Numerical stability of the cyclic Richardson iteration

Abstract

Summary. The succc~s of the cyclic Richardson iteration depends on the properordering of the acceleration parameters. We give a rigorous error analysis to show that, with the proper ordering, the relative error in the iterative method, when properly terminated, is not larger than the error incurred in stable direct methods such as Cholesky factorization. For both the computed approximation fi to u = L-if satisfies Ilu-~ll < cond(L)lluH2 -t and this bound is attainable. We also show that the residual norm IIf -L~II is bounded by HLllcond(L) 11~l12-t. This bound is attainable for a small cycle length N. Our analysis suggests that for a larger cycle length N the residuals are bounded by liE II [I~H2-t. We construct a theoretical example in which this bound is attainable. However we observed in all numerical tests that ultimately the residual norms were of order [ILII 11~112 -t. We explain why in practice even the factor ~ is never encountered. Therefore the residual stopping criterion for the Richardson iteration appears to be very reliable and the method itself appears to be stable.

A cyclic projected gradient method

“…ν i , see [1]. The so-called "Lebedev-Finogenov ordering" of ν i which makes the cyclic Richardson iteration computationally stable was first proposed by and a stability analysis for cycles of lengths n which are powers of two was given in [38]. In [21,22], the following heuristic procedure was suggested to order the values τ i .…”

On some Modifications of the Nekrassov Method for Numerical Solution of Linear Systems of Equations

“…For other contributions see Saad and van der Vorst [14], Freund, Golub and Nachtigal [6], Ishihara, Muroya and Yamamoto [7], Maleev [10], Stork [17], Zawilski [18].…”