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

where,

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

  δ_ij is called the Kronecker delta function