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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 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.
Here is a 1D example. The circles are the sample points and the polygon is a linear interpolation. The blue curve is a smooth approximation of order 3. Consider a function : and a set of sample points = {(,) | =}.
Fitting of a noisy curve by an asymmetrical peak model, with an iterative process (Gauss–Newton algorithm with variable damping factor α).Curve fitting [1] [2] is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, [3] possibly subject to constraints.
set terminal svg fname "Vera" fsize 25 set output "Interpolation_example_linear.svg" set key off set xzeroaxis set xtics axis 1,1 set ytics nomirror 1 set border 2 set samples 2 plot "-" with lines linetype 3 linewidth 4, \ "-" with points linetype 1 pointtype 7 pointsize 0.7, \ 0 with lines linetype -1 linewidth 1 0 0 1 0.8415 2 0.9093 3 0.1411 4 -0.7568 5 -0.9589 6 -0.2794 e 0 0 1 0.8415 2 0 ...
The theory of interpolation of vector spaces began by an observation of Józef Marcinkiewicz, later generalized and now known as the Riesz-Thorin theorem.In simple terms, if a linear function is continuous on a certain space L p and also on a certain space L q, then it is also continuous on the space L r, for any intermediate r between p and q.