Numerical Method 2073

Attempt all questions:

1. Explain the idea of the secant method to estimate the root of any equation. Using the secant method, estimate the root of the equation x² - 4x - 10 = 0 with the initial estimates of x₁ = 4 and x₂ = 2. Do these points bracket a root?(3 + 4 + 1)

2. Given the data

Calculate f(1.35) using Newton's interpolating of order 1 through 3. Choose base points to attain good accuracy. Comment on the accuracy of results on the order of polynomial.(5 + 3)

3. How do you find the derivative if the function values are given in a tabulated form? The distance traveled by a vehicle at the intervals of 2 minutes are given as follows. Evaluate the velocity and acceleration of the 

Vehicle at time T = 5, 10, 13.

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

        3x₁ + 2x₂ + x₃ = 10

        2x₂ + 3x₂ + 2x₃ = 14

        x₁ + 2x₂ + 3x₃ = 14

5. Obtain y(1.5) to the following differential equation using Runge-Kutta 4^th order method.        (8)

    taking h = 0.25 

6. Write the finite difference formula for solving Poisson's equation. Hence solve the Poisson equationover the domain 0 ≤ x ≤ 1.5 and 0 ≤ y ≤ 3 with f = 0 on the boundary and h = 0.5        (1 + 7)

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

OR

Write an algorithm and a C program for the Sinpson's 1/3 rule to integrate a given function.(4 + 8)