In this thesis, our main concern is to develop a new implementation of Karmarkar’s LP Algorithm and compare it with the standard version. In the implementation, the “Simplex Tableau” information is used in the basic step of the algorithm, the projection. Instead of constructing the whole projection matrix, some of the orthogonal feasible directions are obtained by using the Simplex Tableau and to give an idea of its effectiveness, this approximation scheme is compared with the standard implementation of Karmarkar’s Algorithm, by D. Gay. The Simplex Tableau is also used to calculate a basic feasible solution at any iteration with a very modest cost.