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NUM702S - NUMERICAL METHODS 2 - 1ST OPP - NOVEMBER 2024pdf |
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1 n Am I 8 I A Unl VE R5 ITY
OF SCIEnCE
FacultoyfHealthN, atural
ResourceasndApplied
Sciences
Schoool f NaturalandApplied
Sciences
Departmentof Mathematics,
StatisticsandActuarialScience
13JacksonKaujeuaStreet
Private Bag13388
Windhoek
NAMIBIA
T: +264612072913
E: msas@nust.na
W: www.nust.na
QUALIFICATION: BACHELOR OF SCIENCE IN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATION CODE: 07BSAM
LEVEL:7
COURSE:NUMERICAL METHODS 2
DATE: NOVEMBER 2024
DURATION: 3 HOURS
COURSECODE: NUM702S
SESSION: 1
MARKS: 100
EXAMINER:
MODERATOR:
FIRST OPPORTUNITY: QUESTION PAPER
Dr SN NEOSSI-NGUETCHUE
Prof S.S. MOTSA
INSTRUCTIONS:
1. Answer ALL the questions in the booklet provided.
2. Show clearly all the steps used in the calculations. All numerical results must be
given using 5 decimals where necessary unless mentioned otherwise.
3. All written work must be done in blue or black ink and sketches must be done in
pencil
PERMISSIBLE MATERIALS:
1. Non-Programmable Calculator
ATTACHEMENTS
None
This paper consists of 3 pages including this front page
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Problem 1 [19 Marks]
1-1. Find the Pade approximation R2,2 (x) for f(x) = ln(l + x)/x starting with the MacLaurin expansion
f(x)
= 1-
x
-2
+
-x3 2
-
-x43 + -x54 -
··· .
[12]
1-2. Use the result m. 1-1. to establis. h ln(l + x)
fraction form.
R 3 '2
=
30x + 2lx 2 + x 3
30 + 36x + 9x2
and
express
R 3 '2
m•
continued
[7]
Problem 2 [30 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 the following property:
[5]
Tn has n d1. stm. ct zeros xk E [-1, l]: xk = cos ((2k+2l)n1r)
for 0 :S k :Sn - l.
2-2. Compute the expressions of the first five Chebyshev polynomials of the first kind T0 , T1, T2 , T3 and
n.
2-3. Given the trucated power series f(x) = 1 + 2x - x3 + 3x4 .
(i) Economise the power series f(x).
[3]
(ii) Find the Chebyshev series for f(x).
[5]
;;:;-(i{) s_;:__wxx~h•~:~e~o~l:w~n:ftction f is even and use an apprnpriatecesult to find its Fouriecse;::;
2 , for 0 _<X < 7r.
2
(ii) Set x = 0 and conclude that ~ = 1 + }2 + ; 2 + ; 2 + · · ·.
[2]
Problem 3 [27 Marks]
3-1. Given the integral
3
(
Jo
slin+( 2xxs) dx
=
0.6717578646 · · ·
3-1-1. Compute T(J) = R(J, 0) for J = 0, l, 2, 3 using the sequential trapezoidal rule.
[10]
3-1-2. Use the results in 3-1-1. and Romberg's rule to compute the values for the sequential Simpson rule
{R(J, l)}, sequential Boole rule {R(J, 2)} and the third impprovement {R(J, 3)}. Display your results in
a tabular form.
[12]
3-2. State the three-point Gaussian Rule for a continuous function f on the interval [-1, l] and show that
the rule is exact for f (x) = 5x4 .
[5]
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Problem 4 [24 Marks]
4-1. The matrix A and its inverse are A- 1 are given below
[-2-2] A= [1/2-lJ
-1 1 '
A- 1 = -2 -1 .
• Use the power method to find the eigenvalue of the matrix A with the smallest absolute value.
Of Start with the vector x( 0) = (1, and perform two iterations.
[6]
4-2. Use Jacobi's method to find the eigenpairs of the matrix
2] 'A=
./12
./2
3
./2
[ 2 ./2 1
[18]
Goel bless you !!!
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