Finding the polar decomposition of a matrix by an efficient iterative method

Subject Areas : Statistics

F. Kiyoumarsi ¹

1 - Faculty of Basic Science, Shahrekord Branch, Islamic Azad University, Shahrekord, Iran

Received: 2018-06-11 Accepted : 2018-12-07 Published : 2019-02-20

Keywords: تجزیه قطبی, تکرار ماتریسی, عامل قطبی, روش تکراری,

Abstract :

Theobjective in this paper to study and present a new iterative method possessing high convergence order for calculating the polar decompostion of a matrix. To do this, it is shown that the new scheme is convergent and has high convergence. The analytical results are upheld via numerical simulations and comparisons.

References:

 

[1] L. Autonne, Sur les groupes lineaires, reels et orthogonaux, Bull. Sot. Math. France, 30 (1902), 121-134.

 

[2] D.K.R. Babajee, A. Cordero, F. Soleymani, J.R. Torregrosa, On improved three-step schemes with
high efficiency index and their dynamics, Numer. Algor., 65 (2014), 153-169.

 

[3] A. Ben-Israel, T.N.E. Greville, Generalized Inverses Theory and Applications, Second ed., SpringerVerlag, New York, 2003.

 

[4] K. Du, The iterative methods for computing the polar decomposition of rank-deficient matrix, Appl.
Math. Comput. 162 (2005), 95-102.

 

[5] A.A. Dubrulle, Frobenius iteration for the matrix polar decomposition, Hewlett-Packard Company,
1994.

[6] A. A. Dubrulle, An optimum iteration for the matrix polar decomposition, Elect. Trans. Numer. Anal.,
8 (1999), 21-25.

 

[7] W. Gander, On Halley’s iteration method, Amer. Math. Monthly, 92 (1985), 131-134.

[8] W. Gander, Algorithms for the polar decomposition, SIAM J. Sci. Stat. Comput., 11 (1990), 1102-1115.

[9] N.J. Higham, Computing the polar decomposition-with applications, SIAM J. Sci. Stat. Comput., 7
(1986), 1160-1174.

[10] N.J. Higham, The matrix sign decomposition and its relation to the polar decomposition, Lin. Alg.
Appl., 212/213 (1994), 3-20.

[11] N.J. Higham, D.S. Mackey, N. Mackey, F. Tisseur, Computing the polar decomposition and the matrix
sign decomposition in matrix groups, SIAM J. Matrix Anal. Appl., 25 (2004), 1178-1192.

[12] N.J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied
Mathematics, Philadelphia, PA, USA, 2008.

[13] C. Kenney, A.J. Laub, On scaling Newton’s method for polar decomposition and the matrix sign function, SIAM J. Matrix Appl., 13 (1992), 688-706.

[14] B. Laszkiewicz, K. Ziȩtak, Approximation of matrices and a family of Gander methods for polar decomposition, BIT Numerical Mathematics, 46 (2006), 345-366.

[15] M. Trott, The Mathematica Guide-Book for Numerics, Springer, New York, NY, USA, 2006.