Fast transforms for tridiagonal linear equations

Abstract

In this paper we study the use of the Fourier, Sine and Cosine Transform for solving or preconditioning linear systems, which arise from the discretization of elliptic problems. Recently, R. Chan and T. Chan considered circulant matrices for solving such systems. Instead of using circulant matrices, which are based on the Fourier Transform, we apply the Fourier and the Sine Transform directly. It is shown that tridiagonal matrices arising from the discretization of an onedimensional elliptic PDE are connected with circulant matrices by congruence transformations with the Fourier or the Sine matrix. Therefore, we can solve such linear systems directly, using only Fast Fourier Transforms and the Sherman-Morrison-Woodbury formula. The Fast Fourier Transform is highly parallelizable, and thus such an algorithm is interesting on a parallel computer. Moreover, similar relations hold between block tridiagonal matrices and Block Toeplitz-plus-Hankel matrices of ordern ²×n ² in the 2D case. This can be used to define in some sense natural approximations to the given matrix which lead to preconditioners for solving such linear systems.