Sine Transform Matrix for Solving Toeplitz Matrix Problems

Abstract

In recent papers, some authors studied the solutions of symmetric positive definite (SPD) Toeplitz systems $T_nx=b$ by the conjugate gradient method (CG) with different sine transforms based preconditioners. In this paper, we first discuss the properties of eigenvalues for the main known circulant, skew circulant and sine transform based preconditioners. A counter example shows that E. Boman's preconditioner is only positive semi-definite for the banded Toeplitz matrix. To use preconditioner effectively, then we propose a modified Boman's preconditioner and a new Cesaro sum type sine transform based preconditioner. Finally, the results of numerical experimentation with these two preconditioners are presented.