Kreiss matrix theorem
WebIn summary, the theory presented in this paper amounts to a transplantation of the Kreiss matrix theorem from the unit disk (for simple power iterations) to an arbitrary … WebAbstract. Let A be a set of n × n complex matrices A which satisfy the condition ‖ ( I − z A) ‖ ≦ K / ( 1 − z ) α + 1 for some α ≧ 0 and all z < 1. Then it is shown here that there …
Kreiss matrix theorem
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WebKreiss (1962) established an important theorem, called the Kreiss matrix the- orem, which has been one of the fundamental results for establishing numerical stability. Still recently, much research was devoted to this theorem and variants thereof (see, e.g., Giles (1997), Kraaijevanger (1994), Lubich & Nevanlinna (1991), WebIn the Kreiss matrix theorem the power boundedness of N × N matrices is related to a resolvent condition on these matrices. LeVeque and Trefethen proved that the ratio of the constants in these two conditions can be bounded by 2 eN. They conjectured that this bound can be improved to eN. In this note the conjecture is proved to be true.
WebThe first theorem gives a stability estimate which implies that errors in the numerical process cannot grow faster than linearly with s or n. It improves previous results in the literature where various restrictions were imposed on S and ~o(z), including ~J(z) ~= 0 for z E OS and S be bounded. WebIn the Kreiss matrix theorem the power boundedness of N × N matrices is related to a resolvent condition on these matrices. LeVeque and Trefethen proved that the ratio of …
WebClearly, the theorem is proved if we can show that LR(0) _< 4N'MR(O). This is inequality (7). In part 2 of the proof it is shown that this inequality holds if we can prove that the … WebThe Kreiss Matrix Theorem asserts the uniform equivalence over all N x N matrices of power boundedness and a certain resolvent estimate. We show that the ratio of the …
WebThe Kreiss Matrix Theorem asserts the uniform equivalence over all N x N matrices of power boundedness and a certain resolvent estimate. We show that the ratio of the constants in these two conditions grows linearly with N, and we obtain the optimal proportionality factor up to a factor of 2.
Web12 apr. 2024 · We generalize the Kreiss matrix theorem to rational functions on general domains and prove that, for a domain \text {\O}mega bounded by a Jordan curve which is twice continuously differentiable, there exists a constant C such that for all bounded operators T on a Banach space with \sigma (T)\subset {\overline {\text {\O}mega }} and … richmond syracuse basketballWeb1 feb. 1991 · CESIRO MEANS AND THE KREISS MATRIX THEOREM 103 It is essential to include the condition that the Ceso means are bounded for all 0 E R, rather than just 0 = 0, in order to eliminate the possibility of fortuitous cancellation. Here is an informative example. Let .F consist of a single matrix A - 1 -1 _1 0 1 Then powers of A are An ... richmond sydney weather forecastWebis known as Kreiss Matrix Theorem [Kr]. According to Tadmor, it has been shown originally by Kreiss (1962) with the inequality P(T) ≤ Cste(ρ(T))nn. It is useful in proofs of stability theorems for finite difference approximations to partial differential equations. Until 1991, the inequality of Kreiss has been improved successively by Morton, red rocks zip code