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The question now is: what should the result of inv (A) be, with A a sparse singular matrix?
A is a full singular matrix: both Octave and Matlab return inv (A) as a full matrix of the same dimension of A with Inf as elements, that is Inf (size (A)), plus a warning.
A is a sparse singular matrix: Matlab returns, in most cases, a *full* matrix with +/- Inf and/or NaN. Not clear to find a pattern. Octave returns a "division by zero" error.
My proposal, for A sparse singular, is to return a matrix with the same sparsity pattern of A, with Inf as elements, plus a warning. This solution has the advantage to not return a full useless matrix for a sparse singular
input. The drawback is that the inverse of the sparse null matrix would be the input itself. In this specific case, I propose to turn the warning to a dedicated error. Additionally, I propose to extend this last behavior to the full null case. I cannot imagine
a situation in which a user wants to invert the null matrix, except for a coding mistake.