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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

On a generalization of the resolvent condition in the Kreiss matrix theorem


Authors: H. W. J. Lenferink and M. N. Spijker
Journal: Math. Comp. 57 (1991), 211-220
MSC: Primary 65F35; Secondary 15A60, 65L20, 65M06
MathSciNet review: 1079025
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Abstract: This paper deals with a condition on the resolvent of $ s \times s$ matrices A. In one of the equivalent assertions of the Kreiss matrix theorem, the spectral norm of the resolvent of A at $ \zeta $ must satisfy an inequality for all $ \zeta $ lying outside the unit disk in $ \mathbb{C}$. We consider a generalization in which domains different from the unit disk and more general norms are allowed.

Under this generalized resolvent condition an upper bound is derived for the norms of the nth powers of $ s \times s$ matrices B. Here, B depends on A via a relation $ B = \varphi (A)$, where $ \varphi $ is an arbitrary rational function. The upper bound grows linearly with $ s \geq 1$ and is independent of $ n \geq 1$. This generalizes an upper bound occurring in the Kreiss theorem where $ B = A$.

Like the classical Kreiss theorem, the upper bound derived in this paper can be used in the stability analysis of numerical methods for solving differential equations.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1991-1079025-4
PII: S 0025-5718(1991)1079025-4
Article copyright: © Copyright 1991 American Mathematical Society