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The following quote is from "Matrix analysis and applied linear algebra" by Carl D Meyer, p 423-424. "Generalized inverses are useful in formulating theoretica

The following quote is from "Matrix analysis and applied linear algebra" by Carl D Meyer, p 423-424. "Generalized inverses are useful in formulating theoretical statements as those above, ut, just as in the case of the ordinary inverse, generalized inverses are not practical computational tools. In addition to being computationally inefficient, serious numerical problems result from the fact that A+ needs not be a continuous function of the entries of A. ... Not only is A+(x) discontinuous in the sense that lim_{x->0}A+(x)!=A+(0), but it is discontinuous in the worst way because as A(x) comes closer to A(0) the matrix A+(x) moves farther away from A+(0)." Here A+ denotes Moore-Penrose pseudo-inverse and A(x) = [1, 0; 0, x] in Matlab notation (or Numpy? I am very rusty on Matlab). And in case you are wondering why the generalized inverse is plural, there are different kinds of generalized inverses, Drazin inverse being a well defined alternative.

thearn4 12y ago
Pseudo-inverse (and other direct methods) do have a lot going against them, especially as problems scale. In practice, iterative methods introduce their own challenges as well though. Optimal algorithm selection is highly problem dependant, and often hinges on knowing how to generate an appropriate preconditioner. GMRES is still pretty much my favorite general method for inverse problems, but it comes with plenty of nuances that you really don't see when using a direct method on a tiny problem that you can bank on being well-conditioned. It's hard to say how often those kinds of problems exist in the real world (in my case: exceedingly rare), but they are still out there.