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Concept
Linear interpolation
Summary
In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.
If the two known points are given by the coordinates and , the linear interpolant is the straight line between these points. For a value in the interval , the value along the straight line is given from the equation of slopes
which can be derived geometrically from the figure on the right. It is a special case of polynomial interpolation with .
Solving this equation for , which is the unknown value at , gives
which is the formula for linear interpolation in the interval . Outside this interval, the formula is identical to linear extrapolation.
This formula can also be understood as a weighted average. The weights are inversely related to the distance from the end points to the unknown point; the closer point has more influence than the farther point. Thus, the weights are and , which are normalized distances between the unknown point and each of the end points. Because these sum to 1,
yielding the formula for linear interpolation given above.
Linear interpolation on a set of data points (x0, y0), (x1, y1), ..., (xn, yn) is defined as the concatenation of linear interpolants between each pair of data points. This results in a continuous curve, with a discontinuous derivative (in general), thus of differentiability class .
Linear interpolation is often used to approximate a value of some function f using two known values of that function at other points. The error of this approximation is defined as
where p denotes the linear interpolation polynomial defined above:
It can be proven using Rolle's theorem that if f has a continuous second derivative, then the error is bounded by
That is, the approximation between two points on a given function gets worse with the second derivative of the function that is approximated. This is intuitively correct as well: the "curvier" the function is, the worse the approximations made with simple linear interpolation become.
Official source
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