intervalliarvutust

Intervalliarvutus is a field of mathematics that deals with computations involving intervals of real numbers. Instead of using precise numerical values, intervalliarvutus uses closed intervals as its basic data type. This means that any quantity is represented by a range [a, b], where 'a' is the lower bound and 'b' is the upper bound.

The primary motivation for using intervalliarvutus is to handle uncertainty and imprecision in calculations. This can arise from various sources, such as measurement errors, approximations of mathematical constants, or rounding errors in computer arithmetic. By performing calculations with intervals, one can rigorously bound the true result, ensuring that it lies within the computed interval.

Basic arithmetic operations like addition, subtraction, multiplication, and division are defined for intervals. For example, the sum of two intervals [a, b] and [c, d] is the interval [a+c, b+d]. The product of [a, b] and [c, d] is the interval [min(ac, ad, bc, bd), max(ac, ad, bc, bd)]. Other operations, such as square roots and trigonometric functions, are also extended to operate on intervals.

Intervalliarvutus finds applications in several areas, including automatic error analysis, numerical verification of mathematical theorems, computer graphics, control theory, and optimization problems where the reliable bounding of results is crucial. It provides a robust framework for dealing with and quantifying uncertainty in computations.