We describe an algorithm to efficiently compute maximum entropy densities, i.e. densities maximizing the Shannon entropy - under a set of constraints . Our method is based on an algorithm by Zellner and Highfield, which has been found not to converge under a variety of circumstances. To demonstrate that our method overcomes these difficulties, we conduct numerous experiments for the special case g i (x) = x i , n = 4. An extensive table of results for this case and computer code are available on the World Wide Web