A forward or backward error bound is generated when computing the condition number of the solution to a linear system.

If x is the solution to Ax=b, and let x+\delta x be the solution to the perturbed problem A(x+\delta x)=b+\delta b, then

                Ax + A \delta x = b + \delta b,

and it leads to the bound:
                \frac{\parallel \delta x \parallel}{\parallel x \parallel }
\leq
\kappa(A) \frac{\parallel \delta b \parallel}{\parallel b \parallel }.

\kappa(A) is the condition number of a square matrix A.

Similarly, let x+\delta x be the solution to the perturbed problem (A + \delta A)(x+\delta x)=b. It leads to the bound:

                \frac{\parallel \delta x \parallel}{\parallel x \parallel }
\leq
\kappa(A) \frac{\parallel \delta A \parallel}{\parallel A \parallel }.

\frac{\parallel \delta x \parallel}{\parallel x \parallel } is the forward error bound for the computed solution, and \frac{\parallel \delta b \parallel}{\parallel b \parallel } or \frac{\parallel \delta A \parallel}{\parallel A \parallel } is the backward error bound.