Approximation

Approximation ( Latin proximus , “the closest”) is initially a synonym for “approximation”; In mathematics, however, the term is specified as an approximation method .

From a mathematical point of view, there are various reasons to investigate approximations. The most common nowadays are:

• The approximate solution of an equation . If an analytically exact solution of the equation is not available, one wants to find an approximation of the solution in a simple way.
• The approximate representation of functions or numbers . If an explicitly given mathematical object is difficult to handle, an approximation from simple structures is desirable.
• The approximate reconstruction of unknown functions from incomplete data. If the information of the unknown function is only available in discrete form, as function values ​​over certain support points, then a closed representation that defines function values ​​on a continuum is desirable.

In many cases, a numerical method is based on the idea of ​​approximating a complex (and often only implicitly known) function by means of a function that is easy to use. The approximation theory is thus an integral part of modern applied mathematics. It provides a theoretical foundation for many new and established computer-aided solution processes.

Types of approximation

numbers

One of the most common forms of approximation is the representation of an irrational number as a number with a finite number of decimal places and the rounding of a number to a number with fewer decimal places, i.e. the calculation of an approximate value . For example:

${\displaystyle {\sqrt {2}}\approx 1{,}41421356\approx 1{,}41}$

The vast majority of computer programs work with floating point numbers according to the IEEE 754 standard , in which numbers are represented with a finite number of digits, which always requires rounding for irrational numbers and periodic fractions. The accuracy of the representation in the computer is determined by the selected data type .

The theory of Diophantine approximation deals with the approximation of irrational numbers by rational ones .

Geometric objects

Approaching a circle through pentagons, hexagons and octagons

In geometry , complicated objects can often be approximated using polygons . Thus calculated as Archimedes an approximation to the circle figure by a circle with regular polygons approached with more and more corners. ${\displaystyle \pi }$

Functions

The approximation of functions is of particular interest, for example differential equations that cannot be solved exactly for approximate solutions . The most common form is the approximation with polynomials , since these can be easily derived , integrated and calculated. The most common method here is based on the Taylor series expansion . Fourier analysis , in which periodic functions are developed in infinite series of sine and cosine functions , is also of great practical importance .

Many of these approximation methods have their theoretical foundation in the Stone-Weierstraß theorem (named after Marshall Harvey Stone and Karl Weierstraß ) , from which it follows not least that any continuous function on a compact real interval can be approximated with polynomials with equal precision and that in the same way every continuous function which is periodic in the field of real numbers can be approximated equally and with arbitrary precision by trigonometric functions .

The term norm is of central importance for approximations . This is used to quantitatively compare different approximations. In general, the approximate solution for different standards is different. It is important to be able to estimate the error that results from the approximation in order to assess its quality. This is not always easy and is an important task of approximation theory.

Classic examples here are, on the one hand, the Chebyshev approximation, in which continuous real or complex functions are approximated with respect to the supreme norm , and the approximation, in which L ^p functions are approximated with respect to the norm. ${\displaystyle L^{p}}$ ${\displaystyle L^{p}}$

An example of the approximation of functions is the small-angle approximation , in which the sine function is replaced by its angle and the cosine function is replaced by the constant 1. It is valid for small angles and is used, for example, to solve the mathematical pendulum .

Order of approximation

A measure of the quality of the approximation of a function is the order. A -th order approximation is one where the error is of the order of magnitude . A first order approximation is called a linear approximation and a second order approximation is called a quadratic approximation . ${\displaystyle n}$ ${\displaystyle {\mathcal {O}}(x^{n+1})}$

In physics, the linear approximation is often sufficient because it usually has the greatest influence. Higher order terms are important when linear effects are suppressed, such as in nonlinear optics .

Important approximation theorems

Approximation theory and functional analysis

• Approximation theorem for compact operators
• Approximation theorem for uniformly convex spaces
• Approximation theorem for real unitary spaces
• Korowkin's approximation theorem
• Müntz's approximation theorem
• Runge's approximation theorem
• Stone-Weierstrass theorem

Number theory

• Kronecker's approximation theorem
• Liouville's approximation theorem
• Dirichletscher approximation theorem
• Approximation theorem for p-adic numbers
• Hurwitz's theorem
• Thue-Siegel-Roth theorem

Theoretical computer science

Approximations also play a role in theoretical computer science. There are NP-complete optimization problems for which it is not possible to efficiently compute an exact solution. Approximation algorithms can be used here to calculate an approximation. One example is the rucksack problem, in which, from a certain problem size onwards, a great deal of computing effort is required to calculate an optimal solution, but where good approximation algorithms exist that can be used to efficiently calculate approximate solutions.

literature

• Lothar Collatz , Werner Krabs : Approximation Theory. Chebyshev approximation with applications. Teubner, Stuttgart 1973, ISBN 3-519-02041-6 .
• Armin Iske: Approximation (Masterclass). Springer Spectrum, 2018, ISBN 978-3662554647 .
• Günter Meinardus : Approximation of functions and their numerical treatment (=  Springer Tracts in Natural Philosophy . Volume 4 ). Springer Verlag , Berlin, Göttingen, Heidelberg, New York 1964 ( MR0176272 ).
• Manfred W. Müller : Approximation Theory. Akademische Verlags-Gesellschaft, Wiesbaden 1978, ISBN 3-400-00375-1 .
• MJD Powell : Approximation Theory and Methods. Cambridge University Press, Cambridge et al. 1981, ISBN 0-521-22472-1 .
• R. Schaback : Numerical Approximation. (PDF; 7.3 MB) In: Annual report of the German Mathematicians Association. 88, No. 2, 1986, ISSN  0012-0456 , pp. 51-81. => MR0838860
• Holger Wendland : Scattered Data Approximation. Cambridge University Press, 2010, ISBN 978-0521131018 .
• Eberhard Zeidler : Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems . Translated by Peter R. Wadsack. Springer Verlag, New York, Berlin, Heidelberg, Tokyo 1986, ISBN 0-387-90914-1 ( MR0816732 ).