Numerical Method 2070

Attempt all questions:

1. What is bracketing and non-bracketing method? Explain with the help of example. Estimate a real root of following nonlinear equation using bisection method correct up to two significant figures(3 + 5)

                x²sinx + e^-x = 3

2.  Define interpolation. Find the functional value at x=3.6 from the following data using forward difference table:

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

4. What is pivoting? Why is it necessary? Explain. Solve the following set of equations using Gauss elimination or Gauss Seidel method.(3 + 5)

        x₁ + 10x₂ +x₃ = 24

        10x₁ + x₂ + x₃ = 15

        x₁ + x₂ + 10x₃ = 33

5. Compare Euler’s method with Heun’s method for solving differential equation. Obtain y(1.5) from given differential equation using Runge-Kutta 4th order method.(4 + 4)

OR

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

6. Solve the equation over the square domain 0 ≤ x ≤ 1.5 and 0 ≤ y ≤ 1.5 with f=0 on the boundary [Take h = 0.5]. (8)

7. Write an algorithm and C-program to approximate the functional value at any given x from given n no. of data using Lagrange’s interpolation. (5+7)