Madurai Kamraj University (MKU) 2006-2nd Year M.C.A - -exam question paper

MCA 2nd year MAY 2006

PAPER VI - COMPUTER BASED NUMERICAL METHODS

Time : 3 hours Maximum: 100 marks

PART A ans ALL ques.. (8 x five = 40 marks)

1. (a) Use the Secant method to determine the root of the formula x4 - x -10 = 0 .
Or
(b) Apply Newton - Raphson's method to determine a root of the formula x - e -x = 0 .

2. (a) obtain two iterations with the Muller method for the subsequent formula X3 –1/2 =0with X0 = 0.
Or
(b) obtain 2 iterations with the Chebyshev method for finding root of the formula
x = 1/2 + sin x with Xo = 1.

3. (a) Solve by Gauss elimination method for the subsequent
x+y+z=3
2x- y+3z= 16
3x+y-z=-3.
Or
(b) Solve by Triangularization method
x+5y+z=14
2x+ y+ 3z= 13
3x+ y+4z=17.

4. (a) Solve the subsequent system of formula by using Gauss - Seidel method
8x-3y+2z= 20
4x+lly-z = 33
6x+3y+12z= 35.
Or
(b) obtain the inverse of A =
using partition method.

5. (a) Using Lagrange's formula, fit a polynomial to the data.
Or
(b) Prove that _ = 1/2 _2 + _ 1+ _2/4

6. (a) Using Newton's divided difference formula obtain f(8) from the subsequent data:
(b) obtain the approximate value of f' (2.0) and f"(2.0) using the methods based on linear
interpolation

7. (a) calculate r(0.6) from the subsequent table using the formula Richardson extrapolation.
x: 0.2 0.4 0.5 0.6 0.7 0.8 1.0
f(x): 1.42 1.88 2.13 2.39 2.66 2.94 3.56
With h=0.2.
Or
3 -1 1
3 -1 1
5 -2 2
X : 0 one three 4
Y : -12 0 six 12
X : four five seven 10 11 13
F(x): 48 100 294 900 1210 2028
X : 2.0 2.2 2.6
Y : 0.6932 0.7885 0.9555
(b) Solve the formula dy = 1- y provided y(0)= 0
dx
using Euler method for the solutions at x = 0.1, 0.2, 0.3

8. (a) Solve the initial value issue y' = y = (2x/y) y(0) = one for x=01 ,0.2
using backward Euler method
Or
(b) Using mid-point method obtain y(0.1), y(0.2)
provided (dy/dx) = X2 + y2 , y(0)=1.

PART B ans ALL ques.. (5 x 12 = 60 marks)

9. (a) Using Bairstow's method to find the quadratic factor of the formula
X4 - 3X3 + 20X2 + 44x + 54=0 with (p,q) = (2,2) (perform 3 iterations).
Or
(b) Using Graeffe's root squaring method to obtain the roots of X4 - X3 + 3X2 +X-4 =0.

10. (a) obtain the largest eigen value of
and the corresponding eigen vector.
(b) obtain all the eigen values of the matrix.

11. (a) find a linear polynomial approximation
to the function f(x)= X3 on the interval [0,1] using the lowest square approximation.
Or
(b) obtain the lowest squares approximation of 2nd degree for the data.
0.8

12. (a) compute (1 + sin x/ x) dx accurate to 4 decimal places.
0
(Or)
5 5
(b) Evaluate dx dy / (x2 + y2) 1/2 using the trapezoidal rule.
1 1

13. (a) provided the initial value issue u' = t2 + u2,u(O)= 0 obtain the Taylor series for u(t) and
hence find u(0.5)
(b) Solve the initial value issue u'= -2tu2, u(O)=1 with h = 0.1 for x=0.l, 0.2. Use the
fourth order classical Runge -Kutta method.

Earning:   Approval pending.