The paper first describes a version of Lemke's algorithm for the linear complementarity problem based on updating the inverse or factorization of a submatrix rather than the whole tableau. A simple version is then given for a restricted class of matrices which uses only elementary principal pivots, and from this a hybrid algorithm using orthogonal factorizations is developed which processes the same class of problems as the basic version. The algorithms are then extended to deal with a generalized linear complementarity problem involving upper and lower bounds on the variables.