By Louis A. Hageman, David M. Young
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. issues contain polynomial acceleration of simple iterative equipment, Chebyshev and conjugate gradient acceleration strategies appropriate to partitioning the linear approach right into a red/black” block shape, extra. 1981 ed. comprises forty eight figures and 35 tables.
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Additional resources for Applied Iterative Methods (Computer Science and Applied Mathematics)
With OJ = 1, the SOR method reduces to the Gauss-Seidel method. If OJ > 1 or OJ < 1, we have overrelaxation or underrelaxation, respectively. We shall be concerned only with overre1axation. 13). 34) becomes Du(n+l) = OJ(C L u(n+ ll + Cuu(n) + b) + (1 - OJ)Du(n). 3 29 EXAMPLES OF BASIC ITERATIVE METHODS where 2 == (1 - WL)-l(WU + (1 - w)I) and ro k~) == (1 - WL)-l WD- 1 b. 30). The matrix 2 ro is called the SOR iteration matrix. The splitting matrix for the SOR method is (w- 1 D - C L ), which, as in the case of the Gauss-Seidel method, is not SPD.
2) can be solved for /;(n), then we immediately have the solution u = u(n) - /;(n). 2) for /;(n) as it is to solve Au = b for U. On the other hand, it is not always necessary to determine /;(n) with great precision to improve the accuracy of u(n). 1), using u(O) as the initial guess. 2), and set u(O) = U(L) - 8(L). Then go to step (1) again. There are many ways to obtain the approximation 8(L). , see Nicolaides [1975, 1977J and Hackbusch [1977J). Another approach is to obtain the approximation elL) using the method of weighted residuals.
1) n 2:: 1.
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