This invention is directed to a circuit simulation using multi-rate harmonic balancing. Specifically, this invention enables effective reduction of analysis dimensions, e.g. frequency or time. The methodology converts N-dimensional problems to local (N-x)-dimensional problems. The method enables simultaneous solving of all local problems, each of these problems having a dimension less than or equal to N, thus approximating the original system to be solved. In practical situations, N could be the number of independent frequencies in an N-tone harmonic balance analysis.