Monotonicity, Bounds and Acceleration of Block Gauss and Gauss–Radau Quadrature for Computing $$B^T \phi (A) B$$

Source abstract

Abstract In this paper, we explore quadratures for the evaluation of $$B^T \phi (A) B$$ B T ϕ ( A ) B where A is a symmetric positive-definite (s.p.d.) matrix in $$\mathbb {R}^{n \times n}$$ R n × n , B is a tall matrix in $$\mathbb {R}^{n \times p}$$ R n × p , and $$\phi (\cdot )$$ ϕ ( · ) represents a matrix function that is regular enough in the neighborhood of A ’s spectrum, e.g., a Stieltjes or exponential function. These formulations, for example, commonly arise in the computation of multiple-input multiple-output (MIMO) transfer functions for diffusion PDEs. We propose an approximation scheme for $$B^T \phi (A) B$$ B T ϕ ( A ) B leveraging the block Lanczos algorithm and its equivalent representation through Stieltjes matrix continued fractions. We extend the notion of Gauss–Radau quadrature to the block case, facilitating the derivation of easily computable error bounds. For problems stemming from the discretization of self-adjoint operators with a continuous spectrum, we obtain sharp estimates grounded in potential theory for Padé approximations and justify averaging algorithms at no added computational cost. The obtained results are illustrated on large-scale examples of 2D diffusion and 3D Maxwell’s equations and a graph from the SNAP repository. We also present promising experimental results on convergence acceleration via random enrichment of the initial block B .