The precise algorithms for systems with multiple frequencies are presented. Firstly, the concept of consistency condition equation is proposed. Secondly, based on consistency condition equations subject to the system with distinct frequencies, the sensitivity analysis algorithm with respect to all distinct frequencies is presented. And then it turns out that the algorithm proposed is equivalent for the one involving the existing paper. Thirdly, through introducing adjoint vectors associated with right state-space vectors so as to get rid of the influence of degenerate phenomenon of orthogonality between the right state-space vectors, the sensitivity analysis algorithm connected with the multiple frequencies is established explicitly due to the consistency condition equations as well. The procedure of driving the algorithm is easy to understand and suitable for re-analysis program. The algorithms are brief and easy to be implemented. Lastly, the correctness and effectiveness of the methods given here are demonstrated by considering two non-proportionally damped systems respectively, where phenomena of multiple frequencies occur.