Modifications of the Oettli-Prager Theorem with Application to the Eigenvalue Problem
نویسندگان
چکیده
(Jl{. n Xn set of real n x n matrices, ßRn set of real vectors with n components) there sometimes occurs the problem of varying the input data A, b within certain tolerances and looking for the set S of the resulting solutions x*. Examples of this problem are Wilkinson's backward analysis when solving linear systems on a computer (Wilkinson 1963) and an input-output model in economics which is regulated by (2) with input parameters A, band output x (Maier 1985). In the first example one solves (2) on a computer (assuming A to be nonsingular). Due to rounding errors one normally does not obtain the exact solution x* but another vector x. One accepts x as a good approximation of the exact solution x* if it can be interpreted as a solution of a nearby system Ax = b,where'nearby' means lA-AI :Sß, Ib-bl :Sdwith given tolerances 0 :Sß E ßRnxn, 0 :S d E ßRn. (Here and in the sequel, the absolute value I. land the inequality sign ':S' are understood entrywise.) In other words, one considers x as a good approximation for x* if and only if it belongs to the solution set
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