The upper and lower bounds for generalized minimal residual method on a tridiagonal Toeplitz linear system

Doostaki, Reza; Goughery, Hossein Sadeghi

doi:10.1080/00207160.2015.1009901

Cited by 3 publications

(2 citation statements)

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“…For m = 1, A becomes a tridiagonal Toeplitz linear system, and this problem has been studied in [2][3][4][5] and the references therein. Li and Zhang [4] obtained nice upper bounds for GMRES residuals on tridiagonal Toeplitz linear systems.…”

Section: Discussionmentioning

confidence: 99%

Convergence rate of GMRES on tridiagonal block Toeplitz linear systems

Doostaki

2016

Linear and Multilinear Algebra

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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.

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Section: Discussionmentioning

confidence: 99%

Convergence rate of GMRES on tridiagonal block Toeplitz linear systems

Doostaki

2016

Linear and Multilinear Algebra

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“…In this paper, we consider the GMRES method on the tridiagonal block Toeplitz linear system Ax = b, where A is defined as (1.4). For case m = 1, i.e, the tridiagonal Toeplitz linear systems, this problem has been studied previously in [2][3][4][5] and the references therein. 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]).…”

Section: Introductionmentioning

confidence: 99%

GMRES on tridiagonal block Toeplitz linear systems

Doostaki

2018

Glas. Mat. Ser. III

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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.

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