Computation of the perturbation

Whenever some operations over reals are done, we expect to collect the results in some erna objects. These results have to be able to provide approximations of any accuracy, therefore a thorough analysis of the error propagations is needed.

The main questions that have to be answered are:

Let $r_1,\dots,r_k$ be real numbers and $f$ a $k$-variate function over the reals.

Problem 1   (Backward error computation)
Given $\varepsilon>0$, find $\delta_1,\dots,\delta_k>0$ such that, if $r_1^0,\dots,r_k^0\in{\bf Q}$ satisfy:

$$\vert r_i-r_i^0\vert\le \delta_i,\ {\rm for}\ i=1..k,$$

then:

$$\vert f(r_1,\dots,r_k)-f(r_1^0,\dots,r_k^0)\vert\le\varepsilon.$$

Problem 2   (Forward error computation)
Given $\delta_1,\dots,\delta_k>0$ and $r_1^0,\dots,r_k^0\in{\bf Q}$ such that

$$\vert r_i-r_i^0\vert\le \delta_i,\ {\rm for}\ i=1..k,$$

give a (tight) upper bound for

$$\vert f(r_1,\dots,r_k)-f(r_1^0,\dots,r_k^0)\vert.$$

Same questions should be answered when the operands are complex numbers, vectors or matrices.