By Jacob Kogan

ISBN-10: 3540168184

ISBN-13: 9783540168188

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**Sample text**

This is consistent with the theory, which predicts decrease in \\G\\A but not necessarily in | r|| as the iteration progresses. Note that the unpreconditioned iteration is slowly convergent. 15) indicates that convergence will be slow. 9. This example illustrates the importance of a good preconditioner. Even the unpreconditioned iteration, however, is more efficient that the classical stationary iterative methods. For a preconditioner we use a Poisson solver. By this we mean an operator G such that v — Gw is the solution of the discrete form of subject to homogeneous Dirichlet boundary conditions.

Assume that XQ = 0 and that the eigenvalues of A are contained in the interval (9,11). 2 we simply note that ^(A) < 11/9. 7). Note that it is always the case that the spectrum of a spd matrix is contained in the interval [\N, AI] and that ^(A) — \I/\N. A result from [48] (see also [45]) that is, in one sense, the sharpest possible, is In the case of the above example, we can estimate K^(A) by K,

One should also keep in mind that a single Bi-CG iteration requires two matrix-vector products and a GMRES iterate only one, but that the cost of the GMRES iteration increases (in terms of floating-point operations) as the iteration progresses. 2. CGS. A remedy for one of the problems with Bi-CG is the Conjugate Gradient Squared (CGS) algorithm [180]. 17). This explains the name, Conjugate Gradient Squared. GMRES ITERATION 49 The work used in Bi-CG to compute f is now used to update x. CGS replaces the transpose-vector product with an additional matrix-vector product and applies the square of the Bi-CG polynomial to TQ to produce rk.

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