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A better form of the interpolation polynomial for practical (or computational) purposes is the barycentric form of the Lagrange interpolation (see below) or Newton polynomials. Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function.
Barycentric coordinates are strongly related to Cartesian coordinates and, more generally, affine coordinates.For a space of dimension n, these coordinate systems are defined relative to a point O, the origin, whose coordinates are zero, and n points , …,, whose coordinates are zero except that of index i that equals one.
The simplest interpolation method is to locate the nearest data value, and assign the same value. In simple problems, this method is unlikely to be used, as linear interpolation (see below) is almost as easy, but in higher-dimensional multivariate interpolation, this could be a favourable choice for its speed and simplicity.
Barycentric subdivision, a way of dividing a simplicial complex; Barycentric coordinates (mathematics), coordinates defined by the vertices of a simplex; In numerical analysis, Barycentric interpolation formula, a way of interpolating a polynomial through a set of given data points using barycentric weights.
Polynomial interpolation also forms the basis for algorithms in numerical quadrature (Simpson's rule) and numerical ordinary differential equations (multigrid methods). In computer graphics , polynomials can be used to approximate complicated plane curves given a few specified points, for example the shapes of letters in typography .
A ninth order polynomial interpolation (exact replication of the red curve at 10 points) In the mathematical field of numerical analysis , Runge's phenomenon ( German: [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 ...
The Möller–Trumbore ray-triangle intersection algorithm, named after its inventors Tomas Möller and Ben Trumbore, is a fast method for calculating the intersection of a ray and a triangle in three dimensions without needing precomputation of the plane equation of the plane containing the triangle. [1]
Thus, it is appropriate to speak of the "Newton form", or Lagrange form, etc., of the interpolation polynomial. However, different methods of computing this polynomial can have differing computational efficiency. There are several similar methods, such as those of Gauss, Bessel and Stirling.