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The Scheduled Relaxation Jacobi (SRJ) method is an extension of the classicalJacobi iterative method to solve linear systems of equations ($Au=b$)associated with elliptic problems. It inherits its robustness and acceleratesits convergence rate computing a set of $P$ relaxation factors that result froma minimization problem. In a typical SRJ scheme, the former set of factors isemployed in cycles of $M$ consecutive iterations until a prescribed toleranceis reached. We present the analytic form for the optimal set of relaxationfactors for the case in which all of them are different, and find that theresulting algorithm is equivalent to a non-stationary generalized Richardson'smethod. Our method to estimate the weights has the advantage that the explicitcomputation of the maximum and minimum eigenvalues of the matrix $A$ isreplaced by the (much easier) calculation of the maximum and minimumfrequencies derived from a von Neumann analysis. This set of weights is alsooptimal for the general problem, resulting in the fastest convergence of allpossible SRJ schemes for a given grid structure. We also show that with the setof weights computed for the optimal SRJ scheme for a fixed cycle size it ispossible to estimate numerically the optimal value of the parameter $\omega$ inthe Successive Overtaxation (SOR) method in some cases. Finally, we demonstratewith practical examples that our method also works very well for Poisson-likeproblems in which a high-order discretization of the Laplacian operator isemployed. This is of interest since the former discretizations do not yieldconsistently ordered $A$ matrices. Furthermore, the optimal SRJ schemes herededuced, are advantageous over existing SOR implementations for high-orderdiscretizations of the Laplacian operator in as much as they do not need toresort to multi-coloring schemes for their parallel implementation. (abridged)

On the equivalence between the Scheduled Relaxation Jacobi method and Richardson's non-stationary method