Fast computation of error bounds for all eigenpairs of a Hermitian and all singular pairs of a rectangular matrix with emphasis on eigen- and singular value clusters

Abstract

We present verification methods to compute error bounds for all eigenvectors of a Hermitian matrix as well as for all singular vectors of a rectangular real or complex matrix. In case of clusters these are bounds for an orthonormal basis of the invariant subspace or singular vector space, respectively. Individual error bounds for all eigenvalues and singular values including clustered and/or multiple ones are computed as well. The computed bounds do contain the true result with mathematical certainty, and the algorithms apply to interval data as well. In that case the computed bounds are true for every real/complex matrix within the tolerances. The computational complexity to compute inclusions of all eigen/singular pairs of an n×n matrix or m×n matrix is O(n3) or O(mn2) operations, respectively.