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Derivation of the conjugate gradient method : ウィキペディア英語版
Derivation of the conjugate gradient method
In numerical linear algebra, the conjugate gradient method is an iterative method for numerically solving the linear system
:\boldsymbol=\boldsymbol
where \boldsymbol is symmetric positive-definite. The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems.
The intent of this article is to document the important steps in these derivations.
==Derivation from the conjugate direction method==

The conjugate gradient method can be seen as a special case of the conjugate direction method applied to minimization of the quadratic function
:f(\boldsymbol)=\boldsymbol^\mathrm\boldsymbol\boldsymbol-2\boldsymbol^\mathrm\boldsymbol\text

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