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On a conjecture by LeVeque and Trefethen related to the Kreiss matrix theorem. (English) Zbl 0736.15015
Let \(A\) be an \(N\times N\) complex matrix. The condition (1) \(\| A^n\|\leq C_0\) for \(n=1,2,3,\ldots\), for a positive constant \(C_0\) implies the condition (2) \(zI-A\) is regular with \(\|(zI-A)^{- 1}\|\leq C_1(| z|-1)^{-1}\) for all complex \(z\) with \(| z|>1\).
H.-O. Kreiss BIT, Nord. Tidskr. Inf.-behandl. 2, 153–181 (1962; Zbl 0109.34702)] proved that (2) implies (1) with \(C_0\) depending on \(C_1\) and \(N\) only. R. J. LeVeque and L. N. Trefethen proved [BIT 24, 584–591 (1984; Zbl 0559.15018)] that (2) implies (1) if \(C_0=2eNC_1\) and they conjectured that the implication holds for the optimal value \(C_0=eNC_1\).
The author proves this conjecture, his proof relying on a lemma which gives an upper bound for the arc-length of the image of the unit circle in the complex plane under a rational function.

MSC:
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A45 Miscellaneous inequalities involving matrices
65F35 Numerical computation of matrix norms, conditioning, scaling
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[1] Kreiss, H.-O., Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren, BIT 2, 153–181 (1962). · Zbl 0109.34702
[2] LeVeque, R. J., L. N. Trefethen,On the resolvent condition in the Kreiss matrix theorem, BIT 24, 584–591 (1984). · Zbl 0559.15018
[3] Richtmyer, R. D., K. W. Morton,Difference Methods for Initial-value Problems, 2nd Ed., J. Wiley & Sons. New York, London, Sydney, 1967. · Zbl 0155.47502
[4] Smith, J. C.,An inequality for rational functions, The American Mathem. Monthly 92, 740–741 (1985).
[5] Sod, G. A.,Numerical Methods in Fluid Dynamics, Cambridge University Press. Cambridge, 1985. · Zbl 0592.76001
[6] Tadmor, E.,The equivalence of L 2-stability, the resolvent condition, and strict H-stability. Linear Algebra Appl. 41, 151–159 (1981). · Zbl 0469.15011
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