# Conjugate gradient methods

## Basic Concept

For the system of equations:

$A \cdot X = B$

The unpreconditioned conjugate gradient method constructs the ith iterate $x^{(k)}$ as an element of $x^{(k)} + span\left\{ {r^{(0)} ,...,A^{i - 1} r^{(0)} } \right\}$ so that so that $\left( {x^{(0)} - \hat x} \right)^T A\left( {x^{(i)} - \hat x} \right)$ is minimized , where ${\hat x}$ is the exact solution of $AX = B$.

This minimum is guaranteed to exist in general only if A is symmetric positive definite. The preconditioned version of these methods use a different subspace for constructing the iterates, but it satisfies the same minimization property over different subspace. It requires that the preconditioner M is symmetric and positive definite.