10  Exercises (Chapter 3)

10.1 Exercise 1

  1. Suppose f:[a,b]\to\mathbb R. What is the degree 1 interpolating polynomial p_1 on [a,b] with interpolation nodes at the end points?
  2. Suppose that f is twice continuously differentiable. Give an error estimate for \| f - p_1 \|_{L^\infty([a,b])}
  3. Use this bound to estimate the error in a piecewise linear approximation of f on [-1,1].
  4. Let \mathcal P be the set of peicewise linear functions where the first derivative is only not continuous on the set X = \{x_0,x_1,\cdots,x_n\}. The cardinal functions c_j for piecewise linear interpolation on X satisfy (i) c_j \in \mathcal P, and (ii) c_j(x_k) = \delta_{jk}. Sketch these functions and use them to contruct piecewise linear intepolants of f on X.

[notice that \ell_j are cardinal functions for polynomial interpolation]

10.2 Exercsise 2

Fix \{ x_0 < x_1 < \dots < x_n \} and recall that \ell_j(x) = \prod_{k\not= j} \frac{x-x_k}{x_j-x_k} are the Lagrange polynomials. Define,

\begin{align} h_j(x) &= \ell_j(x)^2\big[ 1-2\ell_j'(x) (x-x_j) \big] \nonumber\\ k_j(x) &= \ell_j(x)^2 (x - x_j) \end{align}

  1. What is h_j(x_l)?
  2. What is k_j(x_l)?
  3. What are the degrees of these polynomials?
  4. Use h_j and k_j to construct the Hermite polynomial interpolation of f on X = \{\{ x_0, x_0, x_1,x_1, \dots, x_n, x_n\}\}.

10.3 Exercise 3

Recall that \mathcal P_n is the set of all polynomials of degree less than or equal to n

  1. There exist Legendre polynomials P_n of degreee n such that

\begin{align} (P_n, P_m) := \int_{-1}^{+1} P_n(x) P_m(x) \mathrm{d}x = 0 \end{align}

for all n \not= m. Explain why (P_n, x^m) = 0 for all m < n (that is P_n is orthogonal to x^m) and thus (P_n, q) = 0 for all q \in \mathcal P_{n-1}.

  1. Define \widetilde P_n by dividing P_n by the leading coefficient. Show that \widetilde P_n solves the following problem:

\begin{align} \min_{ p_n = x^n + q \, : \, q \in \mathcal P_{n-1}} \int_{-1}^{+1} \left| p_n(x) \right|^2 \mathrm{d}x \end{align}

[That is Legendre points (zeros of P_n) minimise the L^2 error of the node polynomial]