We present an algorithm to compute bases for the spaces L(G), provided G is a rational divisor over a non-singular absolutely irreducible algebraic curve, and also another algorithm to compute the Weierstrass semigroup at P together with functions for each value in this semigroup, provided P is a rational branch of a singular plane model for the curve. The method is founded on the Brill-Noether algorithm by combining in a suitable way the theory of Hamburger-Noether expansions and the imposition...