# A Parallel Two-Stage Iteration Method for Solving Continuous Sylvester Equations
## 1 Introduction  
The solution of the continuous Sylvester equation  
\[AX+XB=F \tag{1}\]  
with large sparse matrices \(A\in\mathbb{R}^{n\times n}\), \(B\in\mathbb{R}^{n\times n}\), \(X\in\mathbb{R}^{n\times n}\), \(F\in\mathbb{R}^{n\times n}\) and with \(A\), \(B\) positive definite is a common task in numerical linear algebra. It arises in many scientific computing and engineering applications, such as control theory [1; 2], neural networks, model reduction [3], image processing [4], and so on. Therefore, the problem has remained an active area of research. In this context, recent methodological advances have been thoroughly discussed in many papers [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20]. Iterative methods for solving linear or nonlinear equations have seen constant improvement in recent years to reduce the computational time; for example, two multi-step derivative-free iterative methods [5]: block Jacobi two stage method [6] and SYMMLQ  algorithm [7; 8; 9]. In addition, widely-used direct methods are, for instance, the Bartels-Stewart [10] and the Hessenberg-Schur method [11]. The main idea is to transform \(A\) and \(B\) into triangular or Hessenberg form [21] by an orthogonal similarity transformation and then to solve the resulting system of linear equations directly by a back-substitution process. However, this method is not applicable in large-scale problems due to the prohibitive computational issue. In order to overcome this limitation, fast iterative methods have been developed such as the [PERSON] method [12], the alternating direction implicit (ADI) method [13], gradient-based methods [14; 15], and the Krylov subspace-based algorithm [7; 16; 17]. At present, the conjugate gradient (CG) method [7] and the preconditioned conjugate gradient method [18] are popularly used with the advantages of small storage and suitability for parallel computing. Typically, the SYMMLQ algorithm [7; 8; 9] is quite efficient in the case of symmetric coefficient matrices, as it has tremendous advantages in small storage capacity and stable computations. However, it is not a good option for multi-computer systems dueto the high cost of global communication. For asymmetric coefficient matrices, a modified conjugate gradient method (MCG) is useful. However, its convergence speed is slow [22; 23]. Another type of iteration based on splitting methods allows us to better utilize standard methodologies. For instance, [PERSON] et al. [24] proposed Hermitian and skew-Hermitian splitting (HSS) iteration methods for solving systems of linear equations with non-Hermitian positive definite form. This has been studied widely and generalized in [25; 26; 27; 28]. Recently, a Hermitian and skew-Hermitian splitting (HSS) iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semidefinite matrices was discussed in [29]. [PERSON] et al. [30] presented a positive-definite and skew-Hermitian splitting (PSS) iteration method, and in [31] [PERSON] et al. applied the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method to solve the continuous Sylvester equation. [PERSON] and [PERSON] in [32] applied the idea of the normal and skew-Hermitian splitting (NSS) iteration method to continuous Sylvester equations. However, these iteration methods have the common difficulty that there is no accurate formula to determine the positive value of the corresponding parameter in the iteration scheme. In many articles, a large amount of work has been done to address this issue. Unfortunately this estimation methodology is still not fully resolved in practical applications. In addition, their implementations need to solve two continuous Sylvester equations, which results in great additional computational cost. All of these brought about the need for the development and validation of an efficient parallel algorithm.