The conjugate gradient method is an algorithm for finding the nearest local minimum of a function of
variables which presupposes that the gradient of the
function can be computed. It uses conjugate directions instead of the local gradient
for going downhill. If the vicinity of the minimum has
the shape of a long, narrow valley, the minimum is reached
in far fewer steps than would be the case using the method
of steepest descent .
For a discussion of the conjugate gradient method on vector and shared memory computers, see Dongarra et al. (1991). For discussions of the method for more general
parallel architectures, see Demmel et al. (1993) and Ortega (1988) and the
references therein.
See also Biconjugate Gradient Method ,
Biconjugate Gradient Stabilized
Method ,
Chebyshev Iteration ,
Conjugate
Gradient Method on the Normal Equations ,
Conjugate
Gradient Squared Method ,
Flexible
Generalized Minimal Residual Method ,
Generalized
Minimal Residual Method ,
Gradient ,
Linear
System of Equations ,
Local Minimum ,
Method
of Steepest Descent ,
Minimal Residual Method ,
Minimum ,
Nonstationary
Iterative Method ,
Preconditioner ,
Quasi-Minimal
Residual Method ,
Stationary Iterative
Method ,
Symmetric LQ Method ,
Transpose-Free
Quasi-Minimal Residual Method
Portions of this entry contributed by Noel Black and Shirley Moore, adapted from Barrett et al. (1994) (author's link )
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der Vorst, H. Templates
for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed.
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K. W. "Unconstrained Minimization." §3.1.7 in The
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Introduction
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Methods in Optimization." §2.3 in Computational
Methods in Optimization. New York: Academic Press, pp. 44-66, 1971. Press,
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Numerical
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Cambridge University Press, pp. 413-417, 1992. Reid, J. "On
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of Mathematics and Its Applications held in April, 1970 (Ed. J. Reid).
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Math. 48 , 543-560, 1986. Referenced on Wolfram|Alpha Conjugate Gradient Method
Cite this as:
Black, Noel ; Moore, Shirley ; and Weisstein, Eric W. "Conjugate
Gradient Method." From MathWorld --A Wolfram Web Resource.
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