On circulant and skew-circulant splitting algorithms for (continuous) Sylvester equations

Minimal residual Hermitian and skew-Hermitian splitting iteration method for the continuous Sylvester equation

By applying the minimal residual technique to the Hermitian and skew-Hermitian (HSS) iteration scheme, we introduce a non-stationary iteration method named minimal residual Hermitian and skew-Hermitian (MRHSS) iteration method to solve the continuous Sylvester equation. Numerical results verify the effectiveness and robustness of the MRHSS iteration method versus the HSS method for the continuous Sylvester equation. Moreover, by numerical computation, we show that the MRHSS splitting can be used as a splitting preconditioner and induce accurate, robust and effective preconditioned Krylov subsp…

实（反）循环矩阵的本征结构及其一些应用

The eigenstructures of real (skew) circulant matrices with some applications

本文研究实循环和反循环矩阵的实Schur形式，开发了快速算法用于计算循环、反循环和Toeplitz矩阵与实向量的乘法，并提出DCT-DST版本的循环和反循环分裂迭代法，相比FFT版本更高效且节省存储。

A class of multiplicative splitting iterations for solving the continuous Sylvester equation

For solving the continuous Sylvester equation, a class of the multiplicative splitting iteration method is presented. We consider two symmetric positive definite splittings for each coefficient matrix of the continuous Sylvester equations and it can be equivalently written as two multiplicative splitting matrix equations. When both coefficient matrices in the continuous Sylvester equation are (non-symmetric) positive semi-definite, and at least one of them is positive definite; we can choose Hermitian and skew-Hermitian (HS) splittings of matrices $A$ and $B$, in the first equation, and the sp…

Preconditioned MHSS iterative algorithm and its accelerated method for solving complex Sylvester matrix equations

This paper introduces and analyzes a preconditioned modified of the Hermitian and skew-Hermitian splitting (PMHSS). The large sparse continuous Sylvester equations are solved by PMHSS iterative algorithm based on nonHermitian, complex, positive definite/semidefinite, and symmetric matrices. We prove that the PMHSS is converged under suitable conditions. In addition, we propose an accelerated PMHSS method consisting of two preconditioned matrices and two iteration parameters α, \b{eta}. Theoretical analysis showed that the convergence speed of the accelerated PMHSS is faster compared to the PMH…

A fast two-level Strang splitting method for multi-dimensional spatial fractional Allen-Cahn equations with discrete maximum principle

In this paper, we study the numerical solutions of the multi-dimensional spatial fractional Allen-Cahn equations. After semi-discretization for the spatial fractional Riesz derivative, a system of nonlinear ordinary differential equations with Toeplitz structure is obtained. For the sake of reducing the computational complexity, a two-level Strang splitting method is proposed, where the Toeplitz matrix in the system is split into the sum of a circulant matrix and a skew-circulant matrix. Therefore, the proposed method can be quickly implemented by the fast Fourier transform, substituting to ca…