For the design of various structures, the dynamic analysis of the structures is essential. Eigenproblem must be first computed when the mode superposition method is used in the dynamic analysis of a structure. However, since most of solution time is spent on calculating the eigenpairs of a system, the development of more efficient solution method is significantly important. The purpose of this paper is to present an efficient solution method by combining Robinson-Lee's method and accelerated modified Newton-Raphson method to improve numerical stability and increase convergence. Effectiveness of the proposed method is verified through numerical examples.