I have a program that uses JAMA and need to test is a matrix can be inverted. I know that I can just try it and catch an exception but that seems like a bad idea (having a catch block as part of the "normal" code path seems to be bad form).
A test that also returns the inverse (or run in a better O() than the inverse operation) would be preferred.
In general, if you can't solve the matrix, it's singluar (non-invertable). I believe the way that JAMA does this is to try to solve the matrix using LU factorization, and if it fails, it returns "true" for isSingular().
There isn't really a general way to just look at the elements of a matrix and determine if it is singular - you need to check each column to see if it is orthogonal with the others (i.e. the nullspace for the matrix is 0). LU factorization is quite fast, usually... there are times where it takes the bulk of an operation, however.
Do you have an actual speed problem you're trying to overcome?
For a square matrix, you could you just check its determinant --- Matrix.det() returns zero if and only if the matrix is singular. Of course, you will need to watch out for ill-conditioned matrices too.
If an exception is thrown, what is your recovery position?
If you do an LU decomposition and find that it's singular, do you catch the exception and try an SVD (singular value decomposition) instead?
sounds like you would like to estimate the reciprocal of the condition number.
This site looks somewhat promising...
See also Golub and Van Loan, p. 128-130. (If you don't have a copy of it, get one.)
...or Higham, who's an authority on numerical methods. Except that it's kind of hard to parse the math... like walking through a thicket of raspberry bushes. :/
Or maybe check the Octave sources for their version of MATLAB's rcond(). I found this post.