Major Quiz 1 Solution
Q1. The following set of equations is to be solved to get the value of x for a given
𝛿
:
[8]
?
+
?
= 2
?
+ (1 −
𝛿
)
?
= 1
a) For what values of
𝛿
will this problem be wellconditioned?
(5)
Ans
a.
?
+
?
= 2
?
+ (1 −
𝛿
)
?
= 1
Solving for
x
, we get
x= f(δ)=
2𝛿−1
𝛿
The condition number of the problem is C
p
=

𝛿𝑓 ′(𝛿)
𝑓(𝛿)

=

1
2𝛿−1

For the problem to be wellconditioned,
?
𝑝
< 1
(3 marks)
−1 <
1
2δ − 1
< 1
or
−1 > 2δ − 1 > 1
δ < 0
or
δ > 1
(2 marks)
b) Calculate the % relative change in x for a 1% change in
𝛿
, for
𝜹
= 2.
Perform computations using infinite precision but report intermediate steps up to 4 significant
digits with rounding.
(3)
Ans
b.
Calculate the relative change
Approach 1:
∆𝛿 = 0.02
𝛿 = 2
?? =  (∆?/?)/(∆𝛿/𝛿) =
∆𝑥/𝑥
0.02/2
? = ?(𝛿) =
2𝛿−1
𝛿
=
3
2
= 1.50
?
′
= ?(2.02) =
2∗2.02−1
2.02
=
4.04−1
2.02
= 1.50495

∆𝑥
𝑥
 = (1.50495 − 1.50) ∗
100
1.5
= 0.33%
Approach 2:
𝐹?𝑟 𝛿 = 2,
?? =  (∆?/?)/(∆𝛿/𝛿)
∆?/? = ?? ∗ 
∆𝛿
𝛿
 = 
1
2𝛿−1
 ∗ 1%
=
1
3
∗ 1% = 0.33%