determinants

Who would have thunk it: I have spent the last 25 years doing astrophysics in some form or another, and now I am preparing to co-write a paper on computing the determinants of matrices. Foreman-Mackey and I met with Mike O'Neil (NYU) and Sivaram Ambikasaran (NYU) (both Applied Math) today about making determinant calculations fast. The crazy thing is that linear algebra packages out there are happy to make matrix inversion fast, but they uniformly discourage, disparage, or express incredulity about the computation of determinants. I understand the issues—determinants have ungodly units and therefore ungodly magnitudes—but we need to compute them if we are going to compute Gaussian probability densities. Our matrices are somewhat sparse, but the key idea behind Ambikasaran's method is that the matrices are smooth (columns are good at predicting other columns), or, equivalently, that the matrices have low-rank sub-matrices inside them. Plus fastness.