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Preconditioners for nonsymmetric block Toeplitz-like-plus-diagonal linear systems. (English) Zbl 1080.65021

The authors discuss the solution of a linear system of a special form: the coefficient matrix of the system is a nonsymmetric block Toeplitz-like-plus-diagonal matrix (a combination with Kronecker product, sum and multiplication, between Toeplitz matrices and diagonal matrices). Such matrices proceed from the Sinc-Galerkin discretization of ordinary differential equations and boundary value problems. The authors propose efficient preconditioners for the coefficient matrix of the given linear system. As a preconditioner, a block tridiagonal matrix is constructed, and the positivity of this matrix is proved. The convergence of the generalized minimal residual (GMRES) iterative method in this case is checked. To see how the method works, four numerical examples are performed.

MSC:

65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
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