Numerical Methods 2080

Section A
Long Answer Questions
Attempt any TWO questions. [2 × 10 = 20]

1. Write an algorithm and a C program to obtain roots of a non-linear equation using Newton-Raphson Method. [4+6]

2. Solve the following ordinary differential equation using the shooting method.
y'' + xy' - xy = 2x, with boundary conditions y(0) = 1 and y(2) = 10 [10]

3. Compare and contrast between Jacobi iterative methods and Gauss-Seidel method. Solve the following equations using Gauss-Seidel method.
x + 2y + 3z = 5
2x + 8y + 22z = 6
3x + 22y + 82z = -10 [3+7]

Section B
Short Answer Questions
Attempt any EIGHT questions. [8 × 5 = 40]

4. Use the secant method to estimate the root of the equation x^2 - 5x + 6 = 0, with initial estimates x1 = 4 and x2 = 2 (EPS = 0.05). [5]

5. Solve the double integration using Simpson's 1/3 rule.
∫(2 to 2.6) ∫(4 to 4.4) dx dy / xy [5]

6. What are the sources of errors? Discuss various types of errors encountered in numerical computation. [5]

7. Fit a second order polynomial to the data in the table below:

8. Solve the following differential equation dy/dx = 3x + y/2 with y(0) = 1 for x = 0.2 (h = 0.1) using Euler's Method. [5]

9. Why is numerical integration required? Compute the integral: I = ∫(-1 to 1) e^x dx using the composite trapezoidal rule for n = 4. [5]

10. Evaluate dy/dx at x = 5 using Newton's forward interpolation formula using the following table.

11. Find the eigenvalues and eigenvectors of the matrix:
A = [[3, -1], [1, 1]] [5]

12. Solve the Poisson's equation d^2f/dx^2 + d^2f/dy^2 = 2x^2 y^2 over the square domain 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3 with f = 0 on the boundary and h = 1. [5]