Understanding Solutions and Singular Matrices

Hey folks.

I'm working on a project which seems to be encountering a problem.

I took Linear Algebra a few years ago in college, and haven't really applied it very much so I'm at a bit of a loss here.



I have a solution to a iterative nonlinear least squares problem: Trilateration with n distances.

I'm implementing the algorithm found in this PDF in java using the JAMA library.



I don't understand much of the math here, and I need to figure out why my matrices are singular (unfortunately the inverse is part of the equation handed to me).

So I guess I'm asking a few things.



1. For Equation (40) (pg 18) R_{k+1}=R_{k}-(J^T_{k}J_{k})^-1J_{k}f

I'm ending up with a J^T_{k}J_{k} term which is singular. This means I cannot directly solve this equation. But while I'm familiar with solving equations of the form Ax=B I'm not really sure how to find an alternate solution for equation (40).



2. How robust is the JAMA library in general, is there a better linear algebra system for java?



Thanks!