Calculations of transfer matrix by means of symmetric polynomials

Symmetric polynomials of n-th order are defined by the recurrence formulas as a functions of elementary symmetric polynomials of n-th order matrix. The method of symmetric polynomials (MSP) is developed with respect to the calculation of the transfer matrix of waves in layered media. MSP, in contrast to the Lagrange-Sylvester method does not require the computation of eigenvalues of the matrix. The algorithm of numerical calculation of the transfer matrix for layered structures is proposed. Analytical solutions for some of the transfer matrices of second and fourth order for a homogeneous layer and periodic layered structures are found.