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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.
In numerical analysis, multivariate interpolation is interpolation on functions of more than one variable [1] (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 ...
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.
Example of bilinear interpolation on the unit square with the z values 0, 1, 1 and 0.5 as indicated. Interpolated values in between represented by color. In mathematics, bilinear interpolation is a method for interpolating functions of two variables (e.g., x and y) using repeated linear interpolation.
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.
The Chebyshev nodes are important in approximation theory because they form a particularly good set of nodes for polynomial interpolation. Given a function f on the interval [, +] and points ,, …,, in that interval, the interpolation polynomial is that unique polynomial of degree at most which has value () at each 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 ...
The initial slope of the function at the initial value depends on the number and order of zeros and poles that are at values below the initial value, and is found using the first two rules. To handle irreducible 2nd-order polynomials, a x 2 + b x + c {\displaystyle ax^{2}+bx+c} can, in many cases, be approximated as ( a x + c ) 2 {\displaystyle ...