1.2 Interpolation Formulae

Threading a polynomial through several given points $x_{k}, f_{k}$ we arrive at a smooth function which we may then differentiate. In this manner we construct approximations to the derivatives of the tabulated function.

Given some arbitrary point on the $x$-axis, we define

$$u \equiv \frac{x-x_{k}}{\Delta x}$$ (1.1)

as the normalized distance between $x$ and $x_{k}$.

Using the forward, backward, and central differences as defined above we arrive at different interpolation schemes.