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NUM702S - NUMERICAL METHODS 2 - 1ST OPP - NOV 2022 |
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nAmlBIA unlVERSITY
OF SCIEnCE Ano TECHnOLOGY
FACULTYOF HEALTH,APPLIEDSCIENCESAND NATURALRESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION:
Bachelor of Sciencein Applied Mathematics and Statistics
QUALIFICATION CODE: 07BSAM
LEVEL: 7
COURSE CODE: NUM702S
COURSE NAME: NUMERICAL METHODS 2
SESSION:
DURATION:
NOVEMBER 2022
3 HOURS
PAPER: THEORY
MARKS: 100
EXAMINER
MODERATOR:
FIRSTOPPORTUNITY- QUESTION PAPER
Dr S.N. NEOSSNI GUETCHUE
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 5 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.
PERMISSIBLEMATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 2 PAGES (Including this front page)
Attachments
None
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Problem 1 [40 Marks]
1-1. Show that the formula for the best line to fit data (k, Yk) at integers k for 1 ::; k ::; n is y = ax+ b,
[n l l where
6
a= n(n 2 _ l) 2
kyk - (n + 1) n Yk and b = n(n L1. l) [ (2n + 1) n Yk - 3 n kyk
[15]
1-2. Esta bl1. sh t he p ad,e ap- prox1.mat1.0n ex
form.
R~2 ' 2 ( x ) =
12
12
+
-
6x
6x
+
+
x2
X2
ana'
express
R
22
'
m.
contm. ue
d
fractl.on
[10]
1-3. Write down the general formula of Sf (x), the Fourier series of a function f that is 27!"periodic,
piece-wise continuous and defined on (-7r, 7r).
[5]
1-4. Find the Fourier sine series for the 27r-periodic function f(x) = x(1r- x) on (0, 1r). [Hint: Assume f
is an odd function]. Use its Fourier representation to find the value of the infinite series
[10=7+3]
1111
1--+-3-3-+-+5··3· 73 93
Problem 2 [31 Marks]
2-1. Define Tn(x), the nth degree Chebyshev polynomial of the first kind for x E [-1, 1] and show that:
(i) Tk+i(x) = 2xTk(x) -Tk_ 1(x), fork~ l, with T0 (x) = 1, T1(x) = x;
[5]
(ii) Tn has n distinct zeros/roots Xk = cos ( (2k l)1r) for 0::; k::; n - 1.
[7]
2-2. Use the formulae in (i) of question 2-1 to find T2 (x), T3 (x) and then economize
[6]
P(x) = 1 + 2x2 + 3x3 , once.
2-3. Given the integral
3
[
}0
s1in+( 2xxj dx
=
0.6717578646 • • •
2-3-1. Using the sequential trapezoidal rule, state the formula of T(J) = R(J, 0) and then compute its
values for J = 0, l, 2 .
[3+10=13]
Problem 3 [29 Marks]
3-1. For an n-point Gaussian quadrature rule, the Legendre polynomials qn(x), for x E [-1, 1], can be
generated by the recursion formula
qn(x) =
(
-2-n
-
n
1\\I xqn-1(x) -
/
(/-n- - 1\\) qn-2(x)
n1
for n = 2, 3, ...
and qo(x) = 1, q1(x) = x.
3-1-1. Compute q2(x), q3 (x) and determine the zeros of q3 (x).
[2+2+3=7]
3-1-2. Using the zeros of q3 (x) as quadrature nodes, state the associated quadrature formula and deter-
mine the corresponding weights by the method of undetermined coefficients. How do you call the rule thus
obtained?
(2+8+2=12]
65-5] 3-2. Consider the following matrix A= 2 6 -2 . Find its largest eigenvalue (in magnitude) and the
[2 5 -1
corresponding eigenvector after three iterations with the initial vector x<0) = (-1, 1, 1f.
[10]
God bless you !!!