ENGR 6101: COMPUTATIONAL ENGINEERING
Problem Set 1 (due Thursday, 8/28)
Show all your work on all problems. Correct answers without enough work will be graded as 0/3.
Questions:
1. (3 pts.) A computer with 4 digit precision would recognize the given values as follows:
π
→0.3142×101
12347.89 →0.1235×105
0.0001231234 →0.1231×10−3
1→0.1000×101
We can emulate the algebraic operations on such a computer by doing the same operation on a higher
precision machine (eg. a calculator,matlab, etc.) and than decrease the precision as shown above
(rounding to the nearest decimal). But, we have to round after each basic operation, such as sum,
product, square root, etc.
(a) Consider the following expression: √x−3
On such a computer, compute the expression above for x=9.01. 1
(b) Note that
√x−3=(√x−3)(√x+3)
√x+3=x−9
√x+3
On the same computer, evaluate the last expression above, again for x=9.01.
(c) Use a higher precision machine (at least 8 digits) to compute √9.01 −3, and write down this
result. Assuming that this results is “true”, compute the absolute and relative errors (accurate
upto at least 6 digits) for the results you obtained in part (a) and (b).
2. (3 pts.) Consider the following numerical differentiation formula:
f′(x)≈
α
f(x+h) +
β
f(x−2h)
γ
h
Find the appropriate values for
α
,
β
and
γ
for this formula to work (using Taylor series expansion).
What is the order of your approximation?
3. (3 pts) Consider the following numerical differentiation formula:
f′(x)≈−f(x+2h) + 8f(x+h)−8f(x−h) + f(x−2h)
12h
Prove that this is a valid formula (using Taylor’s expansion) and find its order.
1In 1(a) and 1(b), do the operations step by step. For example, in 1(a), first do the square root (on a high precision machine), and
then adjust the precision to four digits. Then do the difference (again on a high precision machine), and again limit the precision of
the result to four digits.
1
