University of Bahrain
College of Science
Department of Mathematics
Second Semester 2006/2007
Math 333
Numerical Analysis III
Final Exam
Date: 20/06/2007
Max. Marks : 100
Time: 08:30-10:30
Duration: 2 hours
Instructor: Dr. Thuraya Juma
Name:
ID Number :
Section:
Instructions
1. Please check that this exam has 6 Questions and
9 Pages
2. Write your name, student number, and section in the above box.
3. Do all your calculations using 4 decimal places.
4. Show all your work clearly and neatly.
Question
Max.
Marks
1
2
3
4
5
6
Total
14
18
18
14
20
16
100
Marks
Obtained
☺☺☺☺
GOOD
LUCK
☺☺☺☺
Page 1
Question 1: [ 14 Marks]
I) Compute three steps of Bisection method to approximate the zero of
f ( x)  3 x 2  e x starting with the interval [0, 2].
Use secant method to approximate the value of
[7]
7 . Start with x0  2 and x1  3 .
II) Construct the divided difference table for the following data
x
0.1 0.2 0.3
f (x ) 0.2 0.24 0.3
hence, approximate the value of f (0.15) using all the possible Newton’s divided
difference interpolating polynomial you can write from the table.
[7]
Page 2
Question 2: [ 18 Marks]
I)
Given the following linear system
x1  4 x2  5 x3  3
3 x1  2 x2  6 x3  4
 x1  x2  3 x3  7
solve the system by Gaussian elimination with partial pivoting. Show your work
clearly.
II) Find the linear least-squares polynomial f ( x)  a x  b for the following data:
0 1 2 3
f ( xk ) 1 2 5 10
xk
[7]
Page 3
Question 3: [ 18 Marks] Given the following linear system
Given the linear system
3x  y  z  5
2x  6 y  z  9
xyz 6
Given X (0)  (0, 0, 0)T
a)
Write the Jacobi iteration matrix TJ . Then calculate T j
[4]
b)
Approximate the solution by applying two iterations of Gauss-Seidel method.

[6]
c)
Does the iterative methods converges to the solution. Justify your answer. [2]
d)
Find X (2)  X (1)
b)
Given f ( x) 
[3]

x2
and the nodes x0  1 , x1  2 and x 2  3
x
a) Find the second degree Lagrange interpolating polynomial, P2 ( x) .
[3]
b) Estimate f (2.5) using P2 ( x) and calculate the actual error in the estimation.[3]
Estimate an upper bound for the error in the approximation of f (2.5) .
[3]
Page 4
Question 4: [ 14 Marks]
 1 0


I) Let B    2 1  . Find the value(s)  and  for which
 0 1 2


a) B is singular.
b) B is positive definite.
c) B is symmetric.
II) Consider the following iteration for calculating
2
:
xn 1  g ( xn )  a xn  b
x n3
Find the values of a and b so that the iteration will converge to
of convergence equal to two (for x0 sufficiently close to  ).
 with order
[7]
Page 5
Question 5: [ 20 Marks]
5 
2

  1  4
Let A  
a) Find the eigenvalues of A by the characteristic polynomial. Hence find the
eigenvectors corresponding to each eigenvalue.
[9]
b) Find the spectral radius of A and the eigenvalues of ( A3  5I ) 1
2
[5]
5
c) Find the dominant eigenvalue of A  
 by applying four steps of the
  1  4
1
power method using X 0    .
[6]
1
Page 6
Question 6: [ 16 Marks] Consider the following fourth order Runge-Kutta method
to solve initial value problems (IVPs) of ordinary differential equations (ODEs) :
1
yi 1  yi  (k1  2 k 2  2k 3  k 4 ) ,
( ** )
6
where
k1  h f ( x i , y i )
k
h
k 2  h f ( xi  , y i  1 )
2
2
k
h
k 3  h f ( xi  , y i  2 )
2
2
k 4  h f ( xi  h, y i  k 3 ).
dy
 y  x2 ,
y (0)  1 , 0  x  0.8 with h  0.2 .
dx
If the first two iterations are y1  1.2186 and y2  1.4682 , find the approximations of
y (0.6) using the fourth order Runge-Kutta method iven by (**).
[4]
For the following IVP :
Draft Page
Page 7