On the Behavior of the Residual in Conjugate Gradient Method

Teruyoshi Washizawa

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DOI: 10.4236/am.2010.13025                

Abstract

In conjugate gradient method, it is well known that the recursively computed residual differs from true one as the iteration proceeds in finite arithmetic. Some work have been devoted to analyze this behavior and to evaluate the lower and the upper bounds of the difference. This paper focuses on the behavior of these two kinds of residuals, especially their lower bounds caused by the loss of trailing digit, respectively.

Keywords

Conjugate Gradient, Residual, Convergence, Finite Arithmetic, Lower Bound

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	T. Ginsburg, “The Conjugate Gradient Method,” Numerische Mathematik, Vol. 5, No. 1, 1963, pp. 191-200.
[2]	J. A. M. Bollen, “Numerical Stability of Descenet Methods for Solving Linear Equations,” Numerische Mathematik, Vol. 43, No. 3, 1984, pp. 361-377.
[3]	H. Wo?niakowski, “Round-off Error Analysis of Iteratinos for Large Linear Systems,” Numeriche Mathematik, Vol. 30, No. 3, 1978, pp. 301-314.
[4]	H. Wo?niakowski, “Roundoff Error Analysis of New Class of Conjugate-Gradient Algorithms,” Linear Algebra and its Applications, Vol. 29, 1980, pp. 507-529.
[5]	A. Greenbaum, “Behavior of Slightly Perturbed Lanczos and Conjugate-Gradient Recurrences,” Linear Algebra and its Applications, Vol. 113, 1989, pp. 7-63.
[6]	A. Greenbaum, “Estimating the Attainable Accuracy of Recursively Computed Residual Methods,” SIAM Journal on Matrix Analysis and Applications, Vol. 18, No. 3, 1997, pp. 535- 551.
[7]	G. Meurant, “The Computation of Bounds for the Norm of the Error in the Conjugate Gradient Algorithm,” Numerical Algorithms, Vol. 16, No. 3-4, 1997, pp. 77-87.
[8]	Z. Strako? and P. Tichy, “On Error Estimation in the Conjugate Gradient Method and Why it Works in Finite Precision Computations,” Electronic Transactions on Numerical Analysis, Vol. 13, 2002, pp. 56-80.
[9]	D. Calvetti, S. Morigi, L. Reichel and F. Sgallari, “Computable Error Bounds and Estimates for the Conjugate Gradient Method,” Numerical Algorithms, Vol. 25, No. 1-4, 2000, pp. 75-88.