A Competitive Error in Variables Approach and Algorithms for Finding Positive Definite Solutions of Linear Systems of Matrix Equations

Abstract

Here, we refine our recent proposed method for computing a positive definite solution to an overdetermined linear system of equations with multiple right-hand sides. This problem is important in several process control contexts including quadratic models for optimal control. The coefficient and the right-hand side matrices are, respectively, named data and target matrices. In several existing approaches, the data matrix is unrealistically assumed to be error free. We have recently presented an algorithm for solving such a problem considering error in measured data and target matrices. We defined a new error in variables (EIV) error function considering error for the variables, the necessary and sufficient optimality conditions and outlined an algorithm to directly compute a solution minimizing the defined error. Moreover, the algorithm was specialized for a special case when the data and target matrices are rank deficient. Here, after giving a detailed review of the proposed algorithms, we rewrite the algorithms by use of the Householder tridiagonalization instead of spectral decomposition. We then show that in case of full rank data and target matrices, the Householder tridiagonalization is more efficient than the previously considered spectral decomposition. A comparison of our proposed approach and two existing methods are provided. The numerical test results and the associated Dolan-Moré performance profiles show the new approach to be more efficient than the two methods and have comparatively smaller standard deviation of error entries and smaller effective rank as features being sought for control problems.