Paraphrase
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Iterative methods for solving general, large, sparse linear systems have been gaining popularity in many areas of scientific computing. This is due in great part to the increased complexity and size of the new generation of linear and nonlinear systems that arise from typical applications. At the same time, parallel computing has penetrated the same application areas, as inexpensive computer power has become broadly available and standard communication languages such as MPI have proved a much needed standardization. This has created an incentive to utilize iterative rather than direct solvers, e.g., Gauss Elimination, because the problems solved are typically from three or more dimensional models for which direct solvers often become ineffective due to the huge sizes of the resulting linear systems. Another incentive is that iterative methods are far easier to implement on parallel computers.
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We are now going to look at some alternative approaches that fall into the category of iterative methods. These techniques can only be applied to square linear systems (n equations in n unknowns), but this is of course a common and important case.
Iterative methods for Ax = b begin with an approximation to the solution, x0, then seek to provide a series of improved approximations x1, x2, … that converge to the exact solution. For the engineer, this approach is appealing because it can be stopped as soon as the approximations xi have converged to an acceptable precision, which might be something as crude as 10−3. With a direct method, bailing out early is not an option; the process of elimination and back-substitution has to be carried right through to completion, or else abandoned altogether. By far the main attraction of iterative methods, however, is that for certain problems (particularly those where the matrix A is large and sparse) they are much faster than direct methods. On the other hand, iterative methods can be unreliable; for some problems they may exhibit very slow convergence, or they may not converge at all.
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