Iterative Splitting Methods for Solving Linear Systems

Core Concepts

This paper introduces a general class of iterative splitting methods for solving linear systems, which include previously proposed methods like Jacobi, Gauss-Seidel, and some recently introduced variants as special cases. The authors analyze the convergence properties of this general class of methods, establishing connections between the partial order of the splittings and the convergence rates.

Abstract

The paper introduces a general class of iterative splitting methods for solving linear systems of the form Ax = b, where A is the coefficient matrix. The key idea is to split the Jacobi iteration matrix BJ = L + U into multiple parts, and perform the iteration in successive steps using these parts.

The main highlights and insights are:

1. The authors define the concept of a splitting mask M(d) and the corresponding splitting B(d) of the Jacobi iteration matrix BJ. This allows them to formulate a general iterative scheme (2.2) that encompasses various existing methods as special cases.

2. The authors prove the cyclicity property of the splittings, showing that cyclic shifts of a splitting lead to spectrum-equivalent iteration matrices.

3. The authors introduce a partial order relation of "refinement" on the set of splittings, and show that more refined splittings typically lead to faster convergence, especially when the Jacobi iteration matrix BJ is nonnegative.

4. The authors extend the classical convergence results under strict diagonal dominance to the whole class of splitting methods proposed.

5. The authors show that the iterative methods proposed by Ahmadi et al. (2021) and Tagliaferro (2022) are special cases of the general class introduced here.

6. The authors propose some new specific splitting methods, such as the "alternate triangular column/row methods", which seem to have good potential for fast convergence.

Overall, the paper provides a unifying framework for analyzing and developing efficient iterative splitting methods for solving large linear systems.

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Key Insights Distilled From

A general class of iterative splitting methods for solving linear systems

by Paolo Novati... at arxiv.org 04-11-2024

https://arxiv.org/pdf/2404.06800.pdf

Deeper Inquiries

How can the proposed general class of splitting methods be extended to handle non-square or non-invertible coefficient matrices A

The proposed general class of splitting methods can be extended to handle non-square or non-invertible coefficient matrices A by considering the matrix splitting in a way that preserves the essential properties required for convergence. Even though the matrix may not be square or invertible, the splitting can still be performed based on suitable criteria such as diagonal dominance or other characteristics that ensure convergence. By adapting the splitting masks and the iterative schemes to accommodate non-square or non-invertible matrices, the general class of methods can still be applied effectively to solve linear systems with such matrices.

What are the implications of the partial order relation on the splittings in terms of the computational complexity and memory requirements of the corresponding iterative schemes

The partial order relation on the splittings has implications on the computational complexity and memory requirements of the corresponding iterative schemes. When a more refined splitting is achieved through the partial order relation, it often leads to faster convergence rates of the iterative methods. This can result in reduced computational time and memory usage per iteration, making the iterative schemes more efficient. By organizing the splittings in a hierarchical manner based on the partial order relation, it becomes possible to optimize the convergence speed while managing computational resources effectively.

Can the ideas presented in this paper be generalized to develop efficient iterative methods for solving nonlinear systems of equations

The ideas presented in this paper can be generalized to develop efficient iterative methods for solving nonlinear systems of equations by adapting the concept of matrix splitting and iterative schemes to handle the nonlinearities in the system. By incorporating nonlinear terms or functions into the iterative process, it is possible to iteratively approximate the solution to nonlinear systems. This may involve techniques such as Newton's method, fixed-point iteration, or other nonlinear solvers within the framework of the general class of splitting methods. By carefully designing the splitting masks and updating rules to account for the nonlinearities, efficient iterative methods can be developed for solving nonlinear systems of equations.