A polynomial-time continuation algorithm is presented for a class of linear complementarity problems with positive semidefinite matrices. The linear extrapolation technique is combined with the Newton iteration in the predictor-corrector procedure of the algorithm to numerically follow the solution curve of the homotopy equations arising from the perturbed Karush-Kuhn-Tucker condition. The convergence rate of the method is proved to be 1 -4/(7&j after each cycle consisting of one extrapolation between two Newton steps.