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A taxonomy for conjugate gradient methods. (English) Zbl 0723.65018
Based on the necessary and sufficient conditions of V. Faber and the second author [ibid. 21, 352-362 (1984; Zbl 0546.65010) and ibid. 24, 170-187 (1987; Zbl 0613.65030)] for existence of CG methods for matrices which are not Hermitian positive definite, a scheme for the development and organization of such CG methods is presented. It is shown that any CG method for \(Ax=b\) can be characterized by an inner product given by a Hermitian positive definite matrix and a left preconditioning matrix. With this approach methods are classified. The connection between CG and Lanczos how to obtain eigenvalue estimates is generalized.

65F10 Iterative numerical methods for linear systems
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
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