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ANA801S - APPLIED NUMERICAL ANALYSIS - 1ST OPP - JUNE 2022 |
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NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH, APPLIED SCIENCES AND NATURAL RESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATION CODE: 35BHAM
LEVEL: 8
COURSE CODE: ANA801S
COURSE NAME: APPLIED NUMERICAL ANALYSIS
SESSION: JUNE 2022
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 120 (to be converted to 100%)
EXAMINERS
MODERATOR: |
FIRST OPPORTUNITY
PROF S. A. REJU
PROF S. MOTSA
EXAMINATION
QUESTION
PAPER
INSTRUCTIONS
1. Attempt ALL the questions.
All written work must be done in blue or black ink and sketches must
be done in pencils.
3. Use of COMMA is not allowed as a DECIMAL POINT.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 3 PAGES (including this front page)
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QUESTION 1 [30 MARKS]
(a) Discuss the general Iterated quadrature rule for obtaining the following integral
[= f. fGdax
(1.1)
and hence state the Iterated Trapezoidal Rule for (1.1).
[5]
(b) From your Composite Trapezoidal rule in (a), state the Romberg’s Method for solving (1.1) and
hence using the unit interval [0, 1] for the integral
b
T(n) =| f(x)dx
and step size
h (b — a)
—
n
obtain the term for the recursive expression T(2”) = T(8) and the expression for R(n, 0) denoting
the Trapezoidal estimate with 2”.
[16]
(c) By just stating the Richardson’s Extrapolation R(n,m) employed in the Romberg’s Table, show
that
R(1,0) = 51R(0,0) +51(6 - af (—)b
(9]
QUESTION 2 [30 MARKS]
(a) Discuss and derive the recursive scheme for the Forward Euler’s Method, using any appropriate
diagram for substantiating your discussion.
[13]
(b) Consider the following IVP:
a dy(t) 2y¥(t)}=3e™a , y(O)=_1
Using a step size of h = 0.1 and tp= 0, employ the method discussed in (1.1) to approximate up to the
5" step, giving your solution in a table showing both the exact and the approximate solution at each
step.
[17]
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QUESTION 3 [30 MARKS]
(a) Discuss the contrast between a quadrature rule and the adaptive rule.
[3]
(b) Consider the integral
[27]
b
3
| a f(x)dx = [ 1 e** sin(3x)dx
Using the Adaptive Simpson’s Method and an error € = 0.2, obtain the approximate value of the
above integral (for computational ease, using where appropriate the following as done in class):
To1]S(a, b) — S5‘(a, at—)h - S_(a—tb. b)
where
[ b renax = S(a,b)-h°TfO, Fe(a.d)
QUESTION 4 [30 MARKS]
(a) State the pseudo code for the Conjugate Gradient Method (CGM) for solving the nxn system of
linear equations:
Ax = b
where A is a symmetric and positive definite matrix.
[10]
(b) Consider the following system of linear equations:
5 -—-2 0 Ly
20
-—2
5
1
rT) =
10
Solve the above system using the Conjugate Gradient Method using the initial vector:
0
lO) = 0
0
[20]
END OF QUESTION PAPER
TOTAL MARKS = 120
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