The convergence result [Y. Ye and P. M. Pardalos, Linear Algebra Appl.,152 (1991), pp. 3–17] of a potential reduction algorithm for the P-matrix linear complementarily problem (LCP) is extended to the $P_0 $-matrix LCP. Computational experience with $PSD$-matrix, P-matrix, and $P_0 $-matrix LCPs is presented to reinforce the theoretical development. These test problems include random test problems, the worst case examples of Murty and Fathi, and various LCPs arising in engineering problems. How row and column scaling may improve the theoretical and practical efficiency of the algorithm is also illustrated.