intervalanalyysit

Interval analyysit, often translated as interval analysis or interval arithmetic, is a branch of mathematics that deals with computations involving intervals rather than single numbers. Instead of representing a quantity by a precise value, interval analysis uses a closed interval $[a, b]$ to represent all possible values that a quantity could take, where $a$ is the lower bound and $b$ is the upper bound. This approach is particularly useful for handling uncertainty, measurement errors, or rounding errors that arise in calculations.

The fundamental operations in interval analysis, such as addition, subtraction, multiplication, and division, are defined to ensure that the resulting interval contains all possible outcomes of the operation applied to any numbers within the input intervals. For example, if we have two intervals $A = [a_1, a_2]$ and $B = [b_1, b_2]$, the sum $A+B$ is defined as $[a_1+b_1, a_2+b_2]$. Similarly, $A \times B = [\min(a_1b_1, a_1b_2, a_2b_1, a_2b_2), \max(a_1b_1, a_1b_2, a_2b_1, a_2b_2)]$.

A key feature of interval analysis is that it guarantees the inclusion property: the true result of a computation lies within the computed interval, even with the presence of rounding errors. This makes it a powerful tool for verification, validation, and robust computation in various fields, including computer-aided design, numerical analysis, control theory, and scientific computing, where the precise boundaries of results are critical.