Let Uq^- be the negative half of a quantum group of finite type. Let P be the transition matrix between the canonical basis and a PBW basis of Uq^-. In the case Uq^- is symmetric, Antor gave a simple algorithm of computing P by making use of monomial bases. By the folding theory, Uq^- (symmetric, with a certain automorphism) is related to a quantum group Uq^- of non-symmetric type. In this paper, we extend the results of Antor to the non-symmetric case, and discuss the relationship between the algorithms for Uq^- and for Uq^-.
Shoji et al. (Sat,) studied this question.