About the sharpness of the stability estimates in the Kreiss matrix theorem

M. N. Spijker, S. Tracogna, Bruno Welfert

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


One of the conditions in the Kreiss matrix theorem involves the resolvent of the matrices A under consideration. This so-called resolvent condition is known to imply, for all n ≥ 1, the upper bounds ||An|| ≤ eK(N + 1) and ||An|| ≤ eK(n + 1). Here K is the spectral norm, K is the constant occurring in the resolvent condition, and the order of A is equal to N + 1 ≥ 1. It is a long-standing problem whether these upper bounds can be sharpened, for all fixed K > 1, to bounds in which the right-hand members grow much slower than linearly with N + 1 and with n + 1, respectively. In this paper it is shown that such a sharpening is impossible. The following result is proved; for each ∈ > 0, there are fixed values C > 0, K > 1 and a sequence of (N + 1) × (N + 1) matrices AN, satisfying the resolvent condition, such that ||(AN|| ≥ C(N + 1)1-∈ = C(n + 1)1-∈ for N = n = 1, 2, 3, .... The result proved in this paper is also relevant to matrices A whose ∈-pseudospectra lie at a distance not exceeding K∈ from the unit disk for all ∈ > 0.

Original languageEnglish (US)
Pages (from-to)697-713
Number of pages17
JournalMathematics of Computation
Issue number242
StatePublished - Apr 2003


  • Kreiss matrix theorem
  • Numerical stability
  • Resolvent condition
  • Stability estimate
  • ∈-pseudospectrum

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics


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