Perturbation Analysis for the Matrix-Scaled Total Least Squares Problem

Qun Wang,Longyan Li,Pingping Zhang

doi:10.4236/apm.2021.112008

Abstract

In this paper, we extend matrix scaled total least squares (MSTLS) problem with a single right-hand side to the case of multiple right-hand sides. Firstly, under some mild conditions, this paper gives an explicit expression of the minimum norm solution of MSTLS problem with multiple right-hand sides. Then, we present the Kronecker-product-based formulae for the normwise, mixed and componentwise condition numbers of the MSTLS problem. For easy estimation, we also exhibit Kronecker-product-free upper bounds for these condition numbers. All these results can reduce to those of the total least squares (TLS) problem which were given by Zheng et al. Finally, two numerical experiments are performed to illustrate our results.

Highlights

Consider the overdetermined linear system Ax ≈ b, where A ∈ m×n and b ∈ m, and the total least squares (TLS) problem can be formulated asmin [E f ], subject to ( A + E ) x = b + f . (1) E, f Fin many linear parameter estimation problems, the error of data matrix A on the left side of the approximate system may be scaled
We extend matrix scaled total least squares (MSTLS) problem with a single right-hand side to the case of multiple right-hand sides
X matrix-scaled total least squares (MSTLS) is a differentiable function with respect to the data, the condition numbers defined in Definition 3.1 can be exactly expressed in derivatives

Summary

Introduction

Consider the overdetermined linear system Ax ≈ b , where A ∈ m×n and b ∈ m , and the total least squares (TLS) problem can be formulated as (see [1]). Meng and Wei gave the exact expressions and their upper bounds of normwise, mixed and componentwise condition numbers of TLS problem with multiple right-hand sides in [16] [17]. These results extend the condition number theory of TLS problem with single right-hand side. All these results can reduce to those of the TLS problem which were given in [16]. We present a power method for calculating the normwise condition number of the MSTLS solution X MSTLS similar to the one in [20]

Preliminaries

Condition Numbers of XMSTLS

Normwise Condition Numbers

Mixed and Componentwise Condition Numbers

Numerical Experiments

Conclusion