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For the discretized linear systems of the spatial fractional diffusion equations, we construct a class of a modified DTS iteration method and give its asymptotic convergence conditions. Then, we design a fast modified DTS preconditioner by replacing Toeplitz matrix T with the τ matrix to accelerate the convergence rates of GMRES method. Theoretically, we show that the spectrum of fast modified DTS preconditioned matrix is clustered around one. Numerical experiments verify the validity of the constructed fast modified DTS preconditioner for GMRES method. [ABSTRACT FROM AUTHOR]

The spatial fractional diffusion equations are discretized by the finite difference method and linear systems are obtained. Transforming the coefficient matrices of these linear systems to obtain the structure of diagonal-plus-Toeplitz. In order to solve this class of linear systems, we construct a two-parameter DTS iteration method on the basis of the DTS iteration method and give its asymptotic convergence conditions. On this basis, we introduce a two-parameter sine-transform-based splitting preconditioner to improve the convergence rate of the generalized minimal residual (GMRES) iteration method, and theoretically prove that the eigenvalues of the corresponding preconditioned matrix are clustered around 1. Finally, numerical experiments demonstrate the efficiency of the new method. [ABSTRACT FROM AUTHOR]

To solve nonsymmetric saddle point problems, a new generalized variant of the DPSS (NGVDPSS) preconditioner is proposed. The invertibility of the NGVDPSS preconditioner, the convergence of the NGVDPSS iteration method and some properties of the corresponding preconditioned matrix are presented. The degree of minimal polynomials of the preconditioned matrix is obtained. Simultaneously, the algorithm implementation of the NGVDPSS preconditioner is discussed. Numerical experiments are performed to illustrate the availability of the new preconditioner. [ABSTRACT FROM AUTHOR]

In this paper, by constructing equivalently the absolute value equation (AVE) as the new block equation of two-by-two, we present a new SOR-like (NSOR) method for solving the AVE. The convergence and optimal parameters of NSOR method for solving the AVE are studied in detail. Numerical experiments are presented to show that the NSOR method is implementable and efficient. [ABSTRACT FROM AUTHOR]

In this paper, we propose two different modified Uzawa methods for solving the saddle point problems, and prove the convergence of the proposed methods. We solve a lower-upper triangular system per iteration in the proposed methods, instead of only solving a lower one. Numerical experiments show that our proposed methods with four parameters perform much faster than previous Uzawa methods with three parameters. [ABSTRACT FROM AUTHOR]