Recursion formulae for the characteristic polynomial of symmetric banded matrices

Abstract

In this article we treat the algebraic eigenvalue problem for real, symmetric, and banded matrices of size N×N, say. For symmetric, tridiagonal matrices, there is a well-known two-term recursion to evaluate the characteristic polynomials of its principal submatrices. This recursion is superfast, i.e. it requires O(N) additions and multiplications. Moreover, it is used as the basis for a numerical algorithm to compute particular eigenvalues of the matrix via bisection. We derive similar recursion formulae also with O(N) numerical operations for symmetric matrices with arbitrary bandwidth, containing divisions. The main results are divisionfree recursions for penta- and heptadiagonal symmetric matrices. These recursions yield similarly as in the tridiagonal case stable and superfast algorithms to compute any particular eigenvalue.