Linear-system residual ($r=b-Ax$)

The residual of an approximate solution $x$ of $Ax=b$ is $r=b-Ax$. It is the unbalanced right-hand side, while the solution error is $x_{\ast }-x=A^{-1}r$. The conjugate gradient method makes residuals orthogonal to successive Krylov subspaces.

The norm of a linear-system residual, used to measure how closely an approximate solution satisfies its linear system. It differs from the solution-error norm $\Vert x_{\ast }-x\Vert$. Their relationship depends on the matrix inverse and the condition number, so a small residual alone is not a scale-free relative-error guarantee.