Numerical Method 2077

Group A

Attempt any Two questions:(10 x 2 = 20)

1. Derive the formula for Newton Raphson Method. Solve the equation x² + 4x - 9 = 0 using Newton Raphson method. Assume error precision is 0.01. Discuss drawbacks of the Newton Raphson method.

2. How interpolation differs from regression? Write down algorithm and program for Lagrange interpolation.

3. Why partial pivoting is used with Naive Gauss Elimination method? Solve the following system of equations using Gauss Elimination method with partial pivoting? How Gauss Jordan method differs from Gauss elimination method?

    2x + 2y - z = 6

    4x + 2y + 3z = 4

    x + y + z = 0

Group B

Attempt any eight questions:(5 x 8 = 40)

4. Define the terms true error and relative error? Use Horner' method to evaluate polynomial 2x³ - 3x² + 5x - 2 at x = 3 and write down its algorithm.

5. Construct Newton's forward difference table for the given data points and approximate the value of f(x) at x = 15.

6. Fit the curve y = ae ^bx through the following data points.

7. Discuss the Doolittle LU decomposition method for matrix factorization.

8. Write down algorithm and program for the differentiating continuous function using three point formula.

9. How Simpson's 1/3 rule differs from trapezoidal rule? Derive the formula for Simpson's 1/3 rule.

10. Appropriate the solution of y' = 2x + y , y(0) = 1 using Eulers method with step size 0.1. Approximate the value of y(0.4).

11. A plate of dimension 18cm x 18cm is subjected to temperatures as follows: left side at 100⁰c, right side at 200⁰C. Upper part at 50⁰ C, and lower at 150⁰C. If square grid length of 6cm x 6cm is assumed, what will be the temperature at the interior nodes?

12. How boundary value problems differs from initial value problems? Discuss shooting method for solving boundary value problem.