Let $A$ be  a  nonsingular  matrix,  let $x_{k+1}$ be  an approximation to the solution of $Ax=b$, and let $r^{k+1}=b-Ax^{k+1}$.Show that if $x^{k+1}$ is $\epsilon$-backward stable approximate of solution $Ax=b$ then $r^{k+1}= \delta b+ \delta A x^{k+1}$
Backwards Stability
numerical analysis;stability