Download e-book for iPad: Applied Iterative Methods by Louis A. Hageman

By Louis A. Hageman

ISBN-10: 0123133408

ISBN-13: 9780123133403

This graduate-level textual content examines the sensible use of iterative equipment in fixing huge, sparse structures of linear algebraic equations and in resolving multidimensional boundary-value difficulties. Assuming minimum mathematical historical past, it profiles the relative benefits of a number of common iterative strategies. themes comprise polynomial acceleration of uncomplicated iterative equipment, Chebyshev and conjugate gradient acceleration methods appropriate to partitioning the linear method right into a “red/black” block shape, adaptive computational algorithms for the successive overrelaxation (SOR) technique, and computational facets within the use of iterative algorithms for fixing multidimensional difficulties. 1981 ed. forty eight figures. 35 tables.

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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.

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Applied Iterative Methods by Louis A. Hageman

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