I know the definition of forward error, backward error and condition number. The following is a backward error analysis I think.
Let $A$ be a square matrix of order $n$ and $\hat{x}$ be a approximate solution of $Ax=b$. We want to know the forward error $\|x-\hat{x}\|$, so approximate the condition number $\kappa$ of $A$ and the backward error $\|b-A\hat{x}\|$.
$$\text{Forward error}\leq \text{Condition number}\times\text{Backward error}$$
Suppose that an approximate value of $\kappa$ was obtained and we will approximate the backward error.
Let $fl(A\hat{x})$ be the value of $A\hat{x}$ computed by floating-point arithmetic. Then $\|b-A\hat{x}\|\leq\|b-fl(A\hat{x})\|+\|fl(A\hat{x})-A\hat{x}\|$ and we can know upper bounds for these.
Is the above a backward error analysis?