The extended symmetric block Lanczos method for matrix-valued Gauss-type quadrature rules

• Published in: Journal of computational and applied mathematics 2022-06, Vol.407, p.114037, Article 114037
• Main Authors: Bentbib, A.H., El Ghomari, M., Jbilou, K., Reichel, L.
• Format: Article
• Language: English
• Subjects: Anti-Gauss quadrature; Extended block Krylov subspace; Gauss quadrature; Laurent polynomial; Mathematics; Matrix function
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2021.114037

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 VTf(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.