Modifications of the calculating scheme 2 that are more resistant to rounding-off errors are known see  , . 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  : 2 The process ends at some for which.
References  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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Conjugate Gradient Method -- from Wolfram MathWorld
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