[SOLVED] MATH 151A Assignment 2

Description

1. Given that each of the following sequences converges to p^∗, show that it converges linearly:
   • The sequence is and the limit is p^∗ = 0;
   • 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|

• Plot the graph of the function f.
• Given n ∈ N\{0}, define for 0 ≤ k ≤ n.

Let g_n(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.

• Plot the sequence {g_n(0.3)}₁≤_n≤₂₀.

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