For this problem you will test a few interpolation approaches for the application of generating interpolated data. We'll do this by interpolating data that is sampled from a known mathematical function. The same function can then be used to compute true values at the interpolated points to use in error calculations.

Consider the following mathematical function $($Runge$\text{'}$s function$)$:

RUNGE's FUNCTION

Write a function mfile that uses this formula to generate a set of data points. Your function should use those points along with the interpolation approaches outlined below to generate interpolated data $($x$\_$i$,$ f$\_$i$)$ on a specified increment. Your function should accept the following inputs $($in order$)$:

A vector of x values at which f$($x$)$ is to be evaluated to generate the sampled data.

A scalar input specifying the increment to use for the interpolated data.

Your function should use the following three approaches to generate interpolated data:

A polynomial of order $($N$-1)$ where N is the number of points in the sampled data

A linear spline

A clamped cubic spline with derivative end conditions set to values equal to the exact analytical derivative of f$($x$)$ evaluated at the end points.

Your function should have the following three outputs $($in order$)$:

The interpolated data $($f$\_$i$)$ generated with polynomial interpolation $($column vector$)$

The interpolated data $($f$\_$i$)$ generated with a linear spline $($column vector$)$

The interpolated data $($f$\_$i$)$ generated with the clamped cubic spline $($column vector$)$

Note this problem is similar to the humps$\_$interp.m example.