We describe a new algorithm to compute Whitney stratifications of real and complex algebraic varieties. This algorithm is a modification of the algorithm of Helmer and Nanda (HN), but is made more efficient by using techniques for equidimensional decomposition rather than computing the set of associated primes of a polynomial ideal at a key step in the HN algorithm. We note that this modified algorithm may fail to produce a minimal Whitney stratification even when the HN algorithm would produce a minimal stratification. We, additionally, present an algorithm to coarsen any Whitney stratification of a complex variety to a minimal Whitney stratification; the theoretical basis for our approach is a classical result of Teissier.