Derivative error bounds for Lagrange interpolation: An extension of Cauchy's bound for the error of Lagrange interpolation

Abstract

It is shown that for any n + 1 times continuously differentiable function f and any choice of n + 1 knots, the Lagrange interpolation polynomial L of degree n satisfies ∥f (n) − L (n)∥ ⩽ ∥ω (n)∥ (n + 1)! ∥tf (n + 1)∥ , where ∥ ∥ denotes the supremum norm. Further, this bound is the best possible. Applications of the above bound to the differencing formula are suggested. It is also shown that for j = 1, 2, …, n − 1, ∥f (j) − L (j)∥ ⩽ ∥ω (j)∥ j!(n + 1 − j)! ∥tf (n + 1)∥ . This formula may be considered as a generalization of a formula due to Ciarlet, Schultz, and Varga (Numerical methods of high-order accuracy, Numer. Math. 9 (1967), 394–430) and may be compared to the conjectured best bound ∥f (j) − L (j)∥ ⩽ ∥ω (j)∥ (n + 1)! ∥tf (n + 1)∥ .