In this paper, a new algorithm for calculating the complex stability radius of a real matrix is presented. The proposed approach is based on finding the values of parameters such that a Hamiltonian matrix of special structure has multiple purely imaginary eigenvalues. Some results allowing one to simplify the calculation of the discriminant for the characteristic polynomial of the matrix are proposed. Newton sums are calculated using a special recurrence formula. The developed algorithm could be used for matrices with entries polynomially dependent on parameters. The relationship between the singular numbers of the matrix and its radius of stability is given. Also, a numerical example shows how the algorithm works.