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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.

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