The extended symmetric block Lanczos method for matrix-valued Gauss-type quadrature rules
Résumé
This paper describes methods based on the extended symmetric block Lanczos process
for computing element-wise estimates of upper and lower bounds for matrix functions
of the form V T f (A)V , where the matrix A ∈ Rn×n is large, symmetric, and nonsingular,
V ∈ Rn×s is a block vector with 1 ≤ s ≪ n orthonormal columns, and f is a function
that is defined on the convex hull of the spectrum of A. Pairs of block Gauss–Laurent
and block anti-Gauss–Laurent quadrature rules are defined and applied to determine
the desired estimates. The methods presented generalize methods discussed by Fenu
et al. (2013), which use (standard) block Krylov subspaces, to allow the application of
extended block Krylov subspaces. The latter spaces are the union of a (standard) block
Krylov subspace determined by positive powers of A and a block Krylov subspace defined
by negative powers of A. Computed examples illustrate the effectiveness of the proposed
method.