Affine arithmetic

Affine arithmetic (AA) is a model for self-validated numerical analysis. In AA, the quantities of interest are represented as affine combinations (affine forms) of certain primitive variables, which stand for sources of uncertainty in the data or approximations made during the computation.

Affine arithmetic is meant to be an improvement on interval arithmetic (IA), and is similar to generalized interval arithmetic, first-order Taylor arithmetic, the center-slope model, and ellipsoid calculus — in the sense that it is an automatic method to derive first-order guaranteed approximations to general formulas.

Affine arithmetic is potentially useful in every numeric problem where one needs guaranteed enclosures to smooth functions, such as solving systems of non-linear equations, analyzing dynamical systems, integrating functions differential equations, etc. Applications include ray tracing, plotting curves, intersecting implicit and parametric surfaces, error analysis (mathematics), process control, worst-case analysis of electric circuits, and more.

In affine arithmetic, each input or computed quantity x is represented by a formula ${\displaystyle x=x_{0}+x_{1}\epsilon _{1}+x_{2}\epsilon _{2}+{}}$${\displaystyle \cdots }$${\displaystyle {}+x_{n}\epsilon _{n}}$ where ${\displaystyle x_{0},x_{1},x_{2},}$${\displaystyle \dots ,}$${\displaystyle x_{n}}$ are known floating-point numbers, and ${\displaystyle \epsilon _{1},\epsilon _{2},\epsilon _{n}}$ are symbolic variables whose values are only known to lie in the range [-1,+1].

...
Wikipedia