Banded matrices and difference equations

In this paper we consider discrete Sturm-Liouville eigenvalue problems of t he form L(y)k := Sigma (n)(u=0)(-Delta)(mu) {r(mu)(k)Delta (mu)y(k+1-mu)} = gimely( k+1) for 0 less than or equal to k less than or equal to N - n with y(1-n) = (.. .) = y(0) = y(N+2-n) = y(N+1) = 0, where N and n are integers with 1 less t han or equal to n less than or equal to N and under the assumption that r(n ) (k) not equal 0 for all k. These problems correspond to eigenvalue proble ms for symmetric, banded matrices A is an element of R(N+1-n)x(N+1-n) with bandwidth 2n + 1. We present the following results: 1. an inversion formula, which shows that every symmetric, banded matrix co rresponds uniquely to a Sturm-Liouville eigenvalue problem of the above for m; 2. a formula for the characteristic polynomial of a, which yields a recursi on for its calculation; 3. an oscillation theorem, which generalizes well-known results on tridiago nal matrices. These new results can be used to treat numerically the algebr aic eigenvalue problem for symmetric, banded matrices without reduction to tridiagonal form. (C) 2001 Elsevier Science Inc. All rights reserved.