Vectors, matrices

In our approach any multidimensional real entity is treated as an erna object in its corresponding approximation space. For instance, $A\in\mathbf{R}^{m\times n}$ is represented by $(A_0, \varepsilon_0)\in \mathbf{Q}^{m\times n}\times \mathbf{Q}_+$, such that $\Vert A-A_0\Vert\leq\varepsilon$, using some consistent norm.

The main problems we have to tackle with are:

• Determine error propagation in nontrivial computations.
• Handle the discontinuity of certain ill-posed problems.

To the first category belongs, for example, the error propagation in Gaussian elimination. A problem in the second category is the computation of the pseudoinverse $A^{\dag }$ of $A\in\mathbf{R}^{m\times n}$.

Definition 2   (Pseudoinverse) Given $A\in\mathbf{R}^{m\times n}$, the pseudoinverse of $A$ is the unique matrix $A^{\dag }\in \mathbf{R}^{n\times m}$ with the property that, for any $b\in \mathbf{R}^m$, the vector $x:=A^{\dag }b\in \mathbf{R}^n$ is the shortest which minimizes $\Vert Ax-b\Vert$.

This problem is ill posed; we use Tikhonov regularization to compute approximate solutions, i.e. we solve the following optimization problem

   for $\alpha>0$ find $x$ such that $$\Vert Ax-b\Vert^2 + \Vert \alpha x\Vert^2$$    is minimal$$.$$

Its solution is

$$\lim_{\alpha\to 0} (A^T A-\alpha I^{n\times n})^{-1}A^T.$$