Accurate Eigenvectors of Oscillatory Matrices

The purpose of this report is to investigate the possibilities for accurate computation of eigenvectors of (unsymmetric) oscillatory matrices. The goal is to decide what “accurate” means, to determine whether the eigenvectors are determined “accurately” in that sense, and to engineer algorithms that will guarantee that accuracy. It is desirable for the computed eigenvector matrix to inherit the mathematical properties of eigenvector matrices of oscillatory matrices. Namely, if λ1 > . . . > λn > 0 are the eigenvalues of an oscillatory matrix A, then 1. the jth eigenvector of A has exactly j − 1 sign changes; 2. the eigenvector matrix V is a γ-matrix, i.e., its LU decomposition is V = LU , where L and U−1 are TN. 3. V is an LTP matrix, i.e., V and V −T are lowerly TP.