Convergence of CG and GMRES on a tridiagonal Toeplitz linear system |
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Authors: | Ren-Cang Li |
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Affiliation: | (1) Department of Mathematics, University of Texas at Arlington, 19408, Arlington, TX 76019-0408, USA |
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Abstract: | The Conjugate Gradient method (CG), the Minimal Residual method (MINRES), or more generally, the Generalized Minimal Residual method (GMRES) are widely used to solve a linear system Ax=b. The choice of a method depends on A’s symmetry property and/or definiteness), and MINRES is really just a special case of GMRES. This paper establishes error bounds on and sometimes exact expressions for residuals of CG, MINRES, and GMRES on solving a tridiagonal Toeplitz linear system, where A is Hermitian or just normal. These expressions and bounds are in terms of the three parameters that define A and Chebyshev polynomials of the first or second kind. AMS subject classification (2000) 65F10, 65N22 |
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Keywords: | Conjugate Gradient method MINRES GMRES Krylov subspace convergence analysis tridiagonal Toeplitz matrix linear system Vandermonde matrix Chebyshev polynomial |
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