Approximate eigenvectors as preconditioner

Abstract

Given approximate eigenvector matrix U ̃ of a Hermitian nonsingular matrix H, the spectral decomposition of H can be obtained by computing H ′= U ̃ *H U ̃ 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 ||||H ′||||= (H ′) 2 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 U ̃ 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.