Simple convergence analysis for the block conjugate gradient method with variable preconditioning

The block conjugate gradient (Bl-CG) method is an effective iterative solver for large sparse symmetric positive definite linear systems with multiple right-hand sides. Variable preconditioning, in which different preconditioners can be applied to each iteration, is a versatile approach for improving the convergence of iterative solvers. However, the convergence properties are obscure when variable preconditioning is used. In this study, we present a simple analysis of the error norm behavior of the Bl-CG method with variable preconditioning. Numerical experiments are conducted to demonstrate the convergence analysis.