Numerical Methods 2078

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

1. Derive the formula for integration using Simpson's 3/8 rule. Use Secant Method to estimate the root of equation x^2 - 4x - 10 = 0, with initial estimate x1 = 4 and x1 = 2. [5+5]

2. What do you mean by boundary value problem? Use shooting method, solve the equation: y'' = 6x^2, with y(0) = 1 and y(1) = 2 in the interval (0,1) for y(0.5) taking h = 0.5. [2+8]

3. Write an algorithm and program to compute the interpolation using Lagrange Interpolation. [10]

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

4. Show that the rate of convergence of Newton's Raphson method is quadratic. [5]

5. The temperature of a metal strip was measured at various time intervals during heating and the values are given in the table below.

6. Given the following set of data points. Obtain the table of divided difference and use that table to estimate the value of f(1.5).

7. Solve the following system of linear equations by Gauss Elimination with Pivoting:
2x + 2y + z = 6
4x + 2y + 3z = 4
x - y + 1 = 0 [5]

8. Determine the Eigen Values and corresponding Eigen Vectors for the matrix.
A = [[1, 6, 1], [1, 2, 0], [0, 0, 3]] [5]

9. The table below gives the values of distance travelled by a car at various time intervals during the initial running.

10. Solve the following integral using trapezoidal rule for n = 8.
I = ∫(2 to 4) (x^4 + 1) dx [5]

11. Given the equation y' = 3x^2 + 1 with y(1) = 2, estimate y(2) by Euler's Method using h = 0.2. [5]

12. Solve the Poisson's Equation ∇^2 f = 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]