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Area of Science:

  • Numerical Analysis
  • Linear Algebra

Background:

  • Large-scale linear systems require efficient solution methods.
  • Backward and Jacobi iteration are common algorithms, with convergence being a critical factor.

Purpose of the Study:

  • To propose a unified backward iterative matrix.
  • To analyze and prove convergence properties of iterative methods.

Main Methods:

  • Development of a unified backward iterative matrix.
  • Theoretical analysis of spectral radii for Jacobi and backward iterative matrices.
  • Conducting numerical experiments to validate the algorithms.

Main Results:

  • The unified matrix unifies several known iterative algorithms.
  • Convergence proofs are established for both Jacobi and backward iterations.
  • The spectral radius of the Jacobi iterative matrix is positive, while the backward iterative matrix's is strongly positive.
  • Both iteration types exhibit simultaneous convergence or divergence.

Conclusions:

  • The proposed unified backward iterative matrix is effective.
  • The theoretical convergence results are validated by numerical experiments.
  • The backward methods offer distinct advantages for solving linear systems.