In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable. A closely related problem is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gain in simplicity may outweigh the loss from interpolation error and give better performance in calculation process.
Example As an example we will use points from the equation f ( x ) = sin ( x ) {\displaystyle f(x)=\sin(x)} to demonstrate various interpolation methods.
Interpolation provides a means of estimating the function at intermediate points, such as x = 2.5. {\displaystyle x=2.5.}
We describe some methods of interpolation, differing in such properties as: accuracy, cost, number of data points needed, and smoothness of the resulting interpolant function.
Piecewise constant 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.
Linear interpolation
One of the simplest methods is linear interpolation (sometimes known as lerp). Consider the above example of estimating f(2.5). Since 2.5 is midway between 2 and 3, it is reasonable to take f(2.5) midway between f(2) = 0.9093 and f(3) = 0.1411, which yields 0.5252. Generally, linear interpolation takes two data points, say (xa,ya) and (xb,yb), and the interpolant is given by:
y = y a + ( y b − y a ) x − x a x b − x a at the point ( x , y ) {\displaystyle y=y_{a}+\left(y_{b}-y_{a}\right){\frac {x-x_{a}}{x_{b}-x_{a}}}{\text{ at the point }}\left(x,y\right)}
y − y a y b − y a = x − x a x b − x a {\displaystyle {\frac {y-y_{a}}{y_{b}-y_{a}}}={\frac {x-x_{a}}{x_{b}-x_{a}}}}
y − y a x − x a = y b − y a x b − x a {\displaystyle {\frac {y-y_{a}}{x-x_{a}}}={\frac {y_{b}-y_{a}}{x_{b}-x_{a}}}}
This previous equation states that the slope of the new line between ( x a , y a ) {\displaystyle (x_{a},y_{a})} and ( x , y ) {\displaystyle (x,y)} is the same as the slope of the line between ( x a , y a ) {\displaystyle (x_{a},y_{a})} and ( x b , y b ) {\displaystyle (x_{b},y_{b})}
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