Numerical Analysis 1 (MATS2207)

Mathematics and Computer Science - MCS

Semester: Second Semester

Level: 200

Year: 2018

Answers (200 )
1
THE UNIVERSITY OF BAMENDA FACULTY OF SCIENCE
FIRST SEMESTER EXAMINATIONS
Department Mathematics Course Instructors: Tanyu nee Nfor Vivian
Month: July Course Code: MATS 2207
Year: 2018 Course Title: Numerical Analysis I
Date: 12/07/2018 Credit Value: Six Credits
Time Allowed: 3 hours
Answer all the questions. All necessary work must be shown and must be
neatly and orderly presented.
1. a) Consider the system of equations
−3x
1
+ 2x
2
— x3 = −1
6x
1
−3x
2
+ 7x
3
= −7
3x
1
− 4x
2
+ 4x
3
= −6
By considering the fact that l
i i
= 0, use the LU decomposition method to solve the system
b) The linear systems
0.00300x
1
+ 59.14x
2
= 59.17
5.291 x
1
− 6.130x
2
= 46.78
has the exact solution x
1
= 10.00 and x
2
= 1.000. What happen to the solutions when the system is
solved using the Gaussian elimination method with four-digit rounding arithmetic? Are the
approximate solutions the same with the exact solution? If not, by solving the system, show how we
can improve on the solution
c) If x = (−3,1,2) in R
3
, find
‖
‖
2
,
‖
‖
d) What is the binary representation for | (8+10+3+4=25 marks)
2. a) State the following theorems
i) Rolle's theorem , ii)Intermediate value theorem, iii) Taylor’s theorem, iv) Mean value theorem
b) Consider the nonlinear equation
− x − 2 = 0
i) Show that there is a root in the interval (1,2).
ii) Estimate how many iterations will be needed in order to approximate this root with and accuracy of
= 0.1 using the bisection method.
iii) Then approximate a with the accuracy of = 0.1 using the bisection method
c) Let f ( x ) =
√
+ 1 + 1. Find the third Taylor polynomial P
3
(x) about x
0
= 0 and use it to
approximate
√
0.5 (4+2+2+6+4=18 marks)
3. a) i) What is an absolute error and a relative error?
ii) When is a problem said to be well-conditioned and ill-conditioned? And what does it mean for
an algorithm to be stable?
b) i) Consider evaluating a function f(x) at a point x = x
0
. If we perturb the input to x = x
0
+ then the
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output is f(x
0
+ ). Show that the condition number, denoted C
f
(x
o
), of f at x
0
is
C
f
(x
o
) =
(
)
(
)
ii) Find the condition number of the function
f(x) =

at the point x = −0.93
iii) Let f(x) =
√
1 + − 1. By evaluating f for x near 0 and with the help of the condition number
of f at 0, determine whether the problem is a well- conditioned problem or an ill-conditioned problem.
What happens to the condition number when the function f is rationalized?
c) Use intermediate value theorem and the Rollers theorem to show that the graph of f (x ) = x
3
+ 2x +
k crosses the x-axis exactly once regardless of the value of the constant k (2+4+4+4+8+5=27 marks)
GOOD LUCK
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