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NUM702S - NUMERICAL METHODS 2 - 2ND OPP - JAN 2023 |
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nAmlBIA unlVERSITY
OF SCIEnCE Ano TECHnOLOGY
FACULTYOF HEALTH,APPLIEDSCIENCESAND NATURAL RESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION:
Bachelor of Sciencein Applied Mathematics and Statistics
QUALIFICATIONCODE: 07BSAM
LEVEL: 7
COURSECODE: NUM702S
COURSENAME: NUMERICAL METHODS 2
SESSION:
DURATION:
JANUARY 2023
3 HOURS
PAPER: THEORY
MARKS: 93
SECONDOPPORTUNITY/SUPPLEMENTARY- QUESTION PAPER
EXAMINER
Dr S.N.NEOSSNI GUETCHUE
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 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 PAPERCONSISTSOF 3 PAGES(Including this front page)
Attachments
None
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Problem 1 [32 Marks]
1-1. Find the best function in the least-squares sense that fits the following data points and is of the form
f(x) = asin(1rx) + bcos(1rx):
[5]
X 1-l 1-1/21 0 11/211
y -l O 1 2 1
1-2. Find the Pade approximation R2,2 (x) for f(x) = tan( ft)/ ,jx starting with the MacLaurin expansion
x 2x2 17x3 62x4
f(x) = l+3+ 15 + 315 + 2835 + ·· ·
[12]
1-3. Use the result in 1-2. to establish tan(x)
R5
4
'
=
0
;
45x
9_5x -
-
42100x5;~3++l5xx5
4
[3]
1-4. Compare the following approximations to f(x) = tan(x)
[12]
Taylor: T9(x)
x 2x2 17x3 62x4
1 + 3 + 15 + 315 + 2835
(given in 1-3.)
on the interval [O,1.4]using 8 equally spaced points xk with h = 0.2. Your results should be correct to 7
significant digits.
Problem 2 [25 Marks]
For any non negative interger n the Chebyshev polynomial of the first kind of degree n is defined as
Tn(x) = cos [ncos- 1 (x)], for x E [-1, l].
2-1. Use the identity/formula:
LN cos(cp+ ka)
k=O
=
sin (N+l)a- cos(cp+ !::!..a.)
2 sm. -°'
2
2
to show that:
[12]
I:N
Tm(Xk)Tn(Xk)= 0, form=/= n,
k=O
l)1r] where xk = cos [ (22(kN++ l) , 0 :Sk :SN, are the roots of TN+I·
2-2. Compute the expressions of the first five Chebyshev polynomials of the first kind T2 , T3 , T4 , T5 and
~-
2-3-1. Find P6 (x) the sixth MacLaurin polynomial for f(x) = xex.
[3]
2-3-2. Use Chebyshev economisation to economise P6 (x) once.
[5]
Problem 3 [36 Marks]
3-1. Determine the number n so that the composite Simpson's rule for 2n subintervals can be used to
compute the following integral with an accuracy of 5 x 10- 9 .
[10]
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3-2. State the three-point Gaussian Rule for a continuous function f on the interval [-1, 1].
[3]
3-3. Use the Composite Simpson's rule with four equal subintervals to approximate the following integral
and compare your result with the one obtained when using the three-point Gaussian Rule
[10]
1-: I=
(2x4 + 5)dx.
3-4. Was the comparison in 3-3. predictable? Justify your answer.
[3]
3-5. The matrix A and its inverse are A- 1 are given below
A= [1/-2lJ
-1 1 '
[-2-2J' A-l = -2 -1 .
Use the power method to find the eigenvalue of the matrix A with the smallest absolute value.
Of Start with the vector x(o) = (1, and perform three iterations.
[10]
God bless you !!!
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