By Louis A. Hageman

ISBN-10: 0123133408

ISBN-13: 9780123133403

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**Example text**

13) in the more computationally convenient form ,,_,. Λ - ( I - * · ) - . 215) n>2, where σ = l/w(l) = (M(G) - m(G))/(2 - M(G) - m(G)). 2 49 OPTIMAL CHEBYSHEV ACCELERATION It can also be shown (Varga, 1962) that limp„ EE pa = 2/(1 + V l - σ2). 17) We now examine the convergence rate of the optimal Chebyshev procedure. )/(l + V1 - ^2)· (4-219) Thus we obtain S(P„(G)) = 2r" /2 /(l + r"). 17) can be expressed in the form Rao(Pn(G))= - i l o g r . 19) that r = p^ — 1. Thus we also have that lim iS(Pn(G))Vln = (pn - 1) 1/2 and that R^P^G)) = - \ logip, - 1).

3) p B + ! 11) with y and pn+i replacing the y and p n + 1 , respectively. 1, the iterates for the polynomial acceleration procedure based on P„jE(x) ma Y be given by uin+1) = Pn+1{y(Gu™ + fc) + (1 - y)u™} + (1 - pn+Mn~l). , and " - ' · " - < ' - « > - . 8) In the above discussion, we have tacitly assumed that ME Φ mE. If ME = mE,theny = 1/(1 — ME\px = p2 = · · · = landσ E = 0. 13). We shall not consider this case separately in the balance of this section. The correct formulas can be obtained by a limiting process.

Let such a matrix Q be defined by the splitting A = Q - R for the coefficient matrix A. 1) X l with the corresponding iteration matrix G = I — Q~ A = Q~ R. We now consider those methods for which the matrix Q has the form Q = HK. " Moreover, to maximize the rate of convergence, H and K should also be chosen such that the spectral radius of the iteration matrix G is as small as possible. If A is symmetric and positive definite, one choice is to let H = LT and K = L, where L is the upper triangular matrix defined by the Cholesky decomposition A = LTL of A.

### Applied Iterative Methods by Louis A. Hageman

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