Solving the interval eigenvalue problem via optimization and constraint interval theory

Abstract

This study develops the theory and numerical methods for solving the interval eigenvalue problem by optimizing the Constraint Interval Representation (CIR) of the problem that improves the computed results with respect to the accuracy compared to previous methods. The CIR is able to represent how the elements in the matrix are connected. Short example problems demonstrate the accuracy of the theory to find boundaries for interval eigenvalues and their associated matrices. While the real interval eigenvalue problem is a subset of the complex interval eigenvalue case, we develop each case separately. Indications of how to go from interval methods to the fuzzy eigenvalue problem are mentioned but not developed. Issues of complexity are discussed in terms of the complexity of the optimization methods that are used.