The paper reviews and compares some new methods which have recently been proposed for updating the inverse in the simplex method. All of these new methods assume an elimination form of inverse, which has lately been shown to be superior to the old product form. Some computational experiments comparing what appears to be the most promising of these new methods with the standard product form method are described and the results appear encouraging. Some suggestions for implementation of this method in a large-scale production LP code are also given. (Author)