About the sharpness of the stability estimates in the Kreiss matrix theorem
نویسندگان
چکیده
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 ‖ · ‖ 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 )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.
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عنوان ژورنال:
- Math. Comput.
دوره 72 شماره
صفحات -
تاریخ انتشار 2003