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NUM702S- NUMERICAL METHODS 2- JAN 2020 |
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NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH AND APPLIED SCIENCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION:
Bachelor of science in Applied Mathematics and Statistics
QUALIFICATION CODE: 35BAMS
LEVEL: 7
COURSE CODE: NUM702S
COURSE NAME: NUMERICAL METHODS 2
SESSION:
DURATION:
JANUARY 2020
3 HOURS
PAPER: THEORY
MARKS: 90
SECOND OPPORTUNITY/SUPPLEMENTARY EXAMINATION QUESTION PAPER
EXAMINER
Dr S.N. NEOSS] NGUETCHUE
MODERATOR:
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 4 to 5 decimals where necessary unless specified 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 without a cover.
THIS QUESTION PAPER CONSISTS OF 3 PAGES (Including this front page)
Attachments
None
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Problem 1 [20 Marks]
1-1. Find the Padé approximation Ro9(x) for f(x) = In(1+<2)/z starting with the MacLaurin expansion
fel e1-244 2S boas,
[19]
1-2. Use the result in 1-1.
fraction form.
to establish
In(1+ <2) =
R32
=
30x + 21x? + x3
30 + 36x + 9x?
and
express
Az.
in continued
[8]
Problem 2 [25 Marks]
For any non negative interger n we define Chebyshev polynomial of the first kind as
T,(x) = cos(n@), where 6 = arccos(zx), for x € [-1, 1].
2-1. Show the following property:
[5]
T, has n distinct zeros x; € [—1, 1] : x, = cos (2Sk+e1n )
for0<k<n-l1.
2-2. Show that the Chebyshev polynomial T;, is a solution of the differential equation:
[8]
d* f if
(1 = ‘a3 — a
+n’f =0.
N
+ (N+1)a
2-3. Use the identity /formula: Ss; cos(y + ka) = I
N
AL
k=
sin 2
to show that:
[12]
N
S > Tin(k)Tn (ae) = 0, for m 7 n,
k=0
(2k + 1)r
where x, = cos Sa | ,0<k<VN, are the roots of Ty41.
Problem 3 [45 Marks]
3-1. Given the integral
Zs
/ sin(22) 1. — 0.6717578646---
0 1+ x
3-1-1. Compute T(J) = R(J,0) for J = 0,1,2,3 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,1)}, 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, 1] and show that
the rule is exact for f(x) = 527.
[5]
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3-3. Use Jacobi’s method to find the eigenpairs of the matrix
1
A= |-2
4
-2 4
5 -2
-2 1
[18]
God bless you !!!