论文标题
一类马鞍点矩阵的Quasi-Perron-Frobenius属性
Quasi-Perron-Frobenius property of a class of saddle point matrices
论文作者
论文摘要
许多科学计算场引起的马鞍点矩阵具有块结构$ w = \ left（\ begin {array} {cc} a＆b \\ b^t＆c \ end {array} \ right）$，其中sub-block $ a $ a $是对称的和正确定的，$ c $是对称的，是对称的和半否。在本文中，我们报告了一个不引人注目但潜在的有价值的结论，即在某些条件下，尤其是当$ c $为零矩阵时，$ w $的频谱半径必须是$ w $的最大特征值。这种特征近似于著名的perron-frobenius属性，在本文中被称为准佩隆 - 弗罗贝尼乌斯属性。在数值测试中，我们观察到从一些用于计算固定stokes方程的混合有限元方法得出的鞍点矩阵。数值结果证实了理论分析，还表明使鞍点矩阵具有准佩隆 - 弗罗贝尼属性的假定条件仅足够而不是必要。
The saddle point matrices arising from many scientific computing fields have block structure $ W= \left(\begin{array}{cc} A & B\\ B^T & C \end{array} \right) $, where the sub-block $A$ is symmetric and positive definite, and $C$ is symmetric and semi-nonnegative definite. In this article we report a unobtrusive but potentially theoretically valuable conclusion that under some conditions, especially when $C$ is a zero matrix, the spectral radius of $W$ must be the maximum eigenvalue of $W$. This characterization approximates to the famous Perron-Frobenius property, and is called quasi-Perron-Frobenius property in this paper. In numerical tests we observe the saddle point matrices derived from some mixed finite element methods for computing the stationary Stokes equation. The numerical results confirm the theoretical analysis, and also indicate that the assumed condition to make the saddle point matrices possess quasi-Perron-Frobenius property is only sufficient rather than necessary.
