Interpolation

• The linear interpolant between two points $(x_0,y_0)$ and $(x_1,y_1)$ for a value $x\in (x_0,x_1)$ is the value $y$ on the straight line connecting these points, given either by:
  • $y=y_0+(x-x_0)\frac{y_1-y_0}{x_1-x_0}$ 
  • $y=y_0\left(\frac{x_1-x}{x_1-x_0}\right)+y_1\left(\frac{x-x_0}{x_1-x_0}\right)=y_0{w}_0+y_1{w}_1$ where $w_0$ and $w_1$ are the weights for $y_0$ and $y_1$, respectively. (These satisfy $w_0+w_1=1$)
• Linear interpolation on a data set of points $(x_0,y_0),(x_1,y_1),\dots ,(x_n,y_n)$ is defined as a piecewise linear function resulting from the concatenation of linear interpolants between each adjacent pair of data points. This function is continuous ($C^0$).
• The error of approximating a function $f(x)$ using linear interpolation with points $(x_0,f(x_0))$ and $(x_1,f(x_1))$ is defined as: $R_T=f(x)-p(x)$ where $p(x)$ is the linear interpolation polynomial $p(x)=f(x_0)+\frac{f(x_1)-f(x_0)}{x_1-x_0}(x-x_0)$
• Error Bound Theorem: If a function $f$ has a continuous second derivative on the interval $[x_0,x_1]$, the error $|R_T|$ of linear interpolation for $x\in [x_0,x_1]$ is bounded by $|R_T|\le \frac{(x_1-x_0{)}^2}{8}{\max }_{x_0\le x\le x_1}\left|f^{\prime \prime }(x)\right|$