A new error in variables model for solving positive definite linear system using orthogonal matrix decompositions

Negin Bagherpour,Nezam Mahdavi-Amiri

doi:10.1007/s11075-015-0042-2

Negin Bagherpour, Nezam Mahdavi-Amiri

Open Access

https://doi.org/10.1007/s11075-015-0042-2

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Journal: Numerical Algorithms	Publication Date: Sep 2, 2015
Citations: 27	License type: cc-by

Affiliation: Sharif University of Technology

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-More performance profiles, we show the approach to be more efficient.

Highlights

Computing a symmetric positive definite solution of an overdetermined linear system of equations arises in a number of physical problems such as estimating the mass inertia matrix in the design of controllers for solid structures and robots; see, e.g., [9], [17], [14]
DX ≃ T, where D, T ∈ Rm×n, with m ≥ n, are given and a symmetric positive definite matrix X ∈ Rn×n is to be computed as a solution of (1.1)
A number of least squares formulations have been proposed for physical problems, which may be classified as ordinary and error in variables (EIV) models

Summary

Introduction

Computing a symmetric positive definite solution of an overdetermined linear system of equations arises in a number of physical problems such as estimating the mass inertia matrix in the design of controllers for solid structures and robots; see, e.g., [9], [17], [14]. No EIV model, even the well-known total least squares formulation, is considered for solving the positive definite linear system of equations in the literature. Several approaches have been proposed for this problem, commonly considering the ordinary least squares formulation and minimizing the error ∆T F over all n × n symmetric positive definite matrices, where. Considering this new formulation for E, it can be concluded that by use of our newly defined EIV model, computing a symmetric and positive definite solution to the overdetermined system of equations DX ≃ T is equivalent to computing a nonsingular matrix Y ∈ Rn×n to be the solution of min.

Mathematical Solution