generalised sparse factorisation of periodic tridiagonal matrices was introduced in Evans and Okolie[l]. By applying a similar factorisation strategy in a similarity transformation we obtain a sparse form extension of the Rutishauser's  L.R. and Q.D. schemes to determine the eigenvalues of a wide class of symmetric periodic tridiagonal matrices under certain diagonal dominance conditions. The method proposed is recommended for use in practical applications such as in the solution of the self -adjoint periodic characteristic Sturm Liouville problem.