## The Error Term For Polynomial Approximations

Interpolation may used to approximate either a known function or a set of data for an unknown function. It is not possible to evaluate the error for an unknown function.

If (i.e. is a function times differentiable in with continuous derivatives) and is the polynomial that interpolates then there exists that depends on such that Proof: This result is trivially true for since both sides are zero.

For define so that Define the function which satisfies for and  has distinct roots in By Rolle's theorem, there must be at least one root of between two successive roots of So has at least distinct roots in Apply Rolle's theorem to derivatives of increasing order successively, to show that must have at least one root in Call this point so that The nth derivative of is but is a polynomial of degree so Let so that Hence Using this we can obtain bounds on the accuracy of Lagrange interpolation, provided we can bound the nth derivative, Since the error term involves Lagrange interpolation is exact for polynomials of degree at most.

Example: Estimate the error for f(x)= sin(3x) interpolated for the points at  So, the error in the approximation to is The actual error is However, the function cannot be approximated accurately by an interpolation

polynomial, using equidistant points on the interval Although is differentiable, its derivatives grow as for even so does not converge to 0 while increases (since interval width &gt; 1). 