faster way to compute `LU(A*A')`

Question

A fairly common operation in applied math is to compute the LU decompositiion of a matrix times it's transpose/adjoint. This can be problematic numerically since roundoff errors in A*A' can cause the matrix to not be positive (semi) definite. Also, computing this the obvious way involves 2 (technically not quite) cubic operations. Is there any way of computing this without first computing A*A' (iterative methods are OK).