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I need to compute the Drazin inverse $A^D$ of a singular M-matrix $A$, i.e., a matrix in the form $A=\lambda I -P$, where $P$ has nonnegative entries and $\lambda$ is the spectral radius (Perron value) of $P$. I already know the right and left eigenvectors $v$ and $u^T$ of $P$ with eigenvalue $\lambda$ (that is, the vectors in the left and right kernel of $A$).

1) Is there a Matlab subroutine around for computing Drazin inverses? I can't seem to find any, so I had to create my own (which is probably very inefficient)

2) Is there a way to exploit the knowledge of the two nullspaces (and the fact that they are 1-dimensional) to speed up this computation?

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  • $\begingroup$ Have you seen dx.doi.org/10.1137/S0895479891228279 already? $\endgroup$ – J. M. is not a mathematician Oct 2 '10 at 1:32
  • $\begingroup$ @J. M. I don't have access to the full-text from my institution, but from the abstract it does not seem related to what I'm doing. I already know the Perron right and left eigenvectors, I need to compute the group/Drazin inverse $\endgroup$ – Federico Poloni Oct 2 '10 at 12:34
  • $\begingroup$ By the way, an answer to question (ii) for the Moore-Penrose pseudo-inverse instead of the Drazin inverse would be most welcome as well. $\endgroup$ – Federico Poloni Oct 2 '10 at 12:35
  • $\begingroup$ Hey, I don't know if you've already found an answer to this, but it looks to me that the results of dx.doi.org/10.1007/BFb0120751 and dx.doi.org/10.1137/0131057 might be applicable to your problem. $\endgroup$ – J. M. is not a mathematician Dec 14 '11 at 13:28
  • $\begingroup$ The first link is a "DOI not found". I'll check the second, though, thanks! $\endgroup$ – Federico Poloni Dec 14 '11 at 20:11
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If you have access to Matlab, and the matrix is not gigantic you can use the following to calculate the Drazin inverse:

  1. Determine the index of $A$, i.e, does $\mathrm{rank}(A^k) = \mathrm{rank}(A^{k+1})$

  2. Solve the linear system $A^{k+1} X = A^k$

  3. Use the fact that $A^D$, the Drazin inverse, is given by

$$A^D = A^k X^{k+1}$$

Jim Shoaf, N. C. Central University

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  • $\begingroup$ Thanks! Could you please expand on how to solve that singular linear system? If I am not missing some crucial detail, it could even be $0X=0$, for instance in the case when A=[0 1; 0 0]. $\endgroup$ – Federico Poloni Aug 26 '15 at 22:06
  • $\begingroup$ 1) One way to try to solve A^(k+1)*X = A^k in Matlab is X = A^(k+1) \ A^k , where these powers of A are stored as Ak2 and Ak 2) Another approach to calculating Ad is the Schulzz (Hoteling) iteration: X = .0001*A repeat X = 2*X - XAX (until convergence -- sometimes divergence!!) $\endgroup$ – Jim Shoaf Jul 25 at 4:41
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This one might have some pertinent leads:

http://www.sciencedirect.com/science/article/pii/0024379585902356

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