InterpolationIn the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable.
Polynomial interpolationIn numerical analysis, polynomial interpolation is the interpolation of a given bivariate data set by the polynomial of lowest possible degree that passes through the points of the dataset. Given a set of n + 1 data points , with no two the same, a polynomial function is said to interpolate the data if for each . There is always a unique such polynomial, commonly given by two explicit formulas, the Lagrange polynomials and Newton polynomials.
Lagrange polynomialIn numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a data set of coordinate pairs with the are called nodes and the are called values. The Lagrange polynomial has degree and assumes each value at the corresponding node, Although named after Joseph-Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler.
Newton polynomialIn the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points. The Newton polynomial is sometimes called Newton's divided differences interpolation polynomial because the coefficients of the polynomial are calculated using Newton's divided differences method. Given a set of k + 1 data points where no two xj are the same, the Newton interpolation polynomial is a linear combination of Newton basis polynomials with the Newton basis polynomials defined as for j > 0 and .
Hermite interpolationIn numerical analysis, Hermite interpolation, named after Charles Hermite, is a method of polynomial interpolation, which generalizes Lagrange interpolation. Lagrange interpolation allows computing a polynomial of degree less than n that takes the same value at n given points as a given function. Instead, Hermite interpolation computes a polynomial of degree less than mn such that the polynomial and its m − 1 first derivatives have the same values at n given points as a given function and its m − 1 first derivatives.
Multivariate interpolationIn numerical analysis, multivariate interpolation is interpolation on functions of more than one variable (multivariate functions); when the variates are spatial coordinates, it is also known as spatial interpolation. The function to be interpolated is known at given points and the interpolation problem consists of yielding values at arbitrary points . Multivariate interpolation is particularly important in geostatistics, where it is used to create a digital elevation model from a set of points on the Earth's surface (for example, spot heights in a topographic survey or depths in a hydrographic survey).
Trigonometric interpolationIn mathematics, trigonometric interpolation is interpolation with trigonometric polynomials. Interpolation is the process of finding a function which goes through some given data points. For trigonometric interpolation, this function has to be a trigonometric polynomial, that is, a sum of sines and cosines of given periods. This form is especially suited for interpolation of periodic functions. An important special case is when the given data points are equally spaced, in which case the solution is given by the discrete Fourier transform.
Finite differenceA finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems. The difference operator, commonly denoted is the operator that maps a function f to the function defined by A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives.
Runge's phenomenonIn the mathematical field of numerical analysis, Runge's phenomenon (ˈʁʊŋə) is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation points. It was discovered by Carl David Tolmé Runge (1901) when exploring the behavior of errors when using polynomial interpolation to approximate certain functions. The discovery was important because it shows that going to higher degrees does not always improve accuracy.
Chebyshev polynomialsThe Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshev polynomials of the first kind are defined by Similarly, the Chebyshev polynomials of the second kind are defined by That these expressions define polynomials in may not be obvious at first sight, but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly.
Bilinear interpolationIn mathematics, bilinear interpolation is a method for interpolating functions of two variables (e.g., x and y) using repeated linear interpolation. It is usually applied to functions sampled on a 2D rectilinear grid, though it can be generalized to functions defined on the vertices of (a mesh of) arbitrary convex quadrilaterals. Bilinear interpolation is performed using linear interpolation first in one direction, and then again in another direction.
Finite difference methodIn numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are discretized, or broken into a finite number of steps, and the value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points.
Nearest-neighbor interpolationNearest-neighbor interpolation (also known as proximal interpolation or, in some contexts, point sampling) is a simple method of multivariate interpolation in one or more dimensions. Interpolation is the problem of approximating the value of a function for a non-given point in some space when given the value of that function in points around (neighboring) that point. The nearest neighbor algorithm selects the value of the nearest point and does not consider the values of neighboring points at all, yielding a piecewise-constant interpolant.
Finite element methodThe finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems).
Spline interpolationIn the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. That is, instead of fitting a single, high-degree polynomial to all of the values at once, spline interpolation fits low-degree polynomials to small subsets of the values, for example, fitting nine cubic polynomials between each of the pairs of ten points, instead of fitting a single degree-ten polynomial to all of them.
Cubic Hermite splineIn numerical analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form, that is, by its values and first derivatives at the end points of the corresponding domain interval. Cubic Hermite splines are typically used for interpolation of numeric data specified at given argument values , to obtain a continuous function. The data should consist of the desired function value and derivative at each .
Newton's methodIn numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a single-variable function f defined for a real variable x, the function's derivative f′, and an initial guess x0 for a root of f. If the function satisfies sufficient assumptions and the initial guess is close, then is a better approximation of the root than x0.
Newton's method in optimizationIn calculus, Newton's method (also called Newton–Raphson) is an iterative method for finding the roots of a differentiable function F, which are solutions to the equation F (x) = 0. As such, Newton's method can be applied to the derivative f ′ of a twice-differentiable function f to find the roots of the derivative (solutions to f ′(x) = 0), also known as the critical points of f. These solutions may be minima, maxima, or saddle points; see section "Several variables" in Critical point (mathematics) and also section "Geometric interpretation" in this article.
Disjoint setsIn mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint. This definition of disjoint sets can be extended to families of sets and to indexed families of sets.
Fuzzy setIn mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics , decision-making , and clustering , are special cases of L-relations when L is the unit interval [0, 1].
