Approximationsfehlern

Approximationsfehler, also known as approximation error, refers to the difference between an approximate value of a quantity and its true value. This error arises whenever a complex or continuous quantity is represented by a simpler or discrete one, which is a common practice in many fields of science and engineering. For example, when using a finite number of decimal places to represent a real number, a rounding error, a type of approximation error, is introduced.

Approximation errors can be categorized into several types. Truncation error occurs when an infinite series or a continuous process is approximated by a finite number of terms or steps. For instance, approximating the mathematical constant pi (π) with 3.14 instead of its infinite decimal expansion introduces a truncation error. Rounding error arises from the limited precision of numerical representations in computers or calculators. When a number cannot be represented exactly, it is rounded to the nearest representable value, creating a small error.

Quantifying and controlling approximation errors is crucial for the accuracy and reliability of computations and models. Error analysis techniques are used to estimate the magnitude of these errors and determine their impact on the final results. Understanding the sources and behavior of approximation errors allows for the selection of appropriate approximation methods and numerical algorithms that minimize their effect, ensuring that the computed results are sufficiently close to the true values for practical purposes.