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Abstract
If\( f(x) = \tfrac{1} {2}(Ax,x) - \operatorname{Re} (b,x), A = A^* \in \mathbb{C}^{n \times n} \), then the boundedness off from below is equivalent to the nonnegative definiteness ofA (prove this). Let us assume thatA > 0. In this case, a linear systemAx =b has a unique solutionz, and, for anyx,
z is the single minimum point forf (x). ⇒ A minimization method forf can equally serve as a method of solving a linear system with the Hermitian positively definite coefficient matrix.
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References
A.W. Chou. On the optimality of Krylov information.J. of Complexity 3: 26–40 (1987).
G. I. Marchuk and Yu. A. Kuznetsov. Iterative methods and quadratic functional.Methods of Numerical Mathematics. Novosibirsk, 1975, pp. 4–143.
Y. Saad and M.H. Schultz. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems.SIAM J. Scientific and Stat. Comp. 7: 856–869 (1986).
I first heard about this striking relation from S. A. Goreinov.
L. Zhou and H. F. Walker. Residual smoothing techniques for iterative methods.SIAM 3. on Sci. Comput. 15(2): 297–312 (1994).
R. W. Freund and N. M. Nachtigal. QMR: a quasi-minimal residual method for non-Hermitian linear systems.Numer. Math. 60: 315–339 (1991).
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Authors and Affiliations
Institute of Numerical Mathematics, Russian Academy of Sciences, Leninski Prospekt 32A, 117 334, Moscow, Russia
Eugene E. Tyrtyshnikov
- Eugene E. Tyrtyshnikov
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© 1997 Springer Science+Business Media New York
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Tyrtyshnikov, E.E. (1997). Lecture 19. In: A Brief Introduction to Numerical Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8136-4_19
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