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ANA801S - APPLIED NUMERICAL ANALYSIS - 2ND OPP - JULY 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: JULY 2022
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 120 (to be converted to 100%)
2ND
EXAMINERS
MODERATOR: |
OPPORTUNITY/SUPPLEMENTARY
PROF S. A. REJU
PROF S. MOTSA
EXAMINATION
QUESTION
PAPER
INSTRUCTIONS
1. Attempt ALL the questions.
2. 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]
Discuss exhaustively the Romberg Method Extrapolation process to show that the nth order
extrapolation employed by the method is given by:
I Improved
n
= 4 I More—accura4taen_4~~ I Less accurate
QUESTION 2 [30 MARKS}
(a) Define the Picard Method for solving the following Initial Value Problem (IVP)
Tdy aos (t, y(t), v(to) = y,
and hence derive the Picard Iteration algorithm
[13]
(b) Using the Picard method, find the solution, correct to 3 decimal places, of the following 1*t order
IVP atx =0.1
dSdwx axty?, y(0)=1
with x(0) =x) = 0
[17]
QUESTION 3 [30 MARKS]
(a) Discuss the contrast between a quadrature rule and the adaptive rule.
[3]
(b) Consider the integral
[27]
b
3
| Feoax = | e** sin(3x)dx
a
1
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) — S.(a, a+b )— S{ a+b .b)
where
[ b tear = 60a- .h°F)O, Fe(a.b)
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QUESTION 4 [30 MARKS]
(a) (i) State the Steepest Descent Algorithm
[6]
(ii) State the theorem that guarantees that the Steepest Descent method ensures some progress
in the direction of the minimum of the objective function during each iteration.
[4]
(b) Using the Steepest Descent Method, obtain the minimum of the following function:
f(x,y) = 4x? — 4xy + 2y?
[20]
END OF QUESTION PAPER
TOTAL MARKS = 120
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