Interpolation Using Lagrange Polynomials
Lagrange polynomials,can approximate a function by a polynomial by interpolating the values of the function at a set of pointsFor a given set of distinct pointsand numbersthe Lagrange polynomial is the polynomial of the least degree (forpoints,is a polynomial of degreein general) that at each point assumes the corresponding valueThe interpolating polynomial of least degree is unique.
The polynomials are built up from Lagrange basis polynomialswhich have the property thatandif
because of the factorWe can ensure equality withby writingand the Lagrange polynomial that interpolatesis the sum of terms such as these:
Example: Find the Lagrange polynomial that interpolates the functionat with
It is frequently the case that theare evenly spaced, though this is not always optimal. Obviously, increasing the number of points in general increases the accuracy.