Explicit Characterization of a Class of Parametric Solution SetsEvgenija D. Popova
Consider a parameter-dependent linear system A(p)x = b(p), where the elements of the matrix and the right-hand side vector depend affine-linearly on a tuple of parameters p varying within given intervals. It is a fundamental problem how to describe the solution set of such systems by a logical combination of inequalities. So far, in the general case of arbitrary affine-linear parameter dependencies, the parametric solution set description can be obtained by a lengthy Fourier-Motzkin like elimination process (Alefeld et al., JCAM 152, 2003). We discuss a computer-algebra implementation of the elimination process and demonstrate that it is also not unique. Recently, explicit descriptions of the symmetric and skew-symmetric solution sets were given by M. Hlad'ik.
We introduce a new classification of the parameters 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, a simple explicit characterization of the parametric solution set is derived. The latter generalizes the famous Oettly-Prager theorem for non-parametric linear systems. Numerical examples and a comparison to quantifier elimination, implemented in CAS, are presented.