The preconditioned variational methods for solving large linear systems

The implementation of the Preconditioned Conjugate Gradient method for the solution of large linear systems arising from the discretization of differential operators, requires the predetermination of only one iteration parameter. The numerical determination of the optimal value of this constant parameter, involve the spectral bounds of some matrices and can be obtained in O(N2) sine function evaluations, where 1/N is the discretization mesh size. It is shown that this parameter can be chosen in a stable manner in O(1) operations per iteration, if it is allowed to vary with the iteration index from information derived from the gradient parameters.