An Introduction to Numerical Methods and Analysis, 2nd Edition

APPENDIX A

PROOFS OF SELECTED THEOREMS, AND ADDITIONAL MATERIAL

A.1 PROOFS OF THE INTERPOLATION ERROR THEOREMS

The interpolation error theorems all depend on applying a generalization of Rolle’s Theorem to a very carefully constructed function. First, we will state and prove the general version of Rolle’s Theorem.

Theorem A.1 (Generalized Rolle’s Theorem) Let f Cⁿ([a, b]) be given, and assume that there are n points, z_k, 1 ≤ k ≤ n in [a, b] such that f(z_k) = 0. Then there exists at least one point ξ [a, b] such that f^(n-1)(ξ) = 0.

Proof: By Rolle’s Theorem, there exists at least one point η_k between each z_k and z_k+1 such that f′(η_k) = 0. Thus there are n − 1 points where g₁(x) = f′(x) is zero. By the same argument, then, there are n − 2 points where g₂(x) = f″(x) is zero. Continuing onward, then, we end up with a single point where g_n−1(x) = f⁽ⁿ⁻¹⁾(x) is zero.

This result allows us to prove, rather easily, both of Theorems 4.3 and 4.5.

Theorem A.2 (Lagrange Interpolation Error Theorem) Let f Cⁿ⁺¹([a, b]) and let the nodes x_k [a, b] for 0 ≤ k ≤ n. Then, for each x [a, b], there is ...