$3\mathrm{.}2\mathrm{.}2\mathrm{.}1$ Lagrange interpolation

The principles of Lagrange interpolation were introduced in the lecture. Recall that this takes a set of $n+1$ data points, $(x_k,f_k)$ with $k=0,$dots,$n,$ and defines Lagrange basis polynomials by

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The Lagrange interpolating polynomial is then defined by

$P(x)={\displaystyle \underset{k=0}{\overset{n}{}}}{f}_k{L}_k(x).$

Note that, for convenience in Python, indices in equations $(3\mathrm{.}1)$ and $(3\mathrm{.}2)$ start at $0.$ There are $n+1$ interpolation points and $P(x)$ is therefore of degree $n.$

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CHAPTER $3.$ INTERPOLATION

Write a function lagrange$\_$poly which computes the $k$ th Lagrange basis polynomial. It should take as arguments an array of points $x,$ the list of interpolation points $x_k,$ and an integer $k,$ and return an array of numbers $L_k(x)$ according to $(3\mathrm{.}1).$

Write a function interpolating$\_$poly which computes the Lagrange interpolating polynomial given by $(3\mathrm{.}2).$ It should take as arguments an array of points $x$ and lists of data points $x_k$ and $f_k,$ and call the function lagrange$\_$poly when necessary.