[Solved] MATH 151A Assignment 3

1. Let f(x) = 1/x, x_i = i + 1, 0 i 2, find the Lagrange interpolation polynomial interpolating the points (x_i,f(x_i)) using
   • Lagrange interpolation formula.
   • Nevilles method.
   • the divided difference interpolation.
2. Find the natural cubic spline passing through (1,1) , (0,1), (1,2).
3. Consider Hermite interpolation problem. Prove the following theorem: Let f C¹([a,b]) and x₀,x₁,x_n be n distinct nodes in [a,b], and let

where

Hn,j = [1 2(x xj)L0n,j(xj)]L2n,j(x) Hn,j = (x xj)L2n,j(x).

Show that H(x_i) = f(x_i) and H⁰(x_i) = f⁰(x_i), for all 0 i n. (You do not need to prove the uniqueness of H.)

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4. Let f(x) be a function defined on the interval [x₀h,x₀ +h], and f C³[x₀h,x₀ +h] .

• Let P(x) be the Lagrange interpolation polynomial of f(x) at the nodes x = x₀ h,x₀,x₀ + h. Write down the expression of P(x).
• Write down the error term E(x) := f(x) P(x) in terms of the derivatives of f(x).
• Using the fact that f(x) = P(x)+E(x), calculate the derivatives of f, f⁰(x) at x = x₀.
• If we approximate the derivative f⁰(x₀) by P⁰(x₀), is it true that the above approximation is exact if f is a polynomial of degree less than or equal to 2 ? Why ?
• Write down an error bound of this approximation rule suggested in (d) for a generalfunction f(x) based on the result in (b).

5. (Programming problem)

Let a number of points (x_i,f(x_i)) be given, 0 i n. Let P(x) be its Lagrange interpolation polynomial interpolating the points (x_i,f(x_i)), 0 i n.

Write a program which allow inputs {(x_i,f(x_i))} and a value a, and calculate the value of P(a).

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