Volume 58, pp. 115-135, 2023.

A posteriori superlinear convergence bounds for block conjugate gradient

Christian E. Schaerer, Daniel B. Szyld, and Pedro J. Torres


In this paper, we extend to the block case the a posteriori bound showing superlinear convergence of the conjugate gradient method developed by van der Vorst and Vuik in [J. Comput. Applied Math., 48 (1993), pp. 327–341]. That is, we obtain similar bounds but now for the block conjugate gradient method. We also present a series of computational experiments, illustrating the validity of the bound developed here as well as the bound by Simoncini and Szyld from [SIAM Review, 47 (2005), pp. 247–272] using angles between subspaces. Using these bounds, we make some observations on the onset of superlinearity and how this onset depends on the eigenvalue distribution and the block size.

Full Text (PDF) [533 KB], BibTeX

Key words

superlinear convergence, block conjugate gradient method, a posteriori analysis

AMS subject classifications

65F10, 65B99, 65F30

Links to the cited ETNA articles

[2]Vol. 14 (2002), pp. 1-19 Bernhard Beckermann and Arno B. J. Kuijlaars: Superlinear CG convergence for special right-hand sides
[6]Vol. 47 (2017), pp. 100-126 Andreas Frommer, Kathryn Lund, and Daniel B. Szyld: Block Krylov subspace methods for functions of matrices

< Back