Mixed-precision conjugate gradient algorithm using the groupwise update strategy

The conjugate gradient (CG) method is the most basic iterative solver for large sparse symmetric positive definite linear systems. In finite precision arithmetic, the residual and error norms of the CG method often stagnate owing to rounding errors. The groupwise update is a strategy to reduce the residual gap (the difference between the recursively updated and true residuals) and improve the attainable accuracy of approximations. However, when there is a severe loss of information in updating approximations, it is difficult to sufficiently reduce the true residual and error norms. To overcome this problem, we propose a mixed-precision algorithm of the CG method using the groupwise update strategy. In particular, we perform the underlying CG iterations with the standard double-precision arithmetic and compute the groupwise update with high-precision arithmetic. This approach prevents a loss of information and efficiently avoids stagnation. Numerical experiments using double-double arithmetic demonstrate that the proposed algorithm significantly improves the accuracy of the approximate solutions with a small overhead of computation time. The presented approach can be used in other related solvers as well.