Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A general class of iterative splitting methods for solving linear systems

Paolo Novati, Fulvio Tagliaferro, Marino Zennaro

TL;DR

The paper develops a general framework for iterative methods to solve linear systems by splitting the Jacobi iteration matrix $B_J$. It defines splittings ${\bf B}_{(d)}$ with masks that partition $B_J$ and studies their convergence via a refinement partial order, including the cyclicity of splittings and essentiality concepts. The analysis covers the nonnegative and irreducible $B_J$ case (L-matrices), extends convergence guarantees to strictly diagonally dominant matrices, and shows AMKS and T$_U$-type methods fit into this framework. Several new splitting families (TC(2,2), TR(2,2), AFTC/AFTR, ATC/ATR) are proposed, with numerical experiments illustrating that more refined splittings often yield faster convergence, though performance is problem-dependent. The work thus unifies existing methods, provides a spectrum-based ranking of convergence, and points to promising directions for parallel computation and further refinements.

Abstract

Recently Ahmadi et al. (2021) and Tagliaferro (2022) proposed some iterative methods for the numerical solution of linear systems which, under the classical hypothesis of strict diagonal dominance, typically converge faster than the Jacobi method, but slower than the forward/backward Gauss-Seidel one. In this paper we introduce a general class of iterative methods, based on suitable splittings of the matrix that defines the system, which include all of the methods mentioned above and have the same cost per iteration in a sequential computation environment. We also introduce a partial order relation in the set of the splittings and, partly theoretically and partly on the basis of a certain number of examples, we show that such partial order is typically connected to the speed of convergence of the corresponding methods. We pay particular attention to the case of linear systems for which the Jacobi iteration matrix is nonnegative, in which case we give a rigorous proof of the correspondence between the partial order relation and the magnitude of the spectral radius of the iteration matrices. Within the considered general class, some new specific promising methods are proposed as well.

A general class of iterative splitting methods for solving linear systems

TL;DR

The paper develops a general framework for iterative methods to solve linear systems by splitting the Jacobi iteration matrix . It defines splittings with masks that partition and studies their convergence via a refinement partial order, including the cyclicity of splittings and essentiality concepts. The analysis covers the nonnegative and irreducible case (L-matrices), extends convergence guarantees to strictly diagonally dominant matrices, and shows AMKS and T-type methods fit into this framework. Several new splitting families (TC(2,2), TR(2,2), AFTC/AFTR, ATC/ATR) are proposed, with numerical experiments illustrating that more refined splittings often yield faster convergence, though performance is problem-dependent. The work thus unifies existing methods, provides a spectrum-based ranking of convergence, and points to promising directions for parallel computation and further refinements.

Abstract

Recently Ahmadi et al. (2021) and Tagliaferro (2022) proposed some iterative methods for the numerical solution of linear systems which, under the classical hypothesis of strict diagonal dominance, typically converge faster than the Jacobi method, but slower than the forward/backward Gauss-Seidel one. In this paper we introduce a general class of iterative methods, based on suitable splittings of the matrix that defines the system, which include all of the methods mentioned above and have the same cost per iteration in a sequential computation environment. We also introduce a partial order relation in the set of the splittings and, partly theoretically and partly on the basis of a certain number of examples, we show that such partial order is typically connected to the speed of convergence of the corresponding methods. We pay particular attention to the case of linear systems for which the Jacobi iteration matrix is nonnegative, in which case we give a rigorous proof of the correspondence between the partial order relation and the magnitude of the spectral radius of the iteration matrices. Within the considered general class, some new specific promising methods are proposed as well.
Paper Structure (20 sections, 35 theorems, 269 equations)

This paper contains 20 sections, 35 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. A general class of iterative methods
  3. Cyclicity of the splittings
  4. Refinement of splittings and essentiality
  5. A general necessary condition for convergence
  6. The case of irreducible nonnegative Jacobi iteration matrices
  7. Splittings of nonnegative Jacobi iteration matrices
  8. Monotonicity of the splittings
  9. Acceleration of convergence
  10. Some remarks on the case of nonpositive Jacobi iteration matrices
  11. The case of strictly diagonally dominant matrices
  12. The AMKS-methods
  13. The T$_U$-method and its refinements
  14. Upper and lower triangular column methods
  15. Upper and lower triangular row methods
  16. ...and 5 more sections

Key Result

Proposition 1.4

Given a matrix $B\in {\mathbb R}^{n\times n}$, a splitting (of $B$) of order $d$ is a $d$-tuple of matrices ${\@fontswitch{}{\mathcal{}} B}_{(d)}=\{B_{1},\ldots ,B_{d}\}$, $B_{p} \in {\mathbb R}^{n\times n}$, such that: \newlabelproposition_splitting

Theorems & Definitions (86)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.1
  • Proposition 1.4
  • Remark 1.2
  • Definition 2.1
  • Lemma 2.2: Cyclicity
  • proof
  • Remark 2.1
  • ...and 76 more