Recurrence relations between symmetric polynomials of n-th order

Abstract

The method of symmetric polynomials (MSP) was developed for computation analytical functions of matrices, in particular, integer powers of matrix. MSP does not require for its realization finding eigenvalues of the matrix. A new type of recurrence relations for symmetric polynomials of order n is found. Algorithm for the numerical calculation of high powers of the matrix is proposed.This computational procedure is more accurate in comparison with ordinary matrix multiplication.

Coefficients of elastic wave's conversion by anisotropic layer

“…Transfer matrix T of the sixth order, according to [5,6] can be calculated by means of the formulas 1 :…”

“…and t gj are elements of the transfer matrix (6). The method for calculation of the transfer matrix is considered in the next section.…”

“…There is an algorithm for computation A 2 j by the formula (17) in which N 2 < N 1 if j ≥ 6. P r o o f. An example of such an algorithm for second order matrices was presented in [6]. A similar approach for the matrices of the sixth order gives N 1 /N 2 > 396j/(1563 + 122j).…”