A Higher Order Iterative Method for Computing the Drazin Inverse

A method with high convergence rate for finding approximate inverses of nonsingular matrices is suggested and established analytically. An extension of the introduced computational scheme to general square matrices is defined. The extended method could be used for finding the Drazin inverse. The application of the scheme on large sparse test matrices alongside the use in preconditioning of linear system of equations will be presented to clarify the contribution of the paper.

[1]  Victor Y. Pan,et al.  Homotopic residual correction processes , 2006, Math. Comput..

[2]  Stan Wagon.  Mathematica in action , 1991 .

[3]  V. Pan.  Structured Matrices and Polynomials: Unified Superfast Algorithms , 2001 .

[4]  Jiri Rohn,et al.  INVERSE INTERVAL MATRIX: A SURVEY ∗ , 2011 .

[5]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[6]  Ronald R. Coifman,et al.  Wavelet-Like Bases for the Fast Solution of Second-Kind Integral Equations , 1993, SIAM J. Sci. Comput..

[7]  G. Schulz.  Iterative Berechung der reziproken Matrix , 1933 .

[8]  E. V. Krishnamurthy,et al.  Numerical algorithms : computations in science and engineering , 2008 .

[9]  Guanrong Chen,et al.  Introduction to random signals and applied Kalman filtering, 2nd edn. Robert Grover Brown and Patrick Y. C. Hwang, Wiley, New York, 1992. ISBN 0‐471‐52573‐1, 512 pp., $62.95. , 1992 .

[10]  L. Grosz.  Preconditioning by incomplete block elimination , 2000, Numer. Linear Algebra Appl..

[11]  J. H. Wilkinson.  Note on the practical significance of the Drazin inverse , 1979 .

[12]  Ivan Kyrchei.  Explicit formulas for determinantal representations of the Drazin inverse solutions of some matrix and differential matrix equations , 2013, Appl. Math. Comput..

[13]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[14]  G. Stewart,et al.  On the Numerical Properties of an Iterative Method for Computing the Moore–Penrose Generalized Inverse , 1974 .

[15]  Adi Ben-Israel,et al.  Generalized inverses: theory and applications , 1974 .

[16]  Yimin Wei,et al.  Iterative methods for the Drazin inverse of a matrix with a complex spectrum , 2004, Appl. Math. Comput..

[17]  Yimin Wei.  Index splitting for the Drazin inverse and the singular linear system , 1998, Appl. Math. Comput..

[18]  Yong Zhang,et al.  Chebyshev-type methods and preconditioning techniques , 2011, Appl. Math. Comput..

[19]  M. Drazin.  Pseudo-Inverses in Associative Rings and Semigroups , 1958 .

[20]  Wai-Kai Chen,et al.  The Electrical Engineering Handbook , 2004 .