Suppose that you measured N values of a physical quantity y that depends on parameter x. You seek to find a function that will represent the data at the N points,

and will provide a good estimate for the physical quantity at a general value of x.

The most straight-forward technique is to interpolate between each pair of points with a straight line, as in :

or

which is accurate to O(x²).

For sharp edged functions, this interpolation can actually provide a more accurate approximation than higher order approaches. However, if the function is smooth and curvy then higher order polynomial interpolations can provide a more accurate estimation.

For example, we could use three points to create a 2nd order polynomial:

This is generalized with the Lagrange formula that provides an Nth order polynomial to fit through N points:

References & Web Resources

• Tao Pang, An Introduction to Computational Physics