Abstract: Positioning service is one of the basic services required by practical application of ubiquitous computing and how to obtain location information of an unknown node precisely is a key problem of positioning service in ubiquitous computing. However, positioning error is inevitable due to various potential errors caused by imprecise measuring instruments, improper measuring methods, etc. Firstly, a new error convergence theorem is presented and proved about how to reduce positioning error rapidly. The theorem is composed of three sub-theorems which indicate respectively how to make the smallest initial positioning error, how to reduce the initial location error more quickly by topological replication of reference nodes, and how to converge the initial location error to obtain minimal location error. Secondly, with a view of the actual application in ubiquitous computing, the optimal computing unit of reference nodes selection is proposed and the location reference node selection algorithm is put forward using topological duplication according to chaos triangles based on the presented error convergence theorem. Performance analysis and simulation experiments indicate that the location reference node selection algorithm is more suitably applied in resource-constrained environment of ubiquitous computing with less system cost and faster positioning error convergence than the traditional polygonal positioning algorithm at the same location accuracy.