关于矩阵族稳定的(J)条件及其与Buchanan准则的关系

本文注意到矩阵族稳定的Kreiss定理和Buchanan准则不便于实际应用。文中(§2,§3)从Kreiss定理的豫解条件出发得到了至少对于四阶以下矩阵族较为实用的判别稳定性的(J)条件;并证明了对于其特征值赋套的上三角矩阵族,(J)条件与Buchanan准则的等价性。§4作为(J)条件的应用讨论了逼近于二维、三维波动方程的显式差分方程(其增长矩阵分别是三、四阶矩阵族),得到了稳定的充要条件。

Condition (J) for the stability of a Family of Matrices and Its Relation to the Buchanan Criterion

It is observed that the Kreiss theorem and the Buchanan criterion for the stability of the family of matrices are not convenient to apply in practice. In this paper (§2. §3) we have derived some more practical condition (J) of which the proof follows from the resolvent condition of the Kreiss matrix theorem. Furthermore, we have proved the equivalence between the condition (J) and the Buchanan criterion for the family of matrices with upper triangular form and with their eigenvalues nested. In §4; as an application of condition (J), we have discussed the stability of some explicit ditference equations of the wave equation with 2 or 3 space variables, and provided necessary and sufficient conditions for stability.