 # Interpolation Function

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An interpolation function (or interpolant) approximates a function using a set of known, discrete points (also called nodes or interpolation points). For example, you can use an interpolation function to interpolate values given in a table. A table of discrete values.

The resulting function can be used to describe the function at every point in space.

Interpolation is used to interpolate values from computations. It can also be used to develop schemes for numerical integration and numerical differentiation.

## Interpolation Function vs. Fitting Function

An interpolation function passes through all of the known points. In contrast, a fitting function doesn’t have to pass through all of the points— it can simply find the line that fits the best (Xue & Chen, 2008).

## Types of Interpolation Function A linear interpolation function (blue) fit to two data points (red).

Many different types of interpolation function are available. Popular software packages usually have at least cubic spline interpolation and FFT, but those aren’t the only ones in existence. The more complicated interpolation functions are usually used when the “basic” interpolation methods don’t provide a good fit.

1. Bulirsch-Stoer: Fits a polynomial function or rational function; Provides solutions of ordinary differential equations.
2. Cubic spline: Fits a different cubic function between each pair of known points.
3. FFT: Calculates the Fourier transform of a vector containing values of a periodic function, then calculates the inverse Fourier transform with more points.
4. Lagrange Interpolating Polynomial: Fits a Lagrange polynomial—a linear combination of n degree n – 1 polynomials, where each equals zero at exactly n – 1 points (Dedford, 2020).
5. Linear : fits a linear function to the points.
6. Nearest neighbor: the value of an interpolated point is set to the value of its nearest known neighbor.
7. Neville: interpolates a function at a given point with increasingly higher order Lagrange interpolation polynomials (Mitchell, 2020).
8. Newton: Fits a polynomial function to “difficult” functions; allows for incremental interpolation (Verschelde, 2018).

## References

Dedford, D. (2020). Lagrange Interpolation. Retrieved September 5, 2020 from: https://people.csail.mit.edu/ddeford/Lagrange_Interpolation.pdf
Mitchell, K. (2020). Neville’s Method. Retrieved September 5, 2020 from: http://people.math.sfu.ca/~kevmitch/teaching/316-10.09/neville.pdf
Verschelde, J. (2018). MCS 471 Lecture 7(b). Newton Interpolation. Numerical Analysis. Retrieved September 5, 2020 from:
Westerink, J. (2018). Lecture 2: Introduction to Interpolation. Retrieved September 5, 2020 from: https://coast.nd.edu/jjwteach/www/www/30125/pdfnotes/lecture2__5v14.pdf
Xue, D. & Chen, Y. (2008). Solving Applied Mathematical Problems with MATLAB. CRC Press.

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Stephanie Glen. "Interpolation Function" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/interpolation-function/
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