Condition numbers for the outer inverse and constrained singular linear system

In this paper, we investigate the condition number of the outer inverse AT,S(2) and outer inverse AT,S(2) solution of constrained linear system Ax = b, x ∈ T, where A is a real m × n matrix, b and x are real vectors, T is a subspace. Let α and β be two positive real numbers, when we consider the weighted Frobenius norm ∥[αA,βb]∥QP(F) on the data we get the formula of condition number for the outer inverse AT,S(2) solution of constrained linear system. For the normwise condition number, the sensitivity of the relative condition number itself is studied, and the componentwise perturbation is also investigated.

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

[2]  Hebing Wu,et al.  (T,S) splitting methods for computing the generalized inverse and rectangular systems , 2001, Int. J. Comput. Math..

[3]  Yimin Wei,et al.  On integral representation of the generalized inverse AT, S(2) , 2003, Appl. Math. Comput..

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

[5]  Francis Hsuan,et al.  2 - Inverses and Their Statistical Application , 1988 .

[6]  Y. Wei,et al.  The representation and approximations of outer generalized inverses , 2004 .

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

[8]  Yimin Wei,et al.  A characterization and representation of the generalized inverse A(2)T,S and its applications , 1998 .

[9]  Yimin Wei,et al.  Operators with equal projections related to their generalized inverses , 2004, Appl. Math. Comput..

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

[11]  M. Z. Nashed,et al.  Convergence of Newton-like methods for singular operator equations using outer inverses , 1993 .

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

[13]  Xiaojun Chen,et al.  Convergence of Newton's Method for Singular Smooth and Nonsmooth Equations Using Adaptive Outer Inverses , 1997, SIAM J. Optim..

[14]  Hebing Wu,et al.  The representation and approximation for the generalized inverse AT, S(2) , 2003, Appl. Math. Comput..

[15]  M. Nashed.  Inner, outer, and generalized inverses in banach and hilbert spaces , 1987 .

[16]  H. Diao,et al.  Condition number for the w-weighted drazin inverse and its applications in the solution of rectangular linear system , 2006 .

[17]  Hebing Wu,et al.  On the perturbation and subproper splittings for the generalized inverse AT,S(2) of rectangular matrix A , 2001 .

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

[19]  Yimin Wei,et al.  Condition number related with generalized inverse A T,S (2) and constrained linear systems , 2003 .

[20]  Hua Xiang,et al.  Condition numbers with their condition numbers for the weighted moore-penrose inverse and the weighted least squares solution , 2006 .

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

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

[23]  Yimin Wei,et al.  Condition number for the Drazin inverse and the Drazin-inverse solution of singular linear system with their condition numbers , 2005 .

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