ] j,k ) is the real, symmetric, orthogonal matrix defined before, so that S n = S T n = S −1 n . Furthermore, the matrix S n is associated with the fast sine transform of type I (see [13,14] for several other sine/cosine transforms). Indeed the multiplication of a matrix S n times a real vector can be conducted in O(n log n) real operations and the cost is around half of the celebrated discrete fast Fourier transform [15].…”

Signal flow graph approach to efficient and forward stable DST algorithms

“…Apart from these two main techniques some other authors (see e.g. [17,26]) have used different techniques like displacement approach and polynomial division in matrix form to derive factorizations for DCT and DST. Efficient algorithms for DCT or DST of radix-2 length n require about 2 n log 2 n flops.…”