Complex valued systems with an indefinite matrix term arise in important applications such as for certain time-harmonic partial differential equations such as the Maxwell's equation and for the Helmholtz equation. Complex systems with symmetric positive definite matrices can be solved readily by rewriting the complex matrix system in two-by-two block matrix form with real matrices which can be efficiently solved by iteration using the preconditioned square block (PRESB) preconditioning method and preferably accelerated by the Chebyshev method. The appearances of an indefinite matrix term causes however some difficulties. To handle this we propose different forms of matrix splitting methods, with or without any parameters involved. A matrix spectral analyses is presented followed by extensive numerical comparisons of various forms of the methods.
翻译:具有无限期基质的复杂估值系统在一些重要应用中出现,如某些时-时-时-时-部分方程,如Maxwell的等式和Helmholtz等方程式。具有对称正数确定基质的复杂系统可以通过将复杂的基质系统改写成两、两、二块块的基质系统,以真实的基质系统,通过使用先决条件方块(PRESB)先决条件的迭代法和最好用Chebyshev法加速,有效解决。无限期基质的外观造成了一些困难。为了解决这个问题,我们提出了不同形式的矩阵分解方法,不论是否涉及任何参数。一个矩阵光谱分析随后对各种方法的形式进行了广泛的数字比较。