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NUM702S - NUMERICAL METHODS 2 - 1ST OPP - NOVEMBER 2023 |
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QUALIFICATION: BACHELORof SCIENCEIN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATIONCODE: 07BSAM
LEVEL:7
COURSE:NUMERICAL METHODS 2
COURSECODE: NUM702S
DATE: NOVEMBER 2023
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
EXAMINER:
MODERATOR:
FIRSTOPPORTUNITY:EXAMINATION QUESTION PAPER
Dr S.N. NEOSS/-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.
PERMISSIBLEMATERIALS:
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 + -x2 - -x3 + -x4 - .. · .
23 4 5
[12]
1-2. Use the result in 1-1. to establish ln(l + x)
fraction form.
= R
3
,,
·-
30x
30
++3261xx+2 +9xx-,3,
and
express
R 3'2
in
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, 1].
2-1. Show the following property:
[5]
Tn has n d1.stm. ct zeros Xk E [-1, 1] : Xk = cos ((_2;k__+-1--)-1'-r-) for 0 :5.k :5.n - 1.
2n
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 + 2x - + x3 3x4 •
(i) Economise the power series f(x).
[3]
(ii) Find the Chebyshev series for f(x).
[5]
;;:; =(;{S)~_;:_xwx'.h•~::e~o~I';:! f:,ction f is even and use an apprnpdate result to find its Fourier se;::;
2 , for 0 :5.X < 7r.
(ii) Set x = 0 and conclude that
1r 2
8
=
1+
1
32
+
1
52
+
1
72
+
···.
[2]
Problem 3 [27 Marks]
3-1. Given the integral
3
{
}0
s1in+( 2xx5) dx
= 0.6717578646 · · ·
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) = 5x4 .
[5]
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r
Problem 4 [24 Marks]
4-1. 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( 0) = (1, of and perform two iterations.
[6]
4-2. 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 !!!
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