Stability estimates for families of matrices of nonuniformly bounded order

• Published in: Linear algebra and its applications 1996-05, Vol.239, p.77-102
• Main Authors: Spijker, M.N., Straetemans, F.A.J.
• Format: Article
• Language: English
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/S0024-3795(96)90004-X

This paper addresses the problem of establishing upper bounds for the norm of the nth power of square matrices. The focus is on upper bounds that are valid for families of matrices of finite, but nonuniformly bounded, order. Such upper bounds are relevant to the stability analysis of numerical methods for solving differential equations. In the famous Kreiss matrix theorem a condition, on the resolvent of matrices, occurs which implies that the norm of the nth power does not grow faster than linearly with n. In this paper a slightly stronger version of this resolvent condition is studied, which is often satisfied in cases of practical interest. We prove that this version implies growth at a rate that is essentially lower than in the case of the classical Kreiss resolvent condition. We consider also a condition, on the iterated resolvent of matrices, which is related to one of the assumptions occurring in the Hille-Yosida theorem. This condition is known to imply growth at a rate C(n 1 2 ) . We prove that, when a slightly stronger version of the iterated resolvent condition is in force, there is growth at a rate that is at most C( n p ), with 0 ≤ p < 1 2 .