Ashby, Steven F.; Manteuffel, Thomas A.; Saylor, Paul E. A taxonomy for conjugate gradient methods. (English) Zbl 0723.65018 SIAM J. Numer. Anal. 27, No. 6, 1542-1568 (1990). 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. Reviewer: V.Mehrmann (Bielefeld) Cited in 71 Documents MSC: 65F10 Iterative numerical methods for linear systems 65F15 Numerical computation of eigenvalues and eigenvectors of matrices Keywords:conjugate gradient methods; Lanczos methods; B-normal matrices; preconditioning; eigenvalue estimates Citations:Zbl 0546.65010; Zbl 0613.65030 × Cite Format Result Cite Review PDF Full Text: DOI