On a generalization of the resolvent condition in the Kreiss matrix theorem
HTML articles powered by AMS MathViewer
- by H. W. J. Lenferink and M. N. Spijker PDF
- Math. Comp. 57 (1991), 211-220 Request permission
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
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- Keith Kendig, Elementary algebraic geometry, Graduate Texts in Mathematics, No. 44, Springer-Verlag, New York-Berlin, 1977. MR 0447222
- Heinz-Otto Kreiss, Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren, Nordisk Tidskr. Informationsbehandling (BIT) 2 (1962), 153–181 (German, with English summary). MR 165712, DOI 10.1007/bf01957330
- 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), no. 195, 221–237. MR 1079026, DOI 10.1090/S0025-5718-1991-1079026-6
- Randall J. LeVeque and Lloyd N. Trefethen, On the resolvent condition in the Kreiss matrix theorem, BIT 24 (1984), no. 4, 584–591. MR 764830, DOI 10.1007/BF01934916
- Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, 2nd ed., Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0220455
- R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
- Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
- Gary A. Sod, Numerical methods in fluid dynamics, Cambridge University Press, Cambridge, 1985. Initial and initial-boundary value problems. MR 832441, DOI 10.1017/CBO9780511753138
- Eitan Tadmor, The equivalence of $L_{2}$-stability, the resolvent condition, and strict $H$-stability, Linear Algebra Appl. 41 (1981), 151–159. MR 649723, DOI 10.1016/0024-3795(81)90095-1
- Robert J. Walker, Algebraic curves, Springer-Verlag, New York-Heidelberg, 1978. Reprint of the 1950 edition. MR 513824
Additional Information
- © Copyright 1991 American Mathematical Society
- 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