Numerical Method 2074

Attempt all questions:

1. How can you use bisection method for the solution of nonlinear equations? Discuss with suitable example.(8)

2. Define interpolation. Find the Lagrange interpolation polynomial to fit the following data:

Use the polynomial to estimate the value of e¹⁵.            (2 + 6)

3. Derive trapezoidal rule to evaluate numerical integration. Use this formula to evaluate the integral             

                                                (2 + 6)

4. Discuss Gauss-Jordan method for solving a system of linear equations. Solve the system

        2x₁ + 4x₂ - 6x₃ = -8

        x₁ + 3x₂ +x₃ = 10

        2x₁ - 4x₂ - 2x₃ = -12

Using Gauss-Jordan method.

5. How can you use Taylor series method to find the solution of ordinary differential equation? Use the Taylor method to solve the equation (5 + 3) y' = x² + y² for x = 0.25 and x = 0.5 given y(0) = 1

OR

Define boundary value problem. Use shooting method to solve the equation in the interval(1, 2).

6. Define partial differential equation. Discuss Laplace's equation along with its derivation.(2 + 6)

7. Write an algorithm and program to compute all roots of polynomial using secant method.