Read e-book online Applied Iterative Methods (Computer Science and Applied PDF

By Louis A. Hageman, David M. Young

ISBN-10: 0123133408

ISBN-13: 9780123133403

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. subject matters comprise polynomial acceleration of simple iterative equipment, Chebyshev and conjugate gradient acceleration techniques acceptable 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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Example text

This condition is the only restriction imposed thus far on QnCx). 1) is completely consistent, convergence of the iterates win) to ii is not guaranteed. 1). 1). 3) to obtain u(n) make it necessary to seek alternative, less costly ways to compute ul n). 6) where Yl' P2' Y2"" are real numbers. 6), satisfy Qil) = lfor all n 2 O. 6) is large. , Davis [1963]). 1. 1) be completely consistent. 3) may be obtained using the three-term relation u(1) = Yl(GU(O) U(n+ 1) + k) + (1 - Yl)U(O), == Pn+ 1 {Yn+ 1 (GU(n) + k) + (1 - Yn+ l)U(n)} + (l - Pn+ l)U(n-l) for n 2 1.

1962]) that the parameters can be so chosen that the number of iterations needed for convergence varies as log h- 1 as the mesh size h tends to zero. 39». The basic theory for the Peaceman-Rachford method is valid only if the matrices H and V commute. For elliptic partial differential equations, this requirement implies that the differential equation is separable and that the region is a rectangle (see Birkhoff and Varga [1959] and Birkhoff et al. [1962]). Widlund [1966, 1969] extended the theory to nonseparable equa- .

6) is large. , Davis [1963]). 1. 1) be completely consistent. 3) may be obtained using the three-term relation u(1) = Yl(GU(O) U(n+ 1) + k) + (1 - Yl)U(O), == Pn+ 1 {Yn+ 1 (GU(n) + k) + (1 - Yn+ l)U(n)} + (l - Pn+ l)U(n-l) for n 2 1. 6). 3). 5), that 8(n+l) = Pn+l[Yn+1 G By adding + (1 - Yn+l)I]8(n) + (1- Pn+l)8(n-1). 9), we then obtain u(n+l) = Pn+l[Yn+1 G + (1 - Yn+l)l]u(n) - Pn+1Yn+l(G - I)u. 7). The special case for e(l) follows similarly. 7). 6). 6). 6). In this book, we consider only those polynomial sequences that are associated with the Chebyshev and conjugate gradient acceleration methods.

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Applied Iterative Methods (Computer Science and Applied Mathematics) by Louis A. Hageman, David M. Young


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