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 functiontimes differentiable inwith continuous derivatives) andis the polynomial that interpolatesthenthere existsthat depends onsuch that
Proof: This result is trivially true forsince both sides are zero.
Define the function
hasdistinct roots in
By Rolle's theorem, there must be at least one root ofbetween two successive roots ofSohas at leastdistinct roots in
Apply Rolle's theorem to derivatives of increasing order successively, to show thatmust have at least one root inCall this pointso that
The nth derivative ofisbutis a polynomial of degreeso
Using this we can obtain bounds on the accuracy of Lagrange interpolation, provided we can bound the nth derivative,Since the error term involvesLagrange interpolation is exact for polynomials of degreeat most.
Example: Estimate the error for f(x)= sin(3x) interpolated for the points
So, the error in the approximation tois
The actual error is
However, the functioncannot be approximated accurately by an interpolation
polynomial, using equidistant points on the intervalAlthoughis differentiable, its derivatives grow asforeven sodoes not converge to 0 whileincreases (since interval width > 1).