The need for function approximations arises in many branches of applied mathematics, and computer science in particular. Applied mathematics is a branch of Mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their In general, a function approximation problem asks us to select a function among a well-defined class that closely matches ("approximates") a target function in a task-specific way. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function
One can distinguish two major classes of function approximation problems: First, for known target functions approximation theory is the branch of numerical analysis that investigates how certain known functions (for example, special functions) can be approximated by a specific class of functions (for example, polynomials or rational functions) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc. In Mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with Quantitatively Numerical analysis is the study of Algorithms for the problems of continuous mathematics (as distinguished from Discrete mathematics) Special functions are particular mathematical functions which have more or less established names and notations due to their importance for the Mathematical analysis In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In ).
Second, the target function, call it g, may be unknown; instead of an explicit formula, only a set of points of the form (x, g(x)) is provided. Depending on the structure of the domain and codomain of g, several techniques for approximating g may be applicable. In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, the codomain, or target, of a function f: X → Y is the set For example, if g is an operation on the real numbers, techniques of interpolation, extrapolation, regression analysis, and curve fitting can be used. In Mathematics, the real numbers may be described informally in several different ways In the mathematical subfield of Numerical analysis, interpolation is a method of constructing new data points within the range of a Discrete set of In Mathematics, extrapolation is the process of constructing new data points outside a Discrete set of known data points In statistics regression analysis is a collective name for techniques for the modeling and analysis of numerical data consisting of values of a Dependent variable (response Curve fitting is finding a curve which has the best fit to a series of data points and possibly other constraints If the codomain of g is a finite set, one is dealing with a classification problem instead. Statistical classification is a procedure in which individual items are placed into groups based on quantitative information on one or more characteristics inherent in the items (referred
To some extent the different problems (regression, classification) have received a unified treatment in statistical learning theory, where they are viewed as supervised learning problems. Supervised learning is a Machine learning technique for learning a function from training data
See also
This mathematical analysis-related article is a stub. A radial basis function network is an Artificial neural network that uses Radial basis functions as activation functions Analysis has its beginnings in the rigorous formulation of Calculus. You can help Wikipedia by expanding it. External links
- In Situ Adaptive Tabulation: Nonlinear function approximation with multiple linear regions adapted under error control.
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