Anna univ MA 038 — NUMERICAL METHODS

PART A — (10 ? 2 = 20 marks)
1. If is continuous in , then under what condition the iterative method has a unique solution in ?
2. Compare Gauss–Jacobi and Gauss–Seidel methods for solving linear systems of the form .
3. Construct a linear interpolating polynomial given the points and .
4. Write down the range for for which Stirling’s formula gives most accurate result.
5. Find the error in the derivative of by computing directly and using the approximation at choosing .
6. What are the errors in Trapezoidal and Simpson’s rules of numerical integration?
7. What is a predictor–corrector method?
8. What do we mean by saying that a method is self–starting? Not self–starting?
9. What is the truncation error of the central difference approximation of ?
10. For what value of , the explicit method of solving the hyperbolic equation is stable, where ?
PART B — (5 ? 16 = 80 marks)
11. (i) Consider the non linear system and
. Use Newton–Raphson method with the starting value
and compute and . (8)
(ii) Find all eigen values of the matrix by Jacobi method (Apply only 3 iterations). (8)
12. (a) Find an approximate polynomial for which agrees with the data :





0 1.3 0.62009 – 0.52202
1 1.6 0.45540 – 0.56990
2 1.9 0.28182 – 0.58116
using Hermite’s interpolation. Hence find the approximate value
of . (16)
Or
(b) (i) Use Newton’s backward difference formula to construct an interpolating polynomial of degree 3 for the data :
, , and . Hence find . (8)
(ii) Given :
x
(in degrees) : 0? 5? 10? 15? 20? 25? 30?
:
0 0.0875 0.1763 0.2679 0.3640 0.4663 0.5774
Using Stirling’s formula, find (16?). (8)
13. (a) (i) Consider the following table of data :
x : 0.2 0.4 0.6 0.8 1.0
:
0.9798652 0.9177710 0.8080348 0.6386093 0.3843735
Find using Newton’s forward difference approximation, using Stirling’s approximation and using Newton’s backward difference approximation. (8)
(ii) For the given data :
x : 0.7 0.9 1.1 1.3
:
0.64835 0.91360 1.16092 1.36178
x : 1.5 1.7 1.9 2.1
:
1.49500 1.55007 1.52882 1.44573
Use Simpson’s rule for the first six intervals and trapezoidal rule for the last interval to evaluate . Also use trapezoidal rule for the first interval and Simpson’s rule for the rest of the intervals to evaluate . Comment on the obtained values by comparing with the exact value of the integral which is equal to 1.81759. (8)
Or
(b) (i) Evaluate using the three point Gaussian quadrature. (8)
(ii) Using trapezoidal rule evaluate choosing and . (8)
14. (a) Consider the initial value problem , .
(i) Using the modified Euler method, find .
(ii) Using 4th order Runge–Kutta method, find and .
(iii) Using Adam–Bashforth Predictor–Corrector method, find .
(16)
Or
(b) Consider the second order initial value problem , with and .
(i) Using Taylor series approximation, find .
(ii) Using 4th Order Runge–Kutta method, find . (16)
15. (a) Solve and , using and for two time steps by Crank–Nicholson implicit finite difference method. (16)
Or
(b) Approximate the solution to the wave equation , , , , and , with and for 3 time steps.