An Efficient Iterative Approach for Hermitian Matrices Having a Fourth-Order Convergence Rate to Find the Geometric Mean

The target of this work is to present a multiplication-based iterative method for two Hermitian positive definite matrices to find the geometric mean. The method is constructed via the application of the matrix sign function. It is theoretically investigated that it has fourth order of convergence....

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Bibliographic Details
Published in:Mathematics (Basel) 2024-06, Vol.12 (11), p.1772
Main Authors: Liu, Tao, Li, Ting, Ullah, Malik Zaka, Alzahrani, Abdullah Khamis, Shateyi, Stanford
Format: Article
Language:English
Subjects:
Average
basins of attraction
Convergence
fourth order of convergence
fractal
geometric mean
iterative approach
Iterative methods
Iterative methods (Mathematics)
Mathematical functions
Matrices
Matrices (mathematics)
matrix sign
Methods
Numerical analysis
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Summary:The target of this work is to present a multiplication-based iterative method for two Hermitian positive definite matrices to find the geometric mean. The method is constructed via the application of the matrix sign function. It is theoretically investigated that it has fourth order of convergence. The type of convergence is also discussed, which is global under an appropriate choice of the initial matrix. Numerical experiments are reported based on input matrices of different sizes as well as various stopping termination levels with comparisons to methods of the same nature and same number of matrix–matrix multiplications. The simulation results confirm the efficiency of the proposed scheme in contrast to its competitors of the same nature.
ISSN:2227-7390
2227-7390
DOI:10.3390/math12111772