Approximate eigenvectors as preconditioner Original Research Article

Given approximate eigenvector matrix image of a Hermitian nonsingular matrix H, the spectral decomposition of H can be obtained by computing image and then diagonalizing H′. This work addresses 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 are sufficiently orthonormal and (ii) the matrix image is of the form DAD with diagonal D and well-conditioned A. In that case, H′ can be efficiently and accurately diagonalized by the Jacobi method. If image is computed by fast eigensolver based on tridiagonalization, this procedure usually gives the eigensolution with high relative accuracy and it is more efficient than accurate Jacobi type methods on their own.