Polynomial and Spline Interpolation

Abstract

We determine to given interpolation points (x _i, y _i) a polynomialp with p(x _i) = y _i. We find this polynomial by evaluating Lagrange’s interpolation formula. As impressively simple as it is to determine this interpolation polynomial, as effective is this tool: We will apply this polynomial interpolation several times in later chapters, for example for the numerical approximation of certain integrals or solutions of initial value problems.

Besides polynomial interpolation, we also consider spline interpolation to given grid points. The goal here is not to specify a closed function which interpolates the grid points, but rather to specify a function defined in sections whose graph passes through the given grid points as smoothly as possible.