Computation of the errors for the inverse of a nonsingular square matrix

Let $A\in\mathbf{R}^{n\times n}$ be a nonsingular matrix with a known approximation $(A_0, \delta_0)\in \mathbf{Q}^{n\times n}\times \mathbf{Q}_+$. We determine first the forward error:

$$\begin{split}\Vert A^{-1}-A_0^{-1}\Vert&= \Vert A^{-1} A (A^{... ...delta_0 \Vert A^{-1}\Vert\cdot \Vert A_0^{-1}\Vert. \end{split}$$ (1)

We have to find an upper bound for $\Vert A^{-1}\Vert$, so by using inequality 1 we get

$$\Vert A^{-1}\Vert \leq \Vert A_0^{-1}\Vert + \Vert A^{-1}-A_0^{-1... ...\leq \Vert A_0^{-1}\Vert + \delta_0 \Vert A^{-1}\Vert\cdot \Vert A_0^{-1}\Vert$$

yielding

$$\Vert A^{-1}\Vert\leq \frac{\Vert A_0^{-1}\Vert}{1-\delta_0\Vert A_0^{-1}\Vert},$$

whenever $\delta_0\Vert A_0^{-1}\Vert<1$. Because $A$ is nonsingular, $\Vert A_0^{-1}\Vert$ converges to $A^{-1}\in\mathbf{R}_+$ as $\delta_0$ goes to zero, thus the previous condition can be fulfilled within finitely many iterative refinements of the approximation $A_0$ of $A$.

Assuming that $(A_0, \delta_0)$ is an approximation with $\delta_0\Vert A_0^{-1}\Vert<1$, for the forward error we get

$$\Vert A^{-1}-A_0^{-1}\Vert \leq \frac{\delta_0 \Vert A_0^{-1}\Vert^2}{1-\delta_0\Vert A_0^{-1}\Vert}.$$

In the backward error computation, given $\varepsilon\in\mathbf{Q}_+$, we determine $\delta_{\varepsilon}\in \mathbf{Q}_+$ such that $\Vert A-A_{\varepsilon}\Vert\leq \delta_{\varepsilon}$ implies $\Vert A^{-1}-A_{\varepsilon}^{-1}\Vert\leq \varepsilon$. We apply inequality 1 for $A$ and $A_{\varepsilon}$:

$$\Vert A^{-1}-A_{\varepsilon}^{-1}\Vert\leq \delta_{\varepsilon} \Vert A^{-1}\Vert\cdot \Vert A_{\varepsilon}^{-1}\Vert,$$

and because $\Vert A_{\varepsilon}^{-1}\Vert\leq \Vert A^{-1}\Vert+\varepsilon$, using the upper bound for $\Vert A^{-1}\Vert$, we write:

$$\Vert A^{-1}-A_{\varepsilon}^{-1}\Vert\leq \delta_{\varepsilon} \... ...g(\frac{\Vert A_0^{-1}\Vert}{1-\delta_0 \Vert A_0^{-1}\Vert}+\varepsilon\Big).$$

If we choose $\delta_{\varepsilon}$ such that the above product is less than $\varepsilon$, we can compute an $\varepsilon$-approximate for $A^{-1}$ from $A_{\varepsilon}$. For instance, we may set

$$\delta_{\varepsilon}= \frac{\varepsilon(1-\delta_0\Vert A_0^{-1}\... ...rt^2+\varepsilon\Vert A_0^{-1}\Vert-\varepsilon\delta_0\Vert A_0^{-1}\Vert^2}.$$