Determining the Rolle function in Lagrange interpolatory approximation

Abstract

We determine the Rolle function in Lagrange polynomial approximation using a suitable differential equation. We then propose a device for improving the Lagrange approximation by exploiting our knowledge of the Rolle function.

Determining the Rolle function in Hermite interpolatory approximation by solving an appropriate differential equation

“…We see that the use of H ξ improves the approximation by many orders of magnitude. This effect was also observed in [1]. Note that the degree of the error polynomial E (x) is four plus the degree of H ξ .…”

“…Recently, we reported on a technique for determining the Rolle function in Lagrange interpolation, and how this could lead to an improvement in the accuracy of the approximation [1]. In this short paper, we extend that investigation to include Hermite interpolation.…”

“…In this short paper, we extend that investigation to include Hermite interpolation. We consider the same example as used in [1], and show how significant improvements in approximation accuracy can be achieved once the Rolle function is known.…”

“…We solve this differential equation in a manner similar to that used in [1]: we find an initial value at a point close to the node x 0 = 0 (we cannot find ξ z at any interpolation node, because the factor n k=0 (x − x k ) 2 in (3) ensures that ∆ (x z |H 2n+1 ) = 0 at every interpolation node, regardless of the value of ξ). Call this point x z and choose x z = 10 −5 .…”

“…The second solution, on the other hand, gives an acceptable Rolle function (see Figure 1). The numerical solution was obtained using a seventh-order Runge-Kutta (RK) method [4] with a stepsize of ∼ 5 × 10 −5 , the same stepsize used in [1].…”