By Louis A. Hageman, David M. Young
This graduate-level textual content examines the sensible use of iterative equipment in fixing huge, sparse platforms of linear algebraic equations and in resolving multidimensional boundary-value difficulties. themes contain polynomial acceleration of uncomplicated iterative equipment, Chebyshev and conjugate gradient acceleration approaches 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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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. h, y} Similar arguments are valid for any other pair of adjacent points (Xi> y).
For the model problem, the convergence rates of the best accelerated SSOR procedures are considerably larger than those of any of the other iterative procedures. 51) must be satisfied or nearly satisfied in order to realize this extremely rapid convergence. 51) is not satisfied, as is the case for many practical applications, the SSOR procedures are much less effective. Because of this and because the computational effort for a SSOR iteration step is sometimes twice that required by other methods, the SSOR procedures are not frequently used in the solution of large' general problems.
J 1, }=I A.. } I, J J 1 }=i+1 i = 1, 2, ... n+l) J L,; }=i+1 - w)A-I, I. u\n+ 1/2) , r + F- } I i = q, q - 1, .. " 1. 41) Here one firstsuccessively computes u\n+ 1/ 2). U~ + 1/2), .. 40). /l, ... 41). 13). :) == (I - wU)-lwD-lb. 3 31 EXAMPLES OF BASIC ITERATIVE METHODS where k", == w(2 - w)(I - WU)-I(I - WL)-ID- 1b. 47) One can verify that the splitting matrix for the SSOR method is (1 ) _1(1 Q = 2 _w w w D - C L D ) w D - Cu . 48) Moreover, since (w - 1 D - C L)T = W - 1D - C u , it follows that Q is SPD and hence that the SSOR method is symmetrizable.
Applied Iterative Methods (Computer Science and Applied Mathematics) by Louis A. Hageman, David M. Young