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Iterative Solvers for Very Large Systems of Linear Equations
Assad Oberai and Claudio Rebbi
March 19, 2004 and March 26, 2004
Very large systems of linear equations cannot be solved by standard dense matrix techniques. However, approximate solutions with a high degree of accuracy can often be obtained by projection methods (or Krylov space methods), which build increasingly more accurate approximations to the solution by iteration. The original problem is thus solved by projecting it onto a subspace of smaller dimension. The projection involves the definition of a trial solution and a weighting function space. By selecting these spaces appropriately, several important algorithms are obtained. Out of these, we will describe the Gauss-Jacoby and Gauss-Seidel relaxation techniques, the method of steepest descent and more Krylov subspace methods such as GMRES and conjugate gradients.

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