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ANA801S - APPLIED NUMERICAL ANALYIS - 2ND OPP - JULY 2023 |
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nAmlBIA unlVERSITY
OF SCIEnCE Ano TECHnOLOGY
FACULTYOF HEALTH, NATURAL RESOURCESAND APPLIEDSCIENCES
SCHOOLOF NATURALAND APPLIEDSCIENCES
DEPARTMENTOF MATHEMATICS, STATISTICSAND ACTUARIALSCIENCE
QUALIFICATION: Bachelor of Science Honours in Applied Mathematics
QUALIFICATION CODE: 08BSHM
LEVEL: 8
COURSE CODE: ANA801S
COURSE NAME: APPLIED NUMERICAL ANALYSIS
SESSION: JULY 2023
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 120 (to be converted to 100%)
2ND OPPORTUNITY /SUPPLEMENTARY EXAMINATION QUESTION PAPER
EXAMINERS
PROFS. A. REJU
MODERATOR: PROFS. MOTSA
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]
(a) Discuss the contrast between a quadrature rule and the adaptive rule.
[3]
(b) Consider the integral
[27]
L J, b
3
= f (x)dx
e2x sin(3x)dx
= Using the Adaptive Simpson's Method and an error E 0.2, obtain the approximate value of the
above integral (for computational ease, using where appropriate the following as done in class):
where
I -_1,_S(o. b) - S(o. -o-+) b - S(--o. ,h b)
10
2
:2
QUESTION 2 (30 MARKS]
Discuss exhaustively the Romberg Method Extrapolation process to show that the nth order
extrapolation employed by the method is given by:
/
_ 4n IMore-accurate - ILess accurate
Improved
4n_l
QUESTION 3 (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 2 - 4xy + 2y 2
(20]
QUESTION 4 (30 MARKS]
(a) Define the Picard Method for solving the following Initial Value Problem (IVP)
ddyt = yI (t) = f ( t,y(t) ) ,y(to) = Yo
and hence derive the Picard Iteration algorithm
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(b) Using the Picard method, find the solution, correct to 3 decimal places, of the following pt order
IVPatx=O.l
-ddyx = X + y 2 , y(O) = 1
with x(O) = x0 = 0
(17]
END OF QUESTION PAPER
TOTAL MARKS= 120
31Pagc