A New Error in Variables Model for Solving Positive Definite Linear System Using Orthogonal Matrix Decompositions

TL;DR

This paper introduces a novel method for solving positive definite linear systems with errors in data and targets, using orthogonal matrix decompositions, which improves accuracy and efficiency over existing techniques.

Contribution

The paper presents a new error function-based approach for positive definite systems, deriving optimality conditions and a direct algorithm, considering errors in both data and target matrices.

Findings

The proposed method yields smaller error standard deviations.

It results in a smaller effective rank suitable for control problems.

Numerical tests show improved efficiency using Dolan-Moré profiles.

Abstract

The need to estimate a positive definite solution to an overdetermined linear system of equations with multiple right hand side vectors arises in several process control contexts. The coefficient and the right hand side matrices are respectively named data and target matrices. A number of optimization methods were proposed for solving such problems, in which the data matrix is unrealistically assumed to be error free. Here, considering error in measured data and target matrices, we present an approach to solve a positive definite constrained linear system of equations based on the use of a newly defined error function. To minimize the defined error function, we derive necessary and sufficient optimality conditions and outline a direct algorithm to compute the solution. We provide a comparison of our proposed approach and two existing methods, the interior point method and a method based on quadratic programming. Two important characteristics of our proposed method as compared to the existing methods are computing the solution directly and considering error both in data and target matrices. Moreover, numerical test results show that the new approach leads to smaller standard deviations of error entries and smaller effective rank as desired by control problems. Furthermore, in a comparative study, using the Dolan-Mor\'{e} performance profiles, we show the approach to be more efficient.