Remote Access Mathematics of Computation
Green Open Access

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
DOI: https://doi.org/10.1090/S0025-5718-1991-1079025-4
MathSciNet review: 1079025
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] N. Dunford and J. Schwartz, Linear operators. I, Interscience, New York, London, 1958. MR 0117523 (22:8302)
  • [2] K. Kendig, Elementary algebraic geometry, Springer-Verlag, New York, Heidelberg, Berlin, 1977. MR 0447222 (56:5537)
  • [3] H.-O. Kreiss, Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren, BIT 2 (1962), 153-181. MR 0165712 (29:2992)
  • [4] H. W. J. Lenferink and M. N. Spijker, On the use of stability regions in the numerical analysis of initial value problems, Math. Comp. 57 (1991), 221-237. MR 1079026 (91m:65217)
  • [5] R. J. LeVeque and L. N. Trefethen, On the resolvent condition in the Kreiss matrix theorem, BIT 24 (1984), 584-591. MR 764830 (86c:39004)
  • [6] R. D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, 2nd ed., Wiley, New York, London, Sydney, 1967. MR 0220455 (36:3515)
  • [7] R. T. Rockafellar, Convex analysis, Princeton Univ. Press, Princeton, NJ, 1970. MR 0274683 (43:445)
  • [8] W. Rudin, Functional analysis, McGraw-Hill, New York, 1973. MR 0365062 (51:1315)
  • [9] G. A. Sod, Numerical methods influid dynamics, Cambridge Univ. Press, Cambridge, 1985. MR 832441 (87k:76006)
  • [10] E. Tadmor, The equivalence of $ {L_2}$-stability, the resolvent condition, and strict H-stability, Linear Algebra Appl. 41 (1981), 151-159. MR 649723 (84g:15030)
  • [11] R. J. Walker, Algebraic curves, Springer-Verlag, New York, Heidelberg, Berlin, 1978. MR 513824 (80c:14001)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65F35, 15A60, 65L20, 65M06

Retrieve articles in all journals with MSC: 65F35, 15A60, 65L20, 65M06


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1991-1079025-4
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society