1. Given that each of the following sequences converges to p∗, show that it converges linearly:
(a) The sequence is and the limit is p∗ = 0;
(b) The sequence is and the limit is p∗ = 1;
2. Show that the following sequences converges to p∗, show that it converges quadratically.
3. (a) Use the Lagrange interpolation method to find a polynomial f such that
f(1) = 2, f(2) = 1, f(3) = 4, f(4) = 3.
(b) Use the Neville’s Method instead to find the same polynomial f. 4. Programming problem: Consider the following function f : [−1,1] → R
f(x) = |x|
(a) Plot the graph of the function f.
(b) Given n ∈ N\{0}, define for 0 ≤ k ≤ n.
Let gn(x) be the unique polynomial of degree n which results by interpolating the n + 1 data , i.e. ) for all 0 ≤ k ≤ n. Plot the functions f,g2,g3,g4 and g5 on the same graph.
(c) Plot the sequence {gn(0.3)}1≤n≤20.
