## Abstract

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 ||A^{n}|| ≤ eK(N + 1) and ||A^{n}|| ≤ 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 A_{N}, satisfying the resolvent condition, such that ||(A_{N}|| ≥ 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 language | English (US) |
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Pages (from-to) | 697-713 |

Number of pages | 17 |

Journal | Mathematics of Computation |

Volume | 72 |

Issue number | 242 |

DOIs | |

State | Published - Apr 2003 |

## Keywords

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

## ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics