Approximate eigenvectors as preconditioner

Given approximate eigenvector matrix (U) over tilde of a Hermitian nonsingu lar matrix H, the spectral decomposition of H can be obtained by computing H' = (U) over tilde*H (U) over tilde and then diagonalizing H'. This work a ddresses the issue of numerical stability of the transition from H to H' in finite precision arithmetic. Our analysis shows that the eigenvalues will be computed with small relative error if(i) the approximate eigenvectors ar e sufficiently orthonormal and (ii) the matrix \H'\ = root(H')(2) is of the form DAD with diagonal D and well-conditioned A. In that case, H' can be e fficiently and accurately diagonalized by the Jacobi method. If (U) over ti lde is computed by fast eigensolver based on tridiagonalization, this proce dure usually gives the eigensolution with high relative accuracy and it is more efficient than accurate Jacobi type methods on their own. (C) 2000 Els evier Science Inc. All rights reserved.