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A parallel multisplitting method with self-adaptive weightings for solving H-matrix linear systems.

Ruiping Wen1, Hui Duan2

  • 1Higher Education Key Laboratory of Engineering and Scientific Computing, Taiyuan Normal University, Taiyuan, Shanxi 030012 P.R. China.

Journal of Inequalities and Applications
|May 23, 2017
PubMed
Summary

A new parallel iterative method uses self-adaptive weighting matrices to solve linear systems with H-matrices. This approach optimizes weighting matrices, releasing non-negativity restrictions and proving effective through numerical examples.

Keywords:
H-matrixconvergencelinear systemsparallel multisplittingself-adaptive weightings

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Area of Science:

  • Numerical analysis
  • Linear algebra
  • Computational mathematics

Background:

  • Solving large-scale linear systems is crucial in many scientific and engineering fields.
  • Iterative methods are commonly used for these systems, but their efficiency can be limited by matrix properties.
  • H-matrices offer a compressed representation for certain large matrices, enabling faster computations.

Purpose of the Study:

  • To introduce a novel parallel multisplitting iterative method for solving linear systems with H-matrix coefficient matrices.
  • To develop a self-adaptive weighting strategy for the iterative method, optimizing performance.
  • To establish the convergence theory for the proposed method and demonstrate its effectiveness.

Main Methods:

  • A parallel multisplitting iterative framework is employed.
  • Weighting matrices are utilized, with their zero patterns pre-determined.
  • Non-zero entries of weighting matrices are optimized within a hyperplane of alpha points.
  • The non-negative restriction on weighting matrices is relaxed.

Main Results:

  • The convergence theory for the parallel multisplitting method with self-adaptive weightings is rigorously established.
  • A numerical example demonstrates the practical effectiveness of the proposed method.
  • The method shows improved performance compared to existing techniques for H-matrix systems.

Conclusions:

  • The presented parallel multisplitting iterative method with self-adaptive weighting matrices is an effective approach for solving linear systems involving H-matrices.
  • The relaxation of non-negativity constraints and the adaptive optimization of weighting matrices contribute to the method's efficiency.
  • This work provides a valuable tool for accelerating computations in relevant scientific and engineering domains.