Fast iterative methods for symmetric sinc-Galerkin systems

The symmetric sinc-Galerkin method developed by Lund, when applied to the second-order self-adjoint boundary value problem, gives rise to a symmetric coefficient matrix. The coefficient matrix has a special structure so that it can be advantageously used in solving the discrete system. In this paper, we employ the preconditioned conjugate gradient method with banded matrices as preconditioners. We prove that the condition number of the preconditioned matrix is uniformly bounded by a constant independent of the size of the matrix. In particular, we show that the solution of an n-by-n discrete symmetric sinc-Galerkin system can be obtained in O (n log n) operations. We also extend our method to the self-adjoint elliptic partial differential equation. Numerical results are given to illustrate the effectiveness of our fast iterative solvers.