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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 numerical methods for linear least squares are important because linear regression models are among the most important types of model, both as formal statistical models and for exploration of data-sets. The majority of statistical computer packages contain
Multilinear polynomials are the interpolants of multilinear or n-linear interpolation on a rectangular grid, a generalization of linear interpolation, bilinear interpolation and trilinear interpolation to an arbitrary number of variables. This is a specific form of multivariate interpolation, not to be confused with piecewise linear
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.
Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function. This behaviour tends to grow with the number of points, leading to a divergence known as Runge's phenomenon; the problem may be eliminated by choosing interpolation points at Chebyshev nodes. [5]
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.
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 ...
In the sixth iteration, we cannot use inverse quadratic interpolation because b 5 = b 4. Hence, we use linear interpolation between (a 5, f(a 5)) = (−3.35724, −6.78239) and (b 5, f(b 5)) = (−2.71449, 3.93934). The result is s = −2.95064, which satisfies all the conditions. But since the iterate did not change in the previous step, we ...