Handbook for Automatic Computation: Linear algebra
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AMS :: Mathematics of Computation
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Numerical results show that the most accurate approximation is achieved by the CGM-version where descent directions and residual vectors are orthogonal in the energy and Euclidean metrics, respectively, at each iteration step. A practical stopping criteria for CGM is proposed as a formula that enables one to determine the number of the CGM iteration step, starting with which the progress is no longer being made. The application of the constructed criteria to the solution of specific systems of linear algebraic equations with ill-conditioned matrices is demonstrated.
References:  Hestenes, M.
Methods of conjugate gradients for solving linear systems. Journal of research of the National Bureau of Standards. Handbook for Automatic Computation.
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Linear Algebra. New York: Springer-Verlag; The best of the 20th century: Editors name top 10 algorithms. On the numerical solution of the equation by which in technical questions frequencies of small oscillations of material systems are determined. In Russ. Some history of the conjugate gradient and Lanczos algorithms: — SIAM Review.
Numerical experiments illustrate the viability of the proposed ideas. Conflicts of Interest The authors declare no conflicts of interest. Cite this paper Dax, A. References [ 1 ] Bai, Z.
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James H. Wilkinson
Bai, Z. Bauer, F. Dax, A. Demmel, J. Golub, G.