A medium-shifted splitting iteration method for a diagonal-plus-Toeplitz linear system from spatial fractional Schrödinger equations

• Published in: Boundary value problems Vol. 2018; no. 1; pp. 1 - 17
• Main Authors: Wen, Ruiping, Zhao, Peipei
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 03.04.2018; Hindawi Limited; SpringerOpen
• Subjects: Analysis; Approximations and Expansions; Boundary value problems; Convergence; Difference and Functional Equations; Discretization; Eigenvalues; Iterative methods; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrix methods; Medium-shifting iteration method; Methods; Nonlinear equations; Numerical analysis; Ordinary Differential Equations; Partial Differential Equations; Preconditioning; Schrodinger equation; Spatial fractional Schrödinger equations; Splitting; Toeplitz matrix; Medium-shifting iteration method; Spatial fractional Schrödinger equations; Toeplitz matrix; Preconditioning; Convergence
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/s13661-018-0967-1

The centered difference discretization of the spatial fractional coupled nonlinear Schrödinger equations obtains a discretized linear system whose coefficient matrix is the sum of a real diagonal matrix D and a complex symmetric Toeplitz matrix T̃ which is just the symmetric real Toeplitz T plus an imaginary identity matrix iI . In this study, we present a medium-shifted splitting iteration method to solve the discretized linear system, in which the fast algorithm can be utilized to solve the Toeplitz linear system. Theoretical analysis shows that the new iteration method is convergent. Moreover, the new splitting iteration method naturally leads to a preconditioner. Analysis shows that the eigenvalues of the corresponding preconditioned matrix are tighter than those of the original coefficient matrix  A . Finally, compared with the other algorithms by numerical experiments, the new method is more effective.