 |
NUM702S - NUMERICAL METHODS 2 - 2ND OPP - JANUARY 2024 |
|
|
1 Page 1 |
▲back to top |
nAmlBIA UnlVERSITY
OF SCIEnCEAnDTECHnOLOGY
FacultyofHealth,Natural
ResourceasndApplied
Sciences
Schoolof NaturalandApplied
Sciences
Departmentof Mathematics,
StatisticsandActuarialScience
13JacksonKaujeuaStreet
PrivateBag13388
Windhoek
NAMIBIA
T: •264 612072913
E: msas@nust.na
W: www.nust.na
QUALIFICATION: BACHELOR OF SCIENCEIN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATIONCODE: 07BSAM
LEVEL:7
COURSE:NUMERICAL METHODS 2
COURSECODE: NUM702S
DATE: JANUARY 2024
SESSION: 1
DURATION: 3 HOURS
MARKS: 90
SECOND OPPORTUNITY/ SUPPLEMENTARY: EXAMINATION QUESTION PAPER
EXAMINER:
MODERATOR:
Dr S.N. NEOSSI-NGUETCHUE
Prof S.S. MOTSA
INSTRUCTIONS:
1. Answer all questions on the separate answer sheet.
2. Please write neatly and legibly.
3. Do not use the left side margin of the exam paper. This must be allowed for the
examiner.
4. No books, notes and other additional aids are allowed.
5. Show clearly all the steps used in the calculations. All numerical results must be
given using 5 decimals where necessary unless mentioned otherwise.
6. Mark all answers clearly with their respective question numbers.
PERMISSIBLE MATERIALS:
1. Non-Programmable Calculator
ATTACHEMENTS
None
This paper consists of 3 pages including this front page
|
|
2 Page 2 |
▲back to top |
.-
Problem 1 [25 Marks]
1-1. Show that the formula for the best line to fit data (k, Yk) at integers k for 1 k n is y = ax+ b,
where
l a= n(n26- 1) [2 n kyk - (n + 1) n Yk
l b = n(n 4- 1) [(2n + 1) n Yk - 3 n kyk
[15]
x x: 1-2. Establish the Pade approximation ex R?-,2 (x) = 1122-+6x5+x-+ and express R?-,?- in continued fraction
form.
[10]
Problem 2 [20 Marks]
For any non negative interger n we define Chebyshev polynomial of the first kind as
Tn(x) = cos(n0), where 0 = arccos(x), for x E [-1, l].
2-1. Show that the Chebyshev polynomial Tn is a solution of the differential equation:
[8]
(1 -
x
2
)-
d2 f
dx 2
-
df
x-dx
+n2J =
0.
n.2-2. Compute the expressions of the first five Chebyshev polynomials of the first kind T0 , T1, T2 , T3 and
2-3. Given the trucated power series f(x) = 1 - x - x3 .
(i) Economise the power series J(x).
[3]
(ii) Find the Chebyshev series for f (x).
[5]
Problem 3 [13 Marks]
3-1. Given the integral
13 sin(2x)
o -1--+-,,-x-5dx = 0.6717578646 · · ·
3-1-1. Compute T(J) = R(J, 0) for J = 0, 1, 2, 3 using the sequential trapezoidal rule.
[10]
3-2. State the three-point Gaussian Rule for a continuous function f on the interval [-1, l].
[3]
Problem 4 [32 Marks]
4-1. Assume a 3 x 3 matrix A is known to have three different real eigenvalues >1., >2. and >3.. Assume we
know that >1. is near -2, >2. is near -5 and >.3is near -1.
4-1-1. Explain how the power method can be used to find the values of >1., )..2 and ).3. respectively. [2x3=6]
4-1-2. Discuss how shifting can be used in 4-1-1. to accelerate the convergence of the power method. [2]
1
|
|
3 Page 3 |
▲back to top |
4-2. The matrix A and its inverse are A- 1 are given below
A= [1/-2lJ
-1 1 '
• Use the power method to find the eigenvalue of the matrix A with the smallest absolute value.
Start with the vector x(D) = (1, Of and perform two iterations.
[6]
4-3. Use Jacobi's method to find the eigenpairs of the matrix
l 2] A=
v'2
v'2
3
v'2
[2 v'2 1
[18]
God bless you !!!
2