Block CG Reloaded

• Date: 5. 11. 2026
• Time: 14:00 - 15:30
• Address:
  Sokolovská 83, Praha
• Room: K3
• Speaker: Petr Tichý

In this talk, we analyze DR-BCG, a variant of the block conjugate gradient method introduced by A. Dubrulle for solving linear systems with a symmetric positive definite matrix and multiple right-hand sides. The algorithm preserves the block size in the rank-deficient case and allows computations with singular block coefficients, without requiring deflation or numerical rank detection. We establish its fundamental orthogonality properties and prove that it is breakdown-free and terminates in finitely many iterations in exact arithmetic. We also prove that, at every iteration, it provides a better approximate solution than block CG with exact deflation. These results clarify how DR-BCG benefits from richer approximation spaces even when standard global orthogonality is no longer preserved. We discuss preconditioning and present numerical experiments illustrating the theoretical results and favorable behavior in finite precision arithmetic. These findings support DR-BCG as a simple, efficient, and natural block generalization of the standard CG algorithm.