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 platforms of linear algebraic equations and in resolving multidimensional boundary-value difficulties. themes contain polynomial acceleration of simple iterative tools, Chebyshev and conjugate gradient acceleration methods acceptable to partitioning the linear procedure right into a “red/black” block shape, extra. 1981 ed. contains forty eight figures and 35 tables.

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1) are SPD. In such a case, A 1/2 and Q1 / 2 are symmetrization matrices. Moreover, any matrix W such that Q = WTW is also a symmetrization matrix. The W obtained from the factorization Q = WTW is usually the most computationally convenient choice. In the next section, symmetrization matrices Ware given for some of the basic iterative methods discussed there. 6, computational aspects in the use of symmetrization matrices are discussed. We remark that the symmetrization property need not imply convergence.

L. t The unknown constants determined from the M equations i = 1, ... 4) where the Wi are known weighting vectors. See, for example, de la Vallee Poussin [1968J and Setturi and Aziz [1973]. The methods discussed above are called additive correction acceleration methods. However, multiplicative correction methods have also been used. Multiplicative correction methods attempt to improve the accuracy of U(L) by multiplying U(L) by some matrix E instead of adding a vector elL) as in step (2) above.

In Chapter 9 we give a precise formula for the OJ that maximizes Ra:,(fL' (0); however, this formula for the" optimum" OJ is valid only for a certain class of partitionings. When the partitioning of A is such that a precise formula for the optimum OJ can be given, the SOR method is competitive with the best acceleration methods applied to the Jacobi method. For other partitionings, the SOR method 9'Ormally should not be considered as an effective general solution method. 'We remark that it is important to use an OJ near the optimum value for intrinsically slowly convergent problems.

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