ON A FAST ITERATIVE METHOD FOR APPROXIMATE INVERSE OF MATRICES

- Journal title : Communications of the Korean Mathematical Society
- Volume 28, Issue 2, 2013, pp.407-418
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/CKMS.2013.28.2.407

Title & Authors

ON A FAST ITERATIVE METHOD FOR APPROXIMATE INVERSE OF MATRICES

Soleymani, Fazlollah;

Soleymani, Fazlollah;

Abstract

This paper studies a computational iterative method to find accurate approximations for the inverse of real or complex matrices. The analysis of convergence reveals that the method reaches seventh-order convergence. Numerical results including the comparison with different existing methods in the literature will also be considered to manifest its superiority in different types of problems.

Keywords

Hotelling-Bodewig algorithm;ill-conditioned;approximate inverse;initial matrix;

Language

English

Cited by

1.

2.

3.

4.

5.

6.

7.

9.

10.

References

1.

A. Ben-Israel and D. Cohen, On iterative computation of generalized inverses and associated projections, SIAM J. Numer. Anal. 3 (1966), 410-419.

2.

A. Ben-Israel and T. N. E. Greville, Generalized Inverses, Springer, 2nd edition, 2003.

3.

W. Cao and B. Guo, Preconditioning for the p-version boundary element method in three dimension with tringaular elements, J. Korean Math. Soc. 41 (2004), no. 2, 345-368.

4.

H. Chen and Y. Wang, A family of higher-order convergent iterative methods for computing the Moore-Penrose inverse, Appl. Math. Comput. 218 (2011), no. 8, 4012-4016.

5.

en.wikipedia.org/wiki/Invertible_matrix.

6.

J. M. Garnett, A. Ben-Israel, and S. S. Yau, A hyperpower iterative method for computing matrix products involving the generalized inverse, SIAM J. Numer. Anal. 8 (1971), 104-109.

7.

H. Hotelling, Analysis of a complerx statistocal variable into principal components, J. Educ. Psysh. 24 (1933), 498-520.

8.

E. V. Krishnamurthy and S. K. Sen, Numerical Algorithms, Computations in science and engineering. Affiliated East-West Press Pvt. Ltd., New Delhi, 1986.

9.

H.-B. Li, T.-Z. Huang, Y. Zhang, X.-P. Liu, and T.-X. Gu, Chebyshev-type methods and preconditioning techniques, Appl. Math. Comput. 218 (2011), no. 2, 260-270.

10.

W. Li and Z. Li, A family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non-square matrix, Appl. Math. Comput. 215 (2010), no. 9, 3433-3442.

11.

M. Monsalve and M. Raydan, A new inversion-free method for a rational matrix equation, Linear Algebra Appl. 433 (2010), no. 1, 64-71.

12.

Y. Nakatsukasa, Z. Bai, and F. Gygi, Optimizing Halley's iteration for computing the matrix polar decomposition, SIAM. J. Matrix Anal. Appl. 31 (2010), no. 5, 2700-2720.

13.

V. Pan and R. Schreiber, An improved Newton iteration for the generalized inverse of a matrix, with applications, SIAM J. Sci. Statist. Comput. 12 (1991), no. 5, 1109-1130.

14.

I. Pavaloiu and E. Catina, Remarks on some Newton and Chebyshev-type methods for approximation eigenvalues and eigenvectors of matrices, Comput. Sci. J. Moldova 7 (1999), no. 1, 3-17.

15.

S. M. Rump, Inversion of extremely ill-conditioned matrices in floating-point, Japan J. Indust. Appl. Math. 26 (2009), no. 2-3, 249-277.

17.

S. Wolfram, The Mathematica Book, 5th edition, Wolfram Media, 2003.