\part{Interpolation and Approximation theory}
\chapter{Interpolation}

Often one has a function of discrete 'data points' to evaluate a function from, however if one seeks to evaluate a function at a point between the given data points one resorts to \emph{interpolation}; extending a function of discrete data points to a continuous domain to evaluate points within the range of the data points.

If one wishes to evaluate points above the largest data point or below the smallest data point, this becomes a problem of \emph{extrapolation}.

\section{Lagrange polynomial interpolation}

\[U = \{ (x_1,y_1), \hdots , (x_{n},y_{n}) \} \]
\[\ell_n (x) = \frac{\prod_{i \neq n}(x-x_i)}{\prod_{i \neq n}(x_n-x_i)}\]
\[f(x) = \sum_{k=1}^{n} y_k \ell_k (x)\]


\section{Vandermonde polynomial interpolation}

\[U = \{ (x_1,y_1), \hdots , (x_n,y_n) \} \]
\[ \mathbf{V} = \begin{bmatrix} 1 & x_1 & x_{1}^2 & \hdots & x_{1}^n \\ 1 & x_2 & x_{2}^2 & \hdots & x_{2}^n \\ \vdots & \ddots & \vdots & \vdots \\ 1 & x_n & x_{n}^2 & \hdots & x_{n}^n\end{bmatrix} \]
\[f(x) = \sum^{n}_{k=1} ( \mathbf{V}^{-1}\mathbf{y} )_{k} x^k\]


\section{Spline interpolation}
\chapter{Approximation theory}
\section{Taylor series approximation}
- Taylor series approximation
\[f(x) = \sum^{n}_{k=0} \frac{f^{(k)}(0)}{k!}x^k\]
\section{Padé approximant}
\section{Stirling's formula}
