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B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECMBER 2008
Electronics and Communication Engineering
MA 1251- NUMERICAL METHODS
PART A (2 Marks)
1. Find the positive root of x2+5x-3=0 using fixed point
iteration starting with0.6 as first approximation.2. Find the
inverse of A= 1 3 by Gauss-Jordan method.2 7
3. Derive Newtons Backward difference formula.4. Form the
divided difference table for:x: -1 1 2 4y: -1 5 23 119
5. Evaluate ?1-1 1/(1+x4) dx using Gaussian quadrature with 2
points
6. Write Simpsonss ? rule, assuming 3n intervals.7. Find the
Taylor series upto x3 term satisfying 2y + y = x + 1, y(0) = 1.8.
Write the Adams Predictor-Corrector formulae.
9. Derive Crank-Nicolson scheme.10. Write standard five point
formula and diagonal five point formula used in solvingLaplace
equation uxx + uyy = 0 at the point (i?x,j?y).
PART B (16 Marks)
1. a.i. Find the root which lies between 2 and 3, correct to 3
places of decimalsof the equation x3-5x-7=0 , using the method of
false position.
ii. Solve the given system of equations by Gaussian elimination
method:-x+y+10z = 35.61, 10x+y_z = 11.19 , x+10y+z = 20.08
OR
b.i. Find the numerically largest eigen value and the
corresponding eigen vectorusing Power method, given
A = 5 4 310 8 620 -4 22
Starting vector is (1,1,1).
ii. Solve by Gauss-Seidal iteration the given system of
equations starting with(0,0,0,0) as solution. Do 5 iterations
only.
4a-b-c = 2, -a+4b-d = 2, -a+4c-d = 1, -b-c+4d = 1.
2. a.i. Fit a Lagrangian interpolating polynomial y=f(x) and
find f(5).
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x: 1 3 4 6y: -3 0 30 132
ii. Find y(12) using Newtons forward difference formula
given:
X: 10 20 30 40 50Y: 46 66 81 93 101
OR
b.i. Fit a natural cubic spline for the following data:
x: 0 1 2 3y: 1 4 0 -2
ii. Derive Newtons divided difference formula.
3. a.i. Find dy/dx at x = 1.5 givenx: 1 2 3 4 5y: 77 78 127 248
375
ii. Evaluate 1?2.5 x=2?4 dxdy/(1+x+y2) using ?x = 0.5 = ?y
Simpsons ?rule in x- direction and trapezoidal rule in the y-
direction.
OR
b.i. Evaluate 3?7 dx/(1+x2) using Gaussian quadrature with 3
points.
ii. Find the first and second derivatives of y w.r.t. x at
x=10.
X: 3 5 7 9 11Y: 31 43 57 41 27
4. a. Solve dy/dx = y-x2 , y(0)=1.Find y(0.1) and y(0.2) by R-K
method fo order 4.Find y(0.3)n by Eulers method.Find y(0.4) by
Milnes Predictor Corrector method.
OR
b. Solve y-0.1(1-y2)y + y = 0 subject to y(0) = 0, y(0) = 1
using fourthorder Runge- Kutta method.Find y(0.2) and y(0.2). Use
step size ?x=0.2.
5. a.i. Solve the boundary value nproblem x2y 2y + x = 0,
subject to y(2) = 0,y(3)=0. Find y(2.25), y(2.5), y(2.75).
ii,. Solve the vibration problem ?y/?t = 4 ?2y/?x2 subject to
the boundaryconditions y(0,t) = 0, y(8,0) = 0 and y(x,0) =0.5
x(8-x). Find y at x=0,2,4,6. Choosing ?x= 2 , ?t=0.5. Compute upto
4 time steps.
OR
