Interpolating Polynomials

• We have a table of $x$ and $y$-values.
• Two entries in this table might be
  $y=2.36$ at $x=0.41$ and
  $y=3.11$ at $x=0.52$.
• If we desire an estimate for $y$ at $x=0.43$, we would use the two table values for that estimate.
• Why not interpolate as if $y(x)$ was linear between the two $x$-values (similar triangles)?

  $$y(0.43)\approx 2.36+\frac{2}{11}(3.11-2.36)=2.50~~ \bigg\vert$$

  where

  $$\frac{2}{11} \Longrightarrow \frac{0.43-0.41}{0.52-0.41}$$

• We will be most interested in techniques adapted to situations where the data are far from linear.
• The basic principle is to fit a polynomial curve to the data.