# Linear Interpolation

by admin in Curve Fitting , Interpolation , Math, Statistics, and Optimization , MATLAB Family on April 27, 2019This code conduct an Interpolation for a given set of points using Linear Interpolation method that uses linear polynomials, to construct new data points within the range of a discrete set of known data points.

### Code Outputs:

- Printed Table of Interpolated Values
- Chart Compare of Interpolated Values with Given Points.

Xi F(Xi)

1 11

12 21

19 41

26 32

34 55

### Input Requirements:

- X Axis Points
- Y Axis Points

### About the Method:

**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.

### Linear Interpolation Between Two Known Points

If the two known points are given by the coordinates **(x0, y0)** and **(x1, y1)**, the **linear interpolant** is the straight line between these points. For a value *x* in the interval **(x0, x0)**, the value *y* 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 ** n = 1**.

Solving this equation for *y*, which is the unknown value at *x*, gives

which is the formula for linear interpolation in the interval **(x0, x0)**. 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 **(x – x0) / (x1 – x0)** and **(x1 – x) / (x1 – x0)**, which are normalized distances between the unknown point and each of the end points. Because these sum to 1,

which yields the formula for linear interpolation given above.

### Interpolation of a Data Set:

Linear interpolation on a set of data points **( x_{0}, y_{0}), (x_{1}, y_{1}), …, (x_{n}, y_{n})** 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

**C0**.

### Linear Interpolation as Approximation

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.

### References:

[1] Meijering, Erik (2002), ‘A chronology of interpolation: from ancient astronomy to modern signal and image processing’, *Proceedings of the IEEE*, **90** (3): 319–342, doi:10.1109/5.993400.

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