GMRES on tridiagonal block Toeplitz linear systems

Abstract

We study the generalized minimal residual (GMRES) method for solving tridiagonal block Toeplitz linear system Ax = b with m × m diagonal blocks. For m = 1, these systems becomes tridiagonal Toeplitz linear systems, and for m > 1, A becomes an m-tridiagonal Toeplitz matrix. Our first main goal is to find the exact expressions for the GMRES residuals for b = (B 1 , 0,. .. , 0) T , b = (0,. .. , 0, B N) T , where B 1 and B N are m-vectors. The upper and lower bounds for the GMRES residuals were established to explain numerical behavior. The upper bounds for the GMRES residuals on tridiagonal block Toeplitz linear systems has been studied previously in [1]. Also, in this paper, we consider the normal tridiagonal block Toeplitz linear systems. The second main goal is to find the lower bounds for the GMRES residuals for these systems.