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The original use of interpolation polynomials was to approximate values of important transcendental functions such as natural logarithm and trigonometric functions.Starting with a few accurately computed data points, the corresponding interpolation polynomial will approximate the function at an arbitrary nearby point.
The function depends on three parameters, the input x, the "left edge" and the "right edge", with the left edge being assumed smaller than the right edge. The function receives a real number x as an argument and returns 0 if x is less than or equal to the left edge, 1 if x is greater than or equal to the right edge, and smoothly interpolates ...
This process yields p 0,4 (x), the value of the polynomial going through the n + 1 data points (x i, y i) at the point x. This algorithm needs O(n 2) floating point operations to interpolate a single point, and O(n 3) floating point operations to interpolate a polynomial of degree n.
A description of linear interpolation can be found in the ancient Chinese mathematical text called The Nine Chapters on the Mathematical Art (九章算術), [1] dated from 200 BC to AD 100 and the Almagest (2nd century AD) by Ptolemy. The basic operation of linear interpolation between two values is commonly used in computer graphics.
The Whittaker–Shannon interpolation formula can be used if the number of data points is infinite or if the function to be interpolated has compact support. Sometimes, we know not only the value of the function that we want to interpolate, at some points, but also its derivative. This leads to Hermite interpolation problems.
Radial basis function (RBF) interpolation is an advanced method in approximation theory for constructing high-order accurate interpolants of unstructured data, possibly in high-dimensional spaces. The interpolant takes the form of a weighted sum of radial basis functions .
is a simple IDW weighting function, as defined by Shepard, [3] x denotes an interpolated (arbitrary) point, x i is an interpolating (known) point, is a given distance (metric operator) from the known point x i to the unknown point x, N is the total number of known points used in interpolation and is a positive real number, called the power ...
Trilinear interpolation as two bilinear interpolations followed by a linear interpolation. Trilinear interpolation is a method of multivariate interpolation on a 3-dimensional regular grid . It approximates the value of a function at an intermediate point ( x , y , z ) {\displaystyle (x,y,z)} within the local axial rectangular prism linearly ...