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.