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Math 424: Numerical Analysis Ii Question Paper
Math 424: Numerical Analysis Ii
Course:Bachelor Of Science
Institution: Chuka University question papers
Exam Year:2013
CHUKA
UNIVERSITY
UNIVERSITY EXAMINATIONS
FOURTH YEAR EXAMINATION FOR THE AWARD OF DEGREE OF
BACHELOR OF SCIENCE (GENERAL)
MATH 424: NUMERICAL ANALYSIS II
STREAMS: BSC (GEN) TIME: 2 HOURS
DAY/DATE: TUESDAY 6/8/2013 11.30 AM- 1.30 PM
INSTRUCTIONS:
Answer Question ONE and any other TWO Questions
Adhere to the Instructions on the Answer Booklet
Question One (Compulsory) (30 Marks)
(a) Economize sinx=x-x^3/6+x^5/120+x^7/5040+ . . . . to 3 significant digit accuracy.
[6 marks]
(b) Solve the system of equations below using Gauss-Jacobi method.
?4x?_1+0.24x_2-0.08x_3=8
?0.09x?_1+3x_2-0.15x_3=9
?0.04x?_1-0.08x_2+4x_3=20 [6 marks]
(c) Determine all the eigen values and eigenvectors of the matrix m=[¦(2&1@1&2)]
[8 marks]
(d) Apply Hermite’s interpolation to find a cubic polynomial which meets the following specifications
xi 0 1
yi 0 1
y_i^'' 0 1
[4 Marks]
(e) (i) Explain with an example the term transcendental equation. [2 marks]
(ii) Distinguish between direct and interative methods of solving transcendental equations giving an example of each. [4 marks]
Question Two (20 Marks)
(a) (i) What is an ill-conditioned system of equations. [1 mark]
(ii) Determine whether the system of equations below is ill-conditioned or not.
x-2y=1
-2x+4.01y=1 [3 marks]
(b) (i) The Chebyshev polynomials satisfy the recurrence relation ? T?_(n+1) (x)=2xT_n (x)-? T?_(n-1) (x).
Show that 2(1+x+x^2 )=3 T_(0 ) (x)+2T_1 (x)+T_2 (x). [4 marks]
(ii) Obtain the best lower approximation to the function f(x)=x^3+?2x?^2 on the interval [-1,1] and determine the error on the interval [-1,1] [4 marks]
(c) Obtain an approximation in the sense of the principle of least squares in the form of a polynomial of the degree 2 to the function 1/(1+x^2 ) in the range -1=x=1.
[8 marks]
Question Three (20 Marks)
(a) (i) Solve the system of equations below by matrix inversion method
x_1+?2x?_2+?3x?_3=6
2x_1+?6x?_2+?8x?_3=12
x_1+x_2+x_3=1 [5 marks]
(ii) Use Gauss – elimination method to solve the system of equations below
2x+y +4z=12
8x-3y+2z=20
4x+11y-z=33 [5 marks]
(b) (i) For a given square matrix, define the terms eigenvalue and eigen vector.
[2 marks]
(ii) Find all the eigenvalues of the matrix A=[¦(4&3@1&2)] using Rutishauser method.
[8 marks]
Question Four (20 Marks)
(a) (i) Define the term interpolation and state any two assumptions interpolation is based on. [3 marks]
(ii) Use Hermite’s interpolation formula to obtain a polynomial of degree 4 from the data below:
xi 0 1 2
yi 1 0 9
y_i^'' 0 0 24
(b) Find by the method of inverse interpolation the real root of x^3+x-3=0 which lies between 1.2 and 1.3. [8 marks]
Question Five (20 Marks)
(a) Use Bairstow’s method to obtain the quadratic factor of the following polynomial with two iterations :x^4-?3x?^3+?20x?^2+44x+54=0 with (p,q)=(2,2). [11 marks]
(b) Determine the least square approximation of the type ?ax?^2+bx+c to the function 2^x at the points xi=0,1,2,3,4 . [9 marks]
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