Modifications of the calculating scheme 2 that are more resistant to rounding-off errors are known see [3] , [4]. The Stiefel method is related to the Zukhovitskii method for the minimax solution of a linear system, cf. Modifications of the method of steepest descent can be found in [a1] , [a12]. The classic reference for the conjugate-gradient method is [a4].

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An up-to-date discussion with additional references is [a3]. Its relation with matrix factorization is discussed in [a9]. It appears that J. Reid was the first to use this method as an iterative method cf. Several modifications have been proposed. Extensions to non-symmetric conjugate-gradient methods are discussed in [a1] and [a10]. Log in. Namespaces Page Discussion.

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Jump to: navigation , search. If is an -orthogonal basis of the space, then for any initial approximation , the exact solution of the system can be obtained from the decomposition where is the discrepancy of. In the conjugate-gradient method, the -orthogonal vectors are constructed by -orthogonalizing the discrepancies of the sequence of approximations , given by the formulas The vectors and constructed in this way have the following properties: 1 The conjugate-gradient method is now defined by the following recurrence relations see [1] : 2 The process ends at some for which.

References [1] D. Faddeev, V. Berezin, N. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR Translated from Russian Comments The Stiefel method is related to the Zukhovitskii method for the minimax solution of a linear system, cf. References [a1] O. Axelsson, "Conjugate gradient type methods for unsymmetric and inconsistent systems of linear equations" Lin.

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## A new hybrid conjugate gradient algorithm for unconstrained optimization

Concus, G. Golub, D.

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O'Leary, "A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations" J. Bunch ed.

### Journal of Applied Mathematics

Rose ed. Press [a3] G. Golub, C. Hestenes, "Conjugate directions methods in optimization" , Springer [a5] M. Hestenes, E.

Stiefel, "Methods of conjugate gradients for solving linear systems" J. Manteuffel, "Shifted incomplete Cholesky factorization" I. Duff ed.

## Conjugate Gradient Method -- from Wolfram MathWorld

Stewart ed. Meijerink, H.

Reid, "On the method of conjugate gradients for the solution of large systems of linear equations" J. Cite this paper Li, X. Natural Science , 3 , Journal of Optimization Theory and Applications, 64, Journal of Systems Science and Complexity, 15, Shanghai Scientific and Technical Publishers, Shanghai.

Mathematical Programming, 91, Computer Journal, 7, Mathematical Programming, 78, Journal of Research National Bureau of Standards, 49, Journal of Optimization Theory and Applications, 71, Journal of Computational and Applied Mathematics, 2, Journal of Optimization Theory and Applications, 69, Francaise informat Recherche Operatinelle, 3e Annee, 16, Lecture Notes in Mathematics, , Applied Mathematics and Computation, ,