Lagrange Interpolation

In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of distinct points Xi and numbers Yi, the Lagrange polynomial is the polynomial of the least degree that at each point Xj assumes the corresponding value Yj (i.e. the functions coincide at each point). The interpolating polynomial of the least degree is unique, however, and it is therefore more appropriate to speak of “the Lagrange form” of that unique polynomial rather than “the Lagrange interpolation polynomial”, since the same polynomial can be arrived at through multiple methods.

The Lagrange interpolating polynomial is the polynomial P(X) of degree <=(n – 1) that passes through the n points (x1, y1 = f(x1)),(x2, y2 = f(x2)) ,.., (xn, yn = f(xn)), , and is given by