As one of the most popular methods for direction-of-arrival (DOA) estimation, the root-MUSIC algorithm usually needs to find the roots of a polynomial by iteration with certain initiation in practical applications. In order to improve the computational efficiency and accuracy of polynomial rooting, it is of great importance to choose and to update appropriate initiations for root-MUSIC. To this end, a new technique for setting and updating the initiations of root-MUSIC is proposed in this paper, by making use of the characteristic of root-MUSIC as well as Least Average Distance to Unit Circle (LADTUC) rule. Theoretical analysis as well as simulation results demonstrates that fake roots are avoided and the speed of iteration is efficiently improved by the new approach, which hence provides new schemes for realizing root-MUSIC in practice.