Quadratic Interpolation

Instead of performing linear interpolation by fitting a straight line through two data points, we can use several data points and fit a curve through them. Given any two points, there is just one straight line which passes through them. Similarly, if we are given three points there is just one quadratic curve which passes through all three. By finding the equation of this curve, and evaluating it at our interpolating value  x , we could perform quadratic interpolation.

Graphically :

If (x0, y0), (x1, y1), (x2, y2), are given data points, then the quadratic polynomial passing through these points can be expressed as



The polynomial P(x) given by the above formula is called Lagrange’s interpolating polynomial and the functions L0, L1, L2 are called Lagrange’s interpolating basis functions.


Note:   Note that


and that

  δij is called the Kronecker delta function