Error and Perturbation Bounds for General Absolute Value Equations

Core Concepts

This paper presents a framework for deriving error bounds and perturbation bounds for two types of general absolute value equations, without limiting the matrix types. Computable estimates for the upper bounds are provided, which are shown to be sharper than existing bounds under certain conditions. The results are also applied to provide a new approach for some existing perturbation bounds of the linear complementarity problem.

Abstract

The paper focuses on studying the error and perturbation bounds of two types of general absolute value equations (AVEs): Ax - B|x| = b and Ax - |Bx| = b.
Key highlights:

By introducing a class of absolute value functions, a framework of error bounds for the AVEs is presented. Computable estimates for the upper bounds are provided, which are shown to be sharper than existing bounds under certain conditions.
A framework of perturbation bounds for the AVEs is established, and some computable upper bounds are presented. It is shown that when the nonlinear term B|x| in the first AVE is vanished, the perturbation bounds reduce to classical perturbation bounds for linear systems.
A new approach for some existing perturbation bounds of the linear complementarity problem is provided.
Numerical examples for the AVEs from the linear complementarity problem are investigated to show the feasibility of the proposed perturbation bounds.

Stats

The following sentences contain key metrics or figures:
The AVEs (1.1) has a unique solution for any b ∈ Rn if and only if A - BD is nonsingular for any diagonal matrix D = diag(di) with di ∈ [-1, 1].
The AVEs (1.2) has a unique solution for any b ∈ Rn if and only if A - DB is nonsingular for any diagonal matrix D = diag(di) with di ∈ [-1, 1].

Quotes

"To our knowledge, the error and perturbation bounds of the general absolute value equations are not discussed."
"Without limiting the matrix type, some computable estimates for their upper bounds are given. These bounds are sharper than the existing bounds in [7] under certain conditions."
"It is pointed out that when the nonlinear term B|x| in (1.1) is vanished, the presented perturbation bounds reduce to the classical perturbation bounds for the linear systems Ax = b, like Theorem 1.3 in numerical linear algebra textbooks [22] and Theorem 2.1 [23]."

Key Insights Distilled From

The error and perturbation bounds of the general absolute value equations

by Shi-Liang Wu... at arxiv.org 04-18-2024

https://arxiv.org/pdf/2207.00954.pdf

Deeper Inquiries

How can the error and perturbation bounds be extended to other types of nonlinear equations beyond absolute value equations

To extend the error and perturbation bounds to other types of nonlinear equations beyond absolute value equations, we can explore the use of different function classes that capture the nonlinearity of the equations. By introducing suitable function classes and analyzing their properties, we can derive error and perturbation bounds for a broader range of nonlinear equations. This may involve considering different norms, operators, or function spaces that are appropriate for the specific type of nonlinear equation under consideration. Additionally, techniques from functional analysis, numerical analysis, and optimization theory can be employed to generalize the error and perturbation bounds to various classes of nonlinear equations.

What are the limitations of the assumptions made in this work, and how can they be relaxed or generalized

The limitations of the assumptions made in this work include the specific form of the absolute value equations considered, the properties of the matrices A and B, and the nature of the perturbations introduced. To relax or generalize these assumptions, one could explore more general forms of nonlinear equations, relax the constraints on the matrices A and B, consider different types of perturbations, and investigate the impact of additional factors such as sparsity, structure, or regularity of the equations. By relaxing these assumptions, the error and perturbation bounds could be extended to a wider range of scenarios and applications, making them more versatile and applicable in diverse settings.

What are the potential applications of the derived error and perturbation bounds in other areas of computational mathematics and optimization

The derived error and perturbation bounds have potential applications in various areas of computational mathematics and optimization. Some potential applications include:

Numerical Methods: The error bounds can be used to assess the accuracy and convergence of numerical methods for solving nonlinear equations. By understanding the error behavior, researchers can develop more efficient and reliable numerical algorithms.
Optimization Algorithms: The perturbation bounds can be utilized in optimization algorithms to analyze the sensitivity of solutions to perturbations in the data. This information can guide the development of robust optimization techniques that are less sensitive to noise or uncertainties.
Machine Learning: In machine learning applications, error and perturbation bounds can help in understanding the stability and generalization capabilities of learning models. By quantifying the error and perturbation effects, researchers can improve the performance and reliability of machine learning algorithms.
Signal Processing: Error and perturbation bounds can be applied in signal processing tasks to analyze the impact of noise or disturbances on signal reconstruction or processing algorithms. Understanding the bounds can lead to more robust signal processing techniques.
Overall, the derived bounds can enhance the theoretical understanding and practical implementation of computational methods in various fields.