Numerical Analysis 1 (MATS2207)
Mathematics and Computer Science - MCS
Semester: Second Semester
Level: 200
Year: 2017
UNIVERSITY OF BAMENDA FACULTY OF SCIENCE
SECOND SEMESTER EXAMINATIONS
Department: Mathematics Course Instructors: Tanyu Vivian
Month August Course Code: MAT S2207
Year: 2017 Course Title: Numerical Analysis
Date: 02/08/2017 Credit Value: Six Credits
Time Allowed: 3 hours
Answer all the questions. All necessary work must be shown and must be neatly and orderly presented.
1.
i) State the Taylor’s theorems
ii) Let f(x) =
√
1 +
a)
Find the third Taylor’s polynomial, P
3
(x) of the function f about x
0
= 0
b)
Use P
3
( x ) to approximate
√
0.75
iii) a) Show that the polynomial nesting technique can be applied to the evaluation of .
f(x) = 1.01e
4x
− 4.62e
3x
− 3.11e
2x
+ 12.2e
x
−1.99
b)
Use three-digit rounding arithmetic, the assumption that e
1.53
= 4.62 and the fact that e
n(i.53)
= (e
n(i.53)
)
n
to
evaluate f (1.53) as given in (a)
c)
Redo the calculation in (b) by first nesting the function f ( x )
d)
Compare the approximation in (b) and (c) to the true three-digit result f(1.53) = −7.61
(l,4+3,3+3+3+3,=20marks)
2.
i) a) Define and explain the condition number for a function f at one variable. What does it mean to say that a
problem is well-conditioned and ill-conditioned?
b) For f(x) =
√
+ 1, x = 3, ∆x = 0.1 compute the absolute and the relative errors of the function value
and the condition numbers.
ii) a) Use the Bisection method to compute the first two approximate values for x
4
− 25 = 0. How many
bisection; n (or iterations) are needed to obtain accuracy to within 10
- 4
?
b) Use the Newton’s method to find a root of f(x) = x
2
−2 ; x ∈ [1, 2] . ' -
iii) Find the intervals that contain a solution to the equation x
3
− 2x
2
— 4 x + 3 = 0
(6+6,7+4,6,=29marks)
3.
i) a) Use Gaussian-elimination method to find the solution of the linear system A x = b where
A =
1 2 4
1 3 9
1 4 16
b =
1
3
4
b) Construct the L U decomposition of the matrix A given in part (a) by using l
i i
= 1
ii) Given the matrix
A =
1 1 1
2 3 4
4 9 16
i f t h e v a l u e o f t h e c o n d i t i o n n u m b e r o f t h e g i v e n m a t r i x A i s 4 3 5 , t h e n
c o m p u t e | | A
- 1
| |
∞
iii) What is the binary representation for
(7+6,4,4,=21 marks)
GOOD LUCK
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