Convergence rate of GMRES on tridiagonal block Toeplitz linear systems

Abstract

Iterative methods such as generalized minimal residual (GMRES) method are used to solve large sparse linear systems. This paper is considered the GMRES method for solving N × N tridiagonal block Toeplitz linear systems Ax = b with m × m diagonal blocks, and establishes upper bounds for GMRES residuals. The coefficient matrix A becomes an m-tridiagonal Toeplitz matrix, and tridiagonal toeplitz systems are subcases of these systems. Also, we show that the GMRES method on m N × m N linear system Ax = b computes the exact solution in at most N steps.

GMRES on tridiagonal block Toeplitz linear systems

“…Let the tridiagonal block Toeplitz matrix A be given as (1.4). The next theorem shows that the matrix A is diagonalizable when µ = 0 and ν = 0, and was proven in [1].…”

“…In this section, we start in analogous way as in [1], and major difference between this paper and [1], is in using Chebyshev polynomials of the second kind instead of Chebyshev polynomials of the first kind.…”

“…Different from [1], we use Chebyshev polynomials of the second kind instead of Chebyshev polynomials of the first kind, to simplify (2.7).…”

“…For m > 1, A becomes an m-tridiagonal Toeplitz matrix, and we obtained the upper bounds for the GMRES residuals on linear system Ax = b ([1, Theorem 2.5]). Also, in [1], we showed that the GMRES method on mN × mN linear system Ax = b computes the exact solution in at most N steps. This paper continues our recent work [1], and the first main goal is to find the exact expressions for the GMRES residuals on the tridiagonal block Toeplitz linear system Ax = b, with special right hand side vector (1.9)…”

“…The second main goal in this paper, is to find the lower bounds for the GMRES residuals when solving the tridiagonal block Toeplitz linear system Ax = b, in case of normal matrix A. We start in analogous way as in [1], and use Chebyshev polynomials of the second kind instead of Chebyshev polynomials of the first kind in [1], to achieve these goals. Throughout this paper, we denote the n × n identity matrix by I n , and e j is its jth column.…”