A restarted version of a generalized preconditioned conjugate gradient method

A generalized preconditioned conjugate gradient method for the iterative solution of a preconditioned linear system Bx = b was recently derived and analysed by the author. Here is presented a special version of it which is efficient in cases where the matrix‐vector multiplication with the given matrix is costly, as in various substructuring methods, where the matrix is usually not present in explicit form but the action of it can be computed. For a restarted version, it is shown that the residuals decrease monotonically for any nonsingular matrix B except when a certain inner product is zero. It is shown why it is advantageous to let the maximal number of search directions in each cycle be not smaller than the number of clusters of eigenvalues of B in the complex plane. An upper bound is derived for the rate of convergence of the algorithm when the clusters can be transformed to an ellipse by a polynomial mapping.