An Iterative Algorithm to Compute the Generalized Inverse A(2) T,S Under the Restricted Inner Product

Let T and S be a subspace of Cn and Cm, respectively. Then for A ∈ Cm×n satisfied AT ⊕ S = Cm, the generalized inverse A(2) T,S is given by A(2) T,S = (PS⊥APT )†. In this paper, a finite formulae is presented to compute generalized inverse A(2) T,S under the concept of restricted inner product, which defined as < A,B >T,S=< PS⊥APT,B > for the A,B ∈ Cm×n. By this iterative method, when taken the initial matrix X0 = PTA∗PS⊥, the generalized inverse A(2) T,S can be obtained within at most mn iteration steps in absence of roundoff errors. Finally given numerical example is shown that the iterative formulae is quite efficient.

Authors:

• Xingping Sheng

Keywords:

• Generalized inverse A(2) T
• S
• Restricted inner product
• Iterative method
• Orthogonal projection.

References:

[1] Ben-Israel A, Greville T. Generalized inverse: Theory and Applications.
2nd Edition, NewYork: Springer Verlag, 2003.
[2] Campbell S.L, Meyer C.D, Generalized Inverses of Linear Transformations,
Fearon-Pitman, Belmont, Calif., 1979.
[3] Chen Y. Iterative methods for computing the generalized inverses A(2)
T,S
of a matrix A. Applied Mathematics and Computation 1996;75(2-3):207-
222.
[4] Chen Y, Chen X. Representation and approximation of the outer inverse
A(2)
T,S of a matrix A. Linear Algebra and its Applications 2000; 308:85-
107.
[5] Djordjevi`c DS, Stanimirovi`c PS, Wei Y. The representation and approximations
of outer generalized inverses. Acta Mathematica Sinica,
Hungarica 2004; 104(1-2):1C26.
[6] Liu X, Hu C, Yu Y. Further results on iterative methods for computing
generalized inverses. Journal of Computational and Applied Mathematics
2010; 234:684-694.
[7] Liu X, Yu Y, Hu C. The iterative methods for computing the generalized
inverse A(2)
T,S of the bounded linear operator between Banach spaces.
Applied Mathematics and Computation 2009; 214:391-410.
[8] Liu X, Zhou G, Yu Y, Note on the iterative methods for computing
the generalized inverse over Banach spaces. Numerical Linear Algebra
With Applications, Article first published online: 14 JAN 2011-DOI:
10.1002/nla.763.
[9] S.K.Mitra and R.E.Hartwing, Partial orders based on outer inverse, Linear
Algebra Appl. 1992,176:3-20.
[10] M.Z.Nashed, Generalized Inverse and Applications, Academic Press,
NewYork,1976.
[11] Sheng X, Chen G. Several representations of generalized inverse A(2)
T,S
and their application. International Journal of Computer Mathematics
2008; 85(9):1441-1453.
[12] X.Sheng, G.Chen, The representation and computation of generalized
inverse A(2)
T,S, Journal of Computation and Applied Mathematics,
2008;213:248-257.
[13] X.Sheng, G.Chen, A finite iterative computing formula for M-P inverse
and Weighted M-P inverse, Mathematica Applicata, 20(2)( 2007)336-344
(In Chinese).
[14] Wei Y. A characterization and representation of the generalized inverse
A(2)
T,S and its applications. Linear Algebra and its Applications 1998;
280:87-96.
[15] Wei Y, Wu H. The representation and approximation for the generalized
inverse A(2)
T,S. Applied Mathematics and Computation 2003; 135:263-
276.
[16] Yu Y, Wei Y. The representation and computational procedures for the
generalized inverse A(2)
T,S of an operator A in Hilbert spaces. Numerical
Functional Analysis and Optimization 2009; 30(1-2):168-182.
[17] Saad Y, Numerical Methods for Large Eigenvalue Problems, Theory and
Algorithms, Wiley, New York, 1992.