intervalliarvutus

Intervalliarvutus, also known as interval arithmetic or interval mathematics, is a branch of mathematics that deals with intervals of real numbers. Instead of single numerical values, interval arithmetic uses closed intervals $[a, b]$ to represent uncertain quantities. These intervals can arise from measurement errors, rounding in computations, or imprecise initial data.

The fundamental operations in interval arithmetic are extensions of standard arithmetic operations. For example, the sum of two intervals $[a, b]$ and $[c, d]$ is defined as $[a+c, b+d]$, and the product is the interval whose endpoints are the minimum and maximum of the four possible products $ac, ad, bc, bd$. This ensures that the true result of an operation on the actual values lies within the resulting interval.

A key property of interval arithmetic is that it guarantees the inclusion of the true result. If a computation involves an uncertain value represented by an interval, the result of the computation using interval arithmetic will be an interval that contains the true result of the computation if the uncertain value were known precisely. This property makes interval arithmetic useful for error analysis and rigorous computation.

Applications of interval arithmetic span various fields. It is used in computer-assisted proofs, where guaranteed bounds are necessary to establish the existence or non-existence of solutions. In engineering and physics, it helps in analyzing the tolerance of parameters and the robustness of designs. It also finds applications in optimization problems, robotics, and scientific computing where dealing with uncertainty is crucial. The development of interval arithmetic provides a systematic way to handle and propagate numerical uncertainties in a reliable manner.