Approximationsfehlers

Approximationsfehler refers to the difference between an exact value and its approximation. This error arises when a more complex or precise quantity is replaced by a simpler or less precise one. In mathematics and computation, approximations are often used to simplify calculations or to represent values that cannot be expressed exactly, such as irrational numbers or the results of complex functions.

The magnitude of the approximationsfehler can be expressed in absolute or relative terms. Absolute error is the direct difference between the true value and the approximation. Relative error, on the other hand, is the absolute error divided by the true value, often expressed as a percentage. Relative error is particularly useful when comparing the accuracy of approximations for values of different magnitudes.

There are several sources of approximationsfehler. Truncation error occurs when an infinite series or a continuous process is cut off after a finite number of terms or steps. Rounding error arises from the finite precision of numerical representations, where numbers are rounded to a specific number of digits. Discretization error is common in numerical methods that approximate continuous functions or differential equations with discrete steps. Understanding and quantifying approximationsfehler is crucial for assessing the reliability and accuracy of numerical results in various scientific and engineering disciplines.