This paper presents a new algorithm, namely, cyclic coefficient-parameter continuation (CCPC), that computes only the zero-dimensional finite roots of large systems of polynomial equations, with Bézout numbers of the order of millions, arising in the synthesis of mechanisms. The new method is applied to a well-known problem of nine precision-point path synthesis of four-bar linkages to compare it with the established methods, such as regeneration and finite root generation (FRG). In comparison with the existing methods, the CCPC algorithm is shown to economise computational efforts. The simple termination criterion of the algorithm is useful in estimating the finite root-counts of previously unsolved problems. A new root-count of 5754, as opposed to the previously known root-count of 5743, is estimated to the problem of six precision-point rigid-body guidance of planar Watt-I linkage with its base link specified a priori. Finally, the algorithm is applied to the problem of eight precision-point rigid-body guidance of the Watt-I linkage, of which the solution is not reported in the literature. A finite root-count estimate of 840,300 cognate pairs is obtained. Numerical examples of physical nature are studied and several feasible solutions to these examples are reported. © 2018 Elsevier Ltd