Applied Iterative Methods by Louis A. Hageman

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By Louis A. Hageman

This graduate-level textual content examines the sensible use of iterative tools 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 numerous common iterative methods. subject matters comprise polynomial acceleration of simple iterative equipment, Chebyshev and conjugate gradient acceleration systems acceptable to partitioning the linear method right into a “red/black” block shape, adaptive computational algorithms for the successive overrelaxation (SOR) strategy, and computational elements within the use of iterative algorithms for fixing multidimensional difficulties. 1981 ed. forty eight figures. 35 tables.

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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). 22), it easily follows that Rn(Pn(G)) < R^P^G)) for all finite n. In fact, it can be shown that Rn(Pn(G)) is an increasing function of n. We omit the proof. 1 show that many iterations often are required before the asymptotic convergence is achieved. 1, we tabulate the values of the ratio RjjPJjG)) ÎUW)) -log(2r"/ 2 /(l + PQ) - l o g r»'2 + log(2/(l + PQ) log r^ ^ ' ^ } as a function of r.

31) and that m(if) = 0. (■*·) ~ 1^2,2 2n2h2 for point Gauss-Seidel, ,„_ Îline L ^Gauss-Seidel, „„„ o ^ , ' for * -> 0. m) = M(B)2/(2 - M(B)2). 9q. 34) Here ω is a real number known as the relaxation factor. With ω = 1, the SOR method reduces to the Gauss-Seidel method. If ω > 1 or ω < 1, we have overrelaxation or underrelaxation, respectively. We shall be concerned only with overrelaxation. 13). 30). The matrix jSfω is called the SOR iteration matrix. The splitting matrix for the SOR method is (ω" XD — CL), which, as in the case of the Gauss-Seidel method, is not SPD.

With ω = 1, the SOR method reduces to the Gauss-Seidel method. If ω > 1 or ω < 1, we have overrelaxation or underrelaxation, respectively. We shall be concerned only with overrelaxation. 13). 30). The matrix jSfω is called the SOR iteration matrix. The splitting matrix for the SOR method is (ω" XD — CL), which, as in the case of the Gauss-Seidel method, is not SPD. For ω > 1, the SOR method is not symmetrizable. ω normally has some eigenvalues that are complex. Thus extrapolation based on the SOR method is not applicable for ω > 1.