Complex symmetric Sylvester matrix equations appear in many applications, such as the numerical solution of the complex Helmholtz equations. In this paper, by designing a global complex symmetric $\mathcal{M}$-Lanczos process we develop a global variant of the conjugate A-orthogonal conjugate residual method (Gl-COCR) for solving the Sylvester matrix equation $AX+XB=C$ with complex symmetric coefficient matrices. To obtain the smooth and monotone convergence behavior, we also propose a smoothed Gl-COCR method, denoted by SGl-COCR. Finally, numerical examples are given to illustrate the performances of our methods.

中文翻译：

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复对称Sylvester矩阵方程AX  +  XB  =  C的COCR方法的全局变量

复对称Sylvester矩阵方程出现在许多应用中，例如复Helmholtz方程的数值解。本文通过设计一个全局复对称的$\mathcal{中号}$-Lanczos过程，我们开发了共轭A的全局变量-正交共轭残差方法（G1-COCR），用于求解Sylvester矩阵方程$\mathrm{一种}X+X乙=C$具有复杂的对称系数矩阵。为了获得平滑和单调的收敛行为，我们还提出了一种平滑的G1-COCR方法，用SG1-COCR表示。最后，通过数值例子说明了我们方法的性能。