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Flexible conjugate gradients. (English) Zbl 0980.65030
The author discusses the conjugate gradient method for the numerical solution of a linear large sparse system, using preconditioning technique. To maintain the optimal convergence properties of the method, the author considers a variant of the conjugate gradient method (called flexible conjugate gradient) that performs an explicit orthogonalization of the search directions vectors. The convergence analysis shows that the convergence rate is essentially independent of the variations in the preconditioner. For problems with regular distribution of the eigenvalues for which the fast convergence is due to a nice condition number, inexact preconditioning is used. Numerical results are discussed for problems with no separated eigenvalues and also for problems with well-separated extremal eigenvalues.

65F10 Iterative numerical methods for linear systems
65F50 Computational methods for sparse matrices
65F35 Numerical computation of matrix norms, conditioning, scaling
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