A Multipreconditioned Conjugate Gradient Algorithm

We propose a generalization of the conjugate gradient method that uses multiple preconditioners, combining them automatically in an optimal way. The derivation is described in detail, and analytical observations are made. A short recurrence relation does not hold in general for this new method, but in at least one case such a relation is satisfied: for two symmetric positive definite preconditioners whose sum is the coefficient matrix of the linear system. A truncated version of the method works effectively for other cases as well. The algorithm may be useful for domain decomposition techniques and other problems in which the need for more than one preconditioner arises naturally. We discuss similarities and differences with the standard and block conjugate gradient methods. Numerical examples illustrate and validate the merits of our algorithm.