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. subject matters contain polynomial acceleration of simple iterative equipment, Chebyshev and conjugate gradient acceleration approaches appropriate to partitioning the linear process right into a red/black” block shape, extra. 1981 ed. comprises forty eight figures and 35 tables.
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Extra resources for Applied Iterative Methods (Computer Science and Applied Mathematics)
S·I,). 4) by - h 2 and transfer to the right-hand side those terms involving the known boundary values Uj,j on Sh' we obtain a linear system of the form Au = b. 4) is expressed in matrix form Au = b, it is implied that a correspondence between equations and unknowns exists and that an ordering of the unknowns has been chosen. 4). However, if Ul,l were the second elemept of u, then this correspondence would imply that a2,2 = Pl,l' For both cases, Pl,l is a diagonal element of A. Moreover, with this correspondence between equations and unknowns, it is easy to see that A is symmetric.
3) (V + p~I)u(n+ 1) = b - (H - p~I)u(n+ If2l. Here it is assumed that for any positive numbers Pn and P~, the first system can be solved easily for u(n+ I f2l , given u(n), and that the second can be solved easily for u(n+ 1), given u(n+ 1/ 2). In a typical case involving a linear system arising from an elliptic partial differential equation, H and V might be tridiagonal matrices or at least matrices' with small bandwidths. For finite difference methods over rectangular mesh subdivisions, H is the matrix corresponding to horizontal differences and V is the matrix corresponding to vertical differences.
2, we show that, indeed, the matrix polynomial Qn(G) that minimizes S(QnCG)) can be defined in terms of Chebyshev polynomials. 2. It turns out that the proper application of Chebyshev acceleration requires the use of "iteration parameters" whose optimum values are functions of the extreme eigenvalues M(G) and meG) of G. When optimum iteration parameters are used, we show that Chebyshev acceleration can significantly improve the convergence rate. For most practical applications, however, the optimum parameters will not be known a priori and must be approximated by some means.