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- Applied Numerical Methods (MATH 151A,) Assignment 3

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1. Let f(x) = 1/x, xi = i + 1, 0 ≤ i ≤ 2, find the Lagrange interpolation polynomial

interpolating the points (xi

, f(xi)) using

(a) Lagrange interpolation formula.

(b) Neville’s method.

(c) 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 ∈ C

1

([a, b]) and x0, x1, …xn be n distinct nodes in [a, b], and let

H(x) = Xn

i=0

f(xi)Hn,j (x) +Xn

i=0

f

0

(xi)Hˆ

n,j (x)

where

Hn,j = [1 − 2(x − xj )L

0

n,j (xj )]L

2

n,j (x) Hˆ

n,j = (x − xj )L

2

n,j (x).

Show that H(xi) = f(xi) and H0

(xi) = f

0

(xi), 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 [x0 − h, x0 + h], and f ∈ C

3

[x0 − h, x0 + h] .

(a) 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).

(b) Write down the error term E(x) := f(x) − P(x) in terms of the derivatives of f(x).

(c) Using the fact that f(x) = P(x)+E(x), calculate the derivatives of f, f

0

(x) at x = x0.

(d) If we approximate the derivative f

0

(x0) by P

0

(x0), is it true that the above approximation is exact if f is a polynomial of degree less than or equal to 2 ? Why ?

(e) Write down an error bound of this approximation rule suggested in (d) for a general

function f(x) based on the result in (b).

5. (Programming problem)

Let a number of points (xi

, f(xi)) be given, 0 ≤ i ≤ n. Let P(x) be its Lagrange interpolation polynomial interpolating the points (xi

, f(xi)), 0 ≤ i ≤ n.

Write a program which allow inputs {(xi

, f(xi))} and a value a, and calculate the value of

P(a).

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