Condition number for the Drazin inverse and the Drazin-inverse solution of singular linear system with their condition numbers

In this paper, we investigate the condition number of Drazin inverse and Drazin-inverse solution of singular linear system Ax=b, where A is a nxn rank-deficient matrix and b a real vector of size n, x a real vector. Let @a and @b be two positive real numbers, when we consider the weighted Frobenius norm @?[@aA,@bb]@?"P","Q^(^F^) on the data we get the formula of condition number of the Drazin-inverse solution of singular linear system. For the normwise condition number, the sensitivity of the relative condition number itself is studied, the componentwise perturbation is also investigated.

[1]  Hebing Wu,et al.  Convergence properties of Krylov subspace methods for singular linear systems with arbitrary index , 2000 .

[2]  Avram Sidi,et al.  DGMRES: A GMRES-type algorithm for Drazin-inverse solution of singular non-symmetric linear systems , 2001 .

[3]  Jiri Rohn New condition numbers for matrices and linear systems , 2005, Computing.

[4]  C. D. Meyer,et al.  Generalized inverses of linear transformations , 1979 .

[5]  Wei Yimin,et al.  The perturbation theory for the Drazin inverse and its applications , 1997 .

[6]  Yimin Wei,et al.  The representation and approximation of the Drazin inverse of a linear operator in Hilbert space , 2003, Appl. Math. Comput..

[7]  Desmond J. Higham,et al.  Condition numbers and their condition numbers , 1995 .

[8]  Yi-min Wei,et al.  Structured perturbations of group inverse and singular linear system with index one , 2005 .

[9]  Joan-Josep Climent,et al.  A semi-iterative method for real spectrum singular linear systems with an arbitrary index , 1997, Journal of Computational and Applied Mathematics.

[10]  Yimin Wei,et al.  Additive results for the generalized Drazin inverse , 2002, Journal of the Australian Mathematical Society.

[11]  J. Demmel On condition numbers and the distance to the nearest ill-posed problem , 2015 .

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

[13]  I. Marek,et al.  On the solution of singular linear systems of algebraic equations by semiiterative methods , 1988 .

[14]  Hebing Wu,et al.  ADDITIONAL RESULTS ON INDEX SPLITTINGS FOR DRAZIN INVERSE SOLUTIONS OF SINGULAR LINEAR SYSTEMS , 2001 .

[15]  Yimin Wei,et al.  Stagnation analysis of DGMRES , 2004, Appl. Math. Comput..

[16]  Yimin Wei,et al.  The perturbation theory for the Drazin inverse and its applications II , 2001, Journal of the Australian Mathematical Society.

[17]  Gene H. Golub,et al.  Matrix computations , 1983 .

[18]  S. Qiao,et al.  Displacement rank of the Drazin inverse , 2004 .

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

[20]  Yimin Wei Perturbation bound of singular linear systems , 1999, Appl. Math. Comput..

[21]  Mei Han An,et al.  accuracy and stability of numerical algorithms , 1991 .

[22]  Serge Gratton,et al.  On the condition number of linear least squares problems in a weighted Frobenius norm , 1996 .

[23]  Wei Xu,et al.  Condition number of Bott-Duffin inverse and their condition numbers , 2003, Appl. Math. Comput..

[24]  Yimin Wei,et al.  Condition number of Drazin inverse and their condition numbers of singular linear systems , 2003, Appl. Math. Comput..

[25]  J. Rice A Theory of Condition , 1966 .

[26]  Yimin Wei,et al.  Perturbation analysis of singular linear systems with index one , 2000, Int. J. Comput. Math..