Intervallilaskenta

Intervallilaskenta, known in English as interval arithmetic or interval computation, is a branch of numerical analysis that deals with calculations involving intervals of real numbers. Instead of using single precise values, interval arithmetic uses closed intervals [a, b] where a is the lower bound and b is the upper bound. This approach is particularly useful for quantifying and controlling round-off errors and uncertainties in mathematical computations.

The fundamental operations of arithmetic, such as addition, subtraction, multiplication, and division, are extended to intervals. For instance, the sum of two intervals [a, b] and [c, d] is defined as the interval [a+c, b+d]. Similarly, the product is [min(ac, ad, bc, bd), max(ac, ad, bc, bd)]. These operations are designed to produce an interval that is guaranteed to contain the result of the operation performed with any pair of numbers chosen from the input intervals. This property is known as the hull property.

Intervallilaskenta has applications in various fields. It is widely used in computer-aided verification of mathematical theorems, as it can rigorously prove the existence or non-existence of solutions to equations. It also plays a role in engineering and scientific computing, especially when dealing with uncertain input data or when the propagation of errors needs to be accurately bounded. For example, in robotics and control systems, interval arithmetic can be used to guarantee the safe operating range of a system. The development of interval arithmetic has led to specialized software libraries and programming languages that support these computations.