The generalized discrete variable representation, as opposed to the discrete variable representation, of a Hamiltonian is such that it can give accurate eigenvalues of the Hamiltonian even if non-Gaussian quadrature points and weights are used in its construction. A new method of building up the generalized discrete variable representation of a Hamiltonian has been described and its properties have been analyzed. This new method appears to be optimal, meaning that no other design based on the same points, weights, and basis functions can. be conceived which would give more accurate eigenvalues. Numerical calculations have revealed that, remarkable accuracy can be achieved even with general, non-Gaussian quadrature points and weights.