[KSCI] Korea Science Citation Index Service

A PRECONDITIONER FOR THE NORMAL EQUATIONS

Salkuyeh, Davod Khojasteh (Department of Mathematics, University of Mohaghegh Ardabili)

Publication Information

Journal of applied mathematics & informatics / v.28, no.3_4, 2010 , pp. 687-696 More about this Journal

Abstract

In this paper, an algorithm for computing the sparse approximate inverse factor of matrix $A^{T}\;A$ , where A is an $m\;{\times}\;n$ matrix with $m\;{\geq}\;n$ and rank(A) = n, is proposed. The computation of the inverse factor are done without computing the matrix $A^{T}\;A$ . The computed sparse approximate inverse factor is applied as a preconditioner for solving normal equations in conjunction with the CGNR algorithm. Some numerical experiments on test matrices are presented to show the efficiency of the method. A comparison with some available methods is also included.

Keywords

Inverse factors; Symmetric positive definite; Preconditioning; Incomplete QR; Incomplete C-orthogonalization; CGNR; PCGNR;

Citations & Related Records

Reference

1	O. Axelsson, Iterative solution methods, Cambridge University Press, Cambridge, 1996.
2	M. Benzi, Preconditioning techniques for large linear systems: A survey, J. of Computational Physics 182 (2002), 418-477. DOI ScienceOn
3	M. Benzi and M. Tuma, A comparative study of sparse approximate inverse preconditioners, Appl. Numer. Math. 30 (1999), 305-340. DOI ScienceOn
4	M. Benzi and M. Tuma, A robust preconditioner with low memory requirements for large sparse least squares problems, SIAM J. Sci. Comput. 25 (2003), 499-512. DOI ScienceOn
5	M. Benzi and M. Tuma, A robust incomplete factorization preconditioner for positive definite matrices, Numer. Linear Algebra Appl. 10 (2003), 385-400. DOI ScienceOn
6	A. Bjorck, Numerical methods for least squares problems, SIAM, Philadelphia, 1996.
7	B. N. Datta, Numerical linear algebra and applications, Pacific Grove, CA: Brooks/Cole Publishing Company, 1995.
8	T. Davis, University of Florida Sparse Matrix Collection, http://www.cise.ufl.edu/-davis/sparse/, 2008.
9	A. Jennings and M. A. Ajiz, Incomplete methods for saving $A^{T}$ Ax = b, SIAM J. Sci. Stat. Comput. 5 (1984), 978-987. DOI
10	National Institute of Standards, Matrix Market, http://math.nist.gov/MatrixMarket, 2008.
11	Y. Saad, Iterative Methods for Sparse linear Systems, PWS press, New York, 1995.
12	M. Tismenetsky, Anew preconditioning technique for solving large sparse linear systems, Linear Algebra Appl. 154/156 (1991), 331-353. DOI ScienceOn