Improved seed methods for symmetric positive definite linear equations with multiple right-hand sides

We consider symmetric positive definite systems of linear equations with multiple right-hand sides. The seed conjugate gradient (CG) method solves one right-hand side with the CG method and simultaneously projects over the Krylov subspace thus developed for the other right-hand sides. Then the next system is solved and used to seed the remaining ones. Rounding error in the CG method limits how much the seeding can improve convergence. We propose three changes to the seed CG method: only the first right-hand side is used for seeding, this system is solved past convergence, and the roundoff error is controlled with some reorthogonalization. We will show that results are actually better with only one seeding, even in the case of related right-hand sides. Controlling rounding error gives the potential for rapid convergence for the second and subsequent right-hand sides. Polynomial preconditioning can help reduce storage needed for reorthogonalization. The new seed methods are applied to examples including matrices from quantum chromodynamics. Copyright (c) 2013 John Wiley & Sons, Ltd.