Abstract
In order to estimate error bounds on the computed Drazin inverse of a matrix, we need to establish some perturbation theory for the Drazin inverse which is analogous to that for the Moore–Penrose inverse. In this paper, we present recent results on this topic, three problems are put forward in this direction.
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References
K.E. Brenan, S.L. Campbell and L.P. Petzold, Numerical Solution of Initial-Value Problems in Differential Algebraic Equations, Classics in Applied Mathematics 14 (SIAM, Philadelphia, PA, 1996).
S.L. Campbell and C.D. Meyer, Continuity properties of the Drazin pseudoinverses, Linear Algebra Appl. 10 (1975) 77–83.
S.L. Campbell and C.D.Meyer, Generalized Inverse of Linear Transformations (Pitman, 1979; Dover, New York, 1991).
N. Castro González and J.J. Koliha, Perturbation of Drazin inverse for closed linear operators, Integral Equations Operator Theory 36 (2000) 92–106.
N. Castro González, J.J. Koliha and Y.M. Wei, Perturbation of the Drazin inverse for matrices with equal eigenprojections at zero, Linear Algebra Appl. 312 (2000) 181–189.
Y. Chen, S.J. Kirkland and M. Neumann, Group generalized inverses of M-matrices associated with periodic and nonperiodic Jacobi matrices, Linear and Multilinear Algebra 39 (1995) 325–340.
Y. Chen, S.J. Kirkland and M. Neumann, Nonnegative alternating circulants leading to M-matrix group inverses, Linear Algebra Appl. 233 (1996), 81–97.
M.P. Drazin, Pseudoinverses in associative rings and semigroups, Amer. Math. Monthly 65 (1958) 506–514.
M. Eiermann, I. Marek and W. Niethammer, On the solution of singular linear systems of algebraic equations by semiiterative methods, Numer. Math. 53 (1988) 265–283.
S.J. Kirkland, M. Neumann and B.L. Shader, Applications of Paz's inequality to perturbation bounds for Markov chains, Linear Algebra Appl. 268 (1998) 183–196.
J.J. Koliha and V. Rakočević, Continuity of the Drazin inverse II, Studia Math. 131 (1998) 167–177.
C.D. Meyer, The role of the group generalized inverse in the theory of finite Markov chains, SIAM Rev. 17 (1975) 443–464.
C.D. Meyer, The condition of a finite Markov chain and perturbations bounds for the limiting probabilities, SIAM J. Algebra Discrete Math. 1 (1980) 273–283.
C.D. Meyer, Sensitivity of the stationary distribution of a Markov chain, SIAM J. Matrix Anal. Appl. 15 (1994) 715–728.
C.D. Meyer and J.M. Shoaf, Updating finite Markov chains by using techniques of group inversion, J. Statist. Comput. Simulation 11 (1980) 163–181.
M.Z. Nashed and Y. Zhao, The Drazin inverse for singular evolution equations and partial differential operators, World Sci. Ser. Appl. Anal. 1 (1992) 441–456.
V. Rakočević, Continuity of the Drazin inverse, J. Operator Theory 41 (1999) 55–68.
V. Rakočević and Y.M. Wei, The perturbation theory for the Drazin inverse and its application II, J. Austral. Math. Soc. 70 (2001) 189–197.
G. Rong, The error bound of the perturbation of the Drazin inverse, Linear Algebra Appl. 47 (1982) 159–168.
G.W. Stewart, On the continuity of the generalized inverse, SIAM J. Appl. Math. 17 (1969) 33–45.
Y.M. Wei, Index splitting for the Drazin inverse and singular linear system, Appl. Math. Comput. 95 (1998) 115–124.
Y.M. Wei, Expressions for the Drazin inverse of a 2 × 2 block matrix, Linear and Multilinear Algebra 45 (1998) 131–146.
Y.M. Wei, On the perturbation of the group inverse and oblique projection, Appl. Math. Comput. 98 (1999) 29–42.
Y.M. Wei, The Drazin inverse of updating of a square matrix with application to perturbation formula, Appl. Math. Comput. 108 (2000) 77–83.
Y.M. Wei, Solving singular linear systems and generalized inverse, Ph.D. Thesis, Institute of Mathematics, Fudan University, Shanghai, China (March 1997).
Y.M. Wei, Perturbation analysis of singular linear systems with index one, Internat. J. Comput. Math. 74 (2000) 483–491.
Y.M. Wei and G.R. Wang, The perturbation theory for the Drazin inverse and its application, Linear Algebra Appl. 258 (1997) 179–186.
Y.M. Wei and H.B. Wu, On the perturbation of the Drazin inverse and oblique projection, Appl. Math. Lett. 13 (2000) 77–83.
Y.M. Wei and H.B. Wu, Convergence properties of Krylov subspace methods for singular linear systems with arbitrary index, J. Comput. Appl. Math. 114 (2000) 305–318.
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Wei, Y., Wu, H. Challenging Problems on the Perturbation of Drazin Inverse. Annals of Operations Research 103, 371–378 (2001). https://doi.org/10.1023/A:1012993626289
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DOI: https://doi.org/10.1023/A:1012993626289