Explicit Characterization of a Class of Parametric Solution Sets
نویسنده
چکیده
Consider a linear system A(p) · x = b(p), where the elements of the matrix and the right-hand side vector depend affine-linearly on a m-tuple of parameters p = (p1, . . . , pm) varying within given intervals. It is a fundamental problem how to describe the parametric solution set Σ (A(p), b(p), [p]) := {x ∈ Rn | ∃p ∈ [p], A(p)x = b(p)}. So far, the solution set description can be obtained by a lengthy (and not unique) parameter elimination process. In this paper we introduce a new classification of the parameters with respect to the way they participate in the equations of the system and give numerical characterization for each class of parameters. For a class of parametric linear systems, where each uncertain parameter occurs in only one equation of the system and does not matter how many times within that equation, an explicit characterization of the parametric solution set is derived which generalizes the famous Oettly-Prager theorem for non-parametric linear systems.
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تاریخ انتشار 2009