Assignment # 3 CSE 330

1. (6 marks) One of the Hermite basis element that we discussed during a class is

hk(x) = (

1 − 2

(

x − xk

)

l

′

k

(xk)

)

l

2

k

(x) .

Very that h

′

k

(xj ) = 0 ∀ j, k.

2. A function is given by f(x) = xe−3x + x

2

. Now answer the following up to five significant figures.

(a) (4 marks) Approximate the derivative of f(x) at x0 = 2 with step size h = 0.1 using the central difference

method.

(b) (4 marks) Calculate the truncation error of f(x) at x0 = 2 using h = 0.1 using the central difference method.

(c) (6 marks) Compute D

(1)

0.1

at x0 = 2 using Richardson extrapolation method and calculate the truncation error.

3. During the class, we derived in detail the first order Richardson extrapolated derivative, by using h → h/2,

D

(1)

h ≡ f

′

(x0)– h

4

480

f

(5)(x0) + O(h

6

) .

(a) (6 marks) Using h → h/2, derive the expression for D

(2)

h which is the second order Richardson extrapolation.

(b) (4 marks) If f(x) = x

2

ln x, x0 = 1, h = 0.1, find the upper bound of error for D

(1)

h

.

Motto: Mathematics is NOT difficult, but what is difficult is to believe that mathematics is NOT difficult.