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

By Louis A. Hageman, David M. Young

This graduate-level textual content examines the sensible use of iterative equipment in fixing huge, sparse structures of linear algebraic equations and in resolving multidimensional boundary-value difficulties. subject matters contain polynomial acceleration of uncomplicated iterative equipment, Chebyshev and conjugate gradient acceleration tactics acceptable to partitioning the linear process right into a “red/black” block shape, extra. 1981 ed. contains forty eight figures and 35 tables.

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P2' P~, ... , Pm' P~ of the parameters in a cyclic order. The values of the Pi often dependt on bounds of the eigenvalues of H and V. 7, it can be t For some variants, some care must be taken to ensure that the resulting incomplete factorization Q = H H T is positive definite. See. for example, Kershaw [1978]. t For algorithms to generate the acceleration parameters p; and p;, see Wachspress [1963J and Kellogg and Spanier [1965]. , see Birkhoff et al. [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.

Therefore, the SSOR method with either Chebyshev or conjugate gradient acceleration can be used to solve general problems. 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.

38), it can be shown that the asymptotic rates of convergence for the SOR method satisfy . Rro(Ie rob) ~ for point SOR, for line SOR, {2nh 2nh,/2 h -> O. u\n+1 /2) I, I I = w{- . ~ i;! A- . 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.

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