Numerical Method 2071

Attempt all questions:

1. How is the bisection method convergent to a root of an equation? Apply the bisection method to find a root of the equation (3 + 5)

2. Define interpolation. Find the Lagrange interpolation polynomial to fit the following data. Estimate the value (1 + 6 + 1)

3. Derive Simpson’s 1/3 rule to evaluate numerical integration. Using this formula evaluate (4 + 4)

4. What do you mean by ill-conditioned systems? Solve the following system using Dolittle LU decomposition method.(2 + 6)

        3x₁ + 2x₂ + x₃ = 24

        2x₁ + 3x₂ + 2x₃ = 14

        x₁ + 2x₂ + 3x₃ = 14

5. Solve the following boundary value problem using shooting method.(8)

6. Write the finite difference formula for solving Poisson’s equation. Hence solve the Poisson’s equation∇²f = 2x²y² over the domain 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3 with f = 0 on the boundary and h = 1.        (1 + 7)

7. Write an algorithm and a C-program for the fixed point iteration method to find the roots of non-linear equation. (4+8)

OR

Write an algorithm and a C-program for the Lagrange’s interpolation to approximate the functional value at any given x from given n data. (4+8)