منابع مشابه
Properties of Worst-Case GMRES
In the convergence analysis of the GMRES method for a given matrix A, one quantity of interest is the largest possible residual norm that can be attained, at a given iteration step k, over all unit norm initial vectors. This quantity is called the worst-case GMRES residual norm for A and k. We show that the worst case behavior of GMRES for the matrices A and A is the same, and we analyze proper...
متن کاملThe worst-case GMRES for normal matrices
We study the convergence of GMRES for linear algebraic systems with normal matrices. In particular, we explore the standard bound based on a min-max approximation problem on the discrete set of the matrix eigenvalues. This bound is sharp, i.e. it is attainable by the GMRES residual norm. The question is how to evaluate or estimate the standard bound, and if it is possible to characterize the GM...
متن کاملOn Worst-case Gmres, Ideal Gmres, and the Polynomial Numerical Hull of a Jordan Block
When solving a linear algebraic system Ax = b with GMRES, the relative residual norm at each step is bounded from above by the so-called ideal GMRES approximation. This worstcase bound is sharp (i.e. it is attainable by the relative GMRES residual norm) in case of a normal matrix A, but it need not characterize the worst-case GMRES behavior if A is nonnormal. Characterizing the tightness of thi...
متن کاملWorst-case and ideal GMRES for a Jordan block ⋆
We investigate the convergence of GMRES for an n by n Jordan block J . For each k that divides n we derive the exact form of the kth ideal GMRES polynomial and prove the equality max ‖v‖=1 min p∈πk ‖p(J)v‖ = min p∈πk max ‖v‖=1 ‖p(J)v‖, where πk denotes the set of polynomials of degree at most k and with value one at the origin, and ‖ · ‖ denotes the Euclidean norm. In other words, we show that ...
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Motivated by the current limitations of the existing algorithms for robust-ness analysis and design, in this paper we take a diierent direction which follows the so-called probabilistic approach. That is, we aim to estimate the probability that a control system with uncertain parameters q restricted to a box Q attains a given level of performance. Since this probability depends on the underlyin...
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ژورنال
عنوان ژورنال: SIAM Journal on Matrix Analysis and Applications
سال: 2013
ISSN: 0895-4798,1095-7162
DOI: 10.1137/13091066x