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ANA801S - APPLIED NUMERICAL ANALYIS - 1ST OPP - JUNE 2023 |
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nAmlBIA unlVERSITY
OF SCIEnCE Ano TECHnOLOGY
FACULTYOF HEALTH,NATURALRESOURCESAND 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: JUNE 2023
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 120 (to be converted to 100%)
EXAMINERS
MODERATOR:
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
PROFS. A. REJU
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) Consider following integral:
A= fd f(x)dx
{1.1)
C
State the general Composite rule and hence the Composite Trapezoidal rule and the Romberg's
Method for solving (1.1); and hence using the unit interval [0, 1) for the integral
and step size
b
T(n) = if (x)dx
h = (b - a)
n
obtain the term for the recursive expression T(2n) = T(8) and the expression for R(n, O) denoting
the Trapezoidal estimate with 2n.
[20]
(b) By just stating the Richardson's Extrapolation R(n, m) employed in the Romberg's Table, show
that
[10)
QUESTION 2 [35 MARKS]
(a) Derive the Forward Euler's Method, using any appropriate diagram for substantiating your
discussion.
[13]
(b) Consider the following IVP:
ddyt(t) + ay(t) = r, y(O) = y 0
(2.1)
(i)
State the Euler's method that approximates the derivative in the above equation and hence
state the resulting difference equations with three stepwise increments oft by h from t = 0.
(ii) Taking a= l = rand y 0 = 0, obtain the numerical solutions of (2.1) fort= 0.25, ...,1.5
when h = 0.25 and h = 0.5 (correct to 4 decimal places)
[22]
QUESTION 3 [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]
2 I P ~l g c
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(bl Consider the following system of linear equations:
Solve the above system using the Conjugate Gradient Method using the initial vector:
0
= .r(O)
0
0
[20)
QUESTION 4 [25 MARKS]
(a) Discuss the contrast between a quadrature rule and the adaptive rule.
[3]
(b) Consider the integral
[27]
f J, b
3
a 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
l,. . -10 S(a. b) - S(a. -o-.)+2b ' - .S.o(-+-.h2 b) I
END OF QUESTION PAPER
TOTAL MARKS= 120