What is the fastest algorithm for computing the inverse matrix and its determinant for positive definite symmetric matrices?

For problems I am interested in, the matrix dimension is 30 or less.

As WolfgangBangerth notes, unless you have a large number of these matrices (millions, billions), performance of matrix inversion typically isn't an issue.

Given a positive definite symmetric matrix, what is the fastest algorithm for computing the inverse matrix and its determinant?

If speed is an issue, you should answer the following questions:

• Do you really need the whole inverse? (Many applications don't need to form an explicit inverse.)
• Do you really need the determinant? (Determinants are uncommon, but certainly not unheard of in computational science.)
• Do you need either to high accuracy? (Low accuracy algorithms tend to be faster.)
• Would a probabilistic approximation suffice? (Probabilistic algorithms tend to be faster.)

The standard response to your problem of inverting a small, positive definite matrix and calculating its determinant would be Cholesky decomposition. If $A = LL^{T}$, then $det(A) = prod_{i=1}^{n}l_{ii}^{2}$, and $det(A^{-1}) = prod_{i=1}^{n}l_{ii}^{-2}$.

Assuming $A$ is $n$ by $n$, the Cholesky decomposition can be computed in around $n^{3}/3$ flops, which is about half the cost of an LU decomposition. However, such an algorithm would not be considered "fast". A randomized LU decomposition might be a faster algorithm worth considering if (1) you really do have to factor a large number of matrices, (2) the factorization is really the limiting step in your application, and (3) any error incurred in using a randomized algorithm is acceptable. Your matrices are probably too small for sparse algorithms to be worthwhile, so the only other opportunities for faster algorithms would require additional matrix structure (e.g., banded), or exploiting problem structure (e.g., maybe you can cleverly restructure your algorithm so that you no longer need to calculate a matrix inverse or its determinant). Efficient determinant algorithms are roughly the cost of solving a linear system, to within a constant factor, so the same arguments used for linear systems apply to calculating determinants as well.