Numerical methods for rectangular multiparameter eigenvalue problems, with applications to finding optimal ARMA and LTI models

TL;DR

This paper introduces numerical methods for solving rectangular multiparameter eigenvalue problems (RMEPs), with applications in optimizing ARMA models and LTI system realizations, by transforming them into standard MEPs for efficient computation.

Contribution

It presents new linearizations for quadratic multivariate matrix polynomials and a numerical approach that is more computationally attractive than existing methods.

Findings

Number of solutions for linear and polynomial RMEPs determined.

Transformation into standard MEP enables efficient numerical solutions.

New linearizations improve computational performance.

Abstract

Standard multiparameter eigenvalue problems (MEPs) are systems of $k\ge 2$ linear $k$-parameter square matrix pencils. Recently, a new form of multiparameter eigenvalue problems has emerged: a rectangular MEP (RMEP) with only one multivariate rectangular matrix pencil, where we are looking for combinations of the parameters for which the rank of the pencil is not full. Applications include finding the optimal least squares autoregressive moving average (ARMA) model and the optimal least squares realization of autonomous linear time-invariant (LTI) dynamical system. For linear and polynomial RMEPs, we give the number of solutions and show how these problems can be solved numerically by a transformation into a standard MEP. For the transformation we provide new linearizations for quadratic multivariate matrix polynomials with a specific structure of monomials and consider mixed systems of rectangular and square multivariate matrix polynomials. This numerical approach seems computationally considerably more attractive than the block Macaulay method, the only other currently available numerical method for polynomial RMEPs.