Solving Equations (and Systems of Equations) Under Uncertainty: How Different Practical Problems Lead to Different Mathematical and Computational Formulations

Authors

Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

12-2015

Comments

Technical Report: UTEP-CS-15-64a

To appear in Granular Computing

Abstract

Many practical problems are naturally reduced to solving systems of equations. There are many efficient techniques for solving well-defined systems of equations, i.e., systems in which we know the exact values of all the parameters and coefficients. In practice, we usually know these parameters and coefficients with some uncertainty -- uncertainty usually described by an appropriate granule: interval, fuzzy set, rough set, etc. Many techniques have been developed for solving systems of equations under such granular uncertainty. Sometimes, however, practitioners use previously successful techniques and get inadequate results. In this -- mostly pedagogical -- paper, we explain that to obtain an adequate solution, we need to take into account not only the system of equations and the granules describing uncertainty: we also need to take into account the original practical problem -- and for different practical problems, we get different solutions to the same system of equations with the same granules. This need is illustrated mainly on the example of interval uncertainty, the simplest type of uncertainty.