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A new stable numerical method for matrix inversion demonstrates twelfth-order convergence. This iterative approach is also effective for calculating the Moore-Penrose inverse, showing practical efficiency.

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Area of Science:

  • Numerical Analysis
  • Linear Algebra
  • Computational Mathematics

Background:

  • Matrix inversion is a fundamental operation in numerous scientific and engineering disciplines.
  • Existing numerical methods for matrix inversion may suffer from instability or slow convergence rates.
  • The Moore-Penrose inverse has broad applications in solving systems of linear equations and statistical analysis.

Purpose of the Study:

  • To propose a novel, stable numerical method for matrix inversion.
  • To theoretically prove the twelfth-order convergence of the proposed method.
  • To analytically demonstrate the method's applicability to computing the Moore-Penrose inverse.

Main Methods:

  • Development of a new iterative scheme for matrix inversion.
  • Theoretical analysis to establish the order of convergence.
  • Analytical investigation into the method's suitability for Moore-Penrose inverse computation.
  • Numerical experimentation to validate efficiency and stability.

Main Results:

  • A stable numerical method for matrix inversion is presented.
  • Theoretical proof confirms twelfth-order convergence under specific initial value conditions.
  • The method is analytically shown to be applicable for calculating the Moore-Penrose inverse.
  • Numerical examples demonstrate the method's efficiency.

Conclusions:

  • The proposed iterative method offers a stable and highly convergent approach for matrix inversion.
  • The method provides a robust tool for computing the Moore-Penrose inverse.
  • The findings contribute to the advancement of numerical linear algebra techniques.