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NUM701S - NUMERICAL METHODS 1 - 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: 35BAMS
LEVEL: 6
COURSE CODE: NUM701S
COURSE NAME: NUMERICAL METHODS 1
SESSION:
DURATION:
JULY 2022
3 HOURS
PAPER: THEORY
MARKS: 100
SUPPLEMENTARY/SECOND OPPORTUNITY EXAMINATION QUESTION PAPER
EXAMINER
Dr S.N. NEOSSI NGUETCHUE
MODERATOR:
Prof S.S. MOTSA
INSTRUCTIONS
1. Answer ALL the questions in the booklet provided.
2. Show clearly all the steps used in the calculations. All numerical results must be
given using 4 decimals where necessary unless mentioned otherwise.
3. All written work must be done in blue or black ink and sketches must be done in
pencil.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 3 PAGES (Including this front page)
Attachments
None
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Problem 1. {21 marks]
1-1-1. Why is the nested form of a polynomial important compared to its canonical (original) form?
Give an example illustrating your statement with the number of operations involved (you can use a
third degree polynomial of your choice).
[2+2=4]
1-1-2. Write down a pseudo-code that uses the nested form of a polynomial of degree n and evaluates
it at T= 2.
[3]
1-2. Write down the general formula of the Taylor’s expansion (with integral remainder) of a func-
tion f(x) about x = ap.
[5]
1-3 The nth root of the number N can be found by solving the equation x” — N = 0.
1-3-1 For the above equation, show that Newton’s method gives:
[5]
Lirit.st = i—nl tn |(n— 1))aa;;+ =N|
1-3-2 Use the above result to find (161)'/3 after three iterations with zp = 6.0 as the starting
point.
[4]
Problem 2 [30 marks]
2-1. Write down in details the formulae of the Lagrange and Newton’s form of the polynomial that
interpolates the set of data points (Zo, yo), (21, y1),---5 (Wns Yn):
(7]
2-2. Use the results in 2-1. to determine the Lagrange and Newton’s form of the polynomial that
interpolates the data set (0,2), (1,5) and (2,12).
[18]
2-3. If an extra point say (4,9) is to be added to the above data set, which of the two forms in 2-1.
would be more efficient and why? [Don’t compute the corresponding polynomials.|
[5]
Problem 3. [30 marks]
3-1. Determine the error term for the formula
[5]
F(a) © s[afle + h) — 3f(0) — fle + 2h)]
3-2. Use the above formula to approximate f’(1.8) with f(z) = Inz using h = 0.1, 0.01 and 0.001.
Display your results in a table and then show that the order of accuracy obtained from your results
is in agreement with the theory in question 3-1.
[10}
3-3. Establish the error term for the rule:
[15}
f(a) aalbile +h) — 10 f(x) +12f(@ — h) — 6f(x — 2h) + f(a — 3h)]
Problem 4. [19 marks]
4-1. State the second-order Runge-Kutta algorithm (RK2) in terms of it slopes kl and k2 (or fi
and fz).
[6]
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4-2 Explain how the Runge-Kutta method can be used to produce a table of the values for the
function
.
f(a) = f edt
0
at 100 equally spaced points in the unit interval.
[3]
4-3. Use the procedure explained in 4-2. and adapt it to compute f(0.3) using RK2 with three
iterations, where this time
iif) = =V2T | Jo [7 e! _» dt
using RK2 to approximate y(0.3) with 3 steps.
[10}
God bless you !!!