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

Abstract:

      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.