Editing Interpolation (section)

===Linear interpolation===
[[File:Interpolation example linear.svg|right|thumb|230px|Plot of the data with linear interpolation superimposed]]
{{Main|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 (''x''<sub>''a''</sub>,''y''<sub>''a''</sub>) and (''x''<sub>''b''</sub>,''y''<sub>''b''</sub>), and the interpolant is given by:
:<math> 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) </math>

:<math> \frac{y-y_a}{y_b-y_a} = \frac{x-x_a}{x_b-x_a} </math>

:<math> \frac{y-y_a}{x-x_a} = \frac{y_b-y_a}{x_b-x_a} </math>

This previous equation states that the slope of the new line between <math> (x_a,y_a) </math> and <math> (x,y) </math> is the same as the slope of the line between <math> (x_a,y_a) </math> and <math> (x_b,y_b) </math>

Linear interpolation is quick and easy, but it is not very precise. Another disadvantage is that the interpolant is not [[derivative|differentiable]] at the point ''x''<sub>''k''</sub>.

The following error estimate shows that linear interpolation is not very precise. Denote the function which we want to interpolate by ''g'', and suppose that ''x'' lies between ''x''<sub>''a''</sub> and ''x''<sub>''b''</sub> and that ''g'' is twice continuously differentiable. Then the linear interpolation error is

:<math> |f(x)-g(x)| \le C(x_b-x_a)^2 \quad\text{where}\quad C = \frac18 \max_{r\in[x_a,x_b]} |g''(r)|. </math>

In words, the error is proportional to the square of the distance between the data points. The error in some other methods, including [[polynomial interpolation]] and spline interpolation (described below), is proportional to higher powers of the distance between the data points. These methods also produce smoother interpolants.
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