Leja, Fejér-Leja and ℜ-Leja sequences for Richardson iteration

Abstract

We study Leja sequences on the unit disc and resulting mapped sequences to conventional domain, ellipses and real intervals, for the problem of relaxation of Richardson iteration. Using simple considerations, we establish upper and lower estimates for the growth of associated Newton polynomials and the so-called residual polynomials. These results broaden the understanding of such sequences and add to results established in Calvetti et al. (Numer. Math. 67(1), 21–40, 1994), Calvetti and Reichel (Numerical Algorithms 11(1-4), 79–98, 1996; J. Comput. Appl. Math. 71(2), 267–286 1996), Fischer and Reichel (Numer. Math. 54(2), 225–242, 1989), Nachtigal et al. (SIAM Journal on Matrix Analysis and Applications 13(3), 796–825, 1992), Reichel (SIAM J. Numer. Anal. 25(6), 1359–1368, 1988), Tal-Ezer (J. Sci. Comput. 4(1), 25–60, 1989). We also propose adaptive strategies for the selection of relaxation parameters that build on fast strategies (Balgama et al., Electron. Trans. Numer. Anal. 7, 124–140, 1998; Reichel, Linear Algebra and its Applications 154-156(C), 389–414, 1991) which are known to be suitable for the problem. In particular a Leja ordering is enforced on the mapped sequences. New strategies for conveniently confining the spectrums of certain matrices arising in PDEs discretization is described and demonstrate the flexibility of the Richardson iteration.