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A higher order iterative method for computing the Drazin inverse.

F Soleymani1, Predrag S Stanimirović

  • 1Department of Mathematics, Zahedan Branch, Islamic Azad University, Zahedan, Iran.

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A novel computational method efficiently finds approximate matrix inverses, with extensions for general matrices and applications in solving linear systems.

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Area of Science:

  • Numerical analysis
  • Linear algebra

Background:

  • Approximate matrix inversion is crucial for solving large-scale linear systems.
  • Existing methods may lack convergence speed or applicability to general matrices.

Purpose of the Study:

  • To introduce and analytically establish a high-convergence rate method for approximate matrix inversion.
  • To extend this computational scheme for general square matrices, including Drazin inverse computation.
  • To demonstrate the method's utility in preconditioning linear systems and on large sparse matrices.

Main Methods:

  • Analytical establishment of a computational scheme for approximate matrix inversion.
  • Extension of the scheme to handle general square matrices.
  • Application and testing on large sparse matrices for linear system preconditioning.

Main Results:

  • A method with a high convergence rate for approximate inverses of nonsingular matrices.
  • An extended computational scheme applicable to general square matrices.
  • Demonstrated potential for Drazin inverse computation and preconditioning of linear systems.

Conclusions:

  • The proposed method offers an efficient approach for approximate matrix inversion.
  • The extended scheme broadens applicability to general matrices and Drazin inverse.
  • The technique shows promise for accelerating solutions to linear systems, especially large sparse ones.