Assignment # 3 CSE 330
Instructions for preparing the solution script:
• Write your name, ID#, and Section number clearly in the very front page.
• Write all answers sequentially.
• Start answering a question (not the pat of the question) from the top of a new page.
• Write legibly and in orderly fashion maintaining all mathematical norms and rules. Prepare a single solution file.
The deadline is March 01, 2023 in class.
• Start working right away. There is no late submission form. If you miss the deadline, you need to use the make-up
assignment to cover up the marks.
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.
