Determinants of Interval Matrices

TL;DR

This paper investigates the computational complexity of determining bounds on determinants of interval matrices, introduces a new efficient method based on Cramer's rule, and explores classes of matrices with polynomially computable determinant tasks.

Contribution

It proves NP-hardness of approximation, presents a new Cramer's rule-based method, and generalizes Gerschgorin circles for interval matrices.

Findings

NP-hardness of determinant bounds approximation established

New Cramer-based method achieves similar accuracy with less computation

Polynomial-time results for specific matrix classes like symmetric positive definite

Abstract

In this paper we shed more light on determinants of interval matrices. Computing the exact bounds on a determinant of an interval matrix is an NP-hard problem. Therefore, attention is first paid to approximations. NP-hardness of both relative and absolute approximation is proved. Next, methods computing verified enclosures of interval determinants and their possible combination with preconditioning are discussed. A new method based on Cramer's rule was designed. It returns similar results to the state-of-the-art method, however, it is less consuming regarding computational time. As a byproduct, the Gerschgorin circles were generalized for interval matrices. New results about classes of interval matrices with polynomially computable tasks related to determinant are proved (symmetric positive definite matrices, class of matrices with identity midpoint matrix, tridiagonal H-matrices). The mentioned methods were exhaustively compared for random general and symmetric matrices.