Numerical Method 2075(New Course)

Group A

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

1. What is non-linear equation? Derive the required expression to calculate the root of non-linear equation using secant method. Using this expression find a root of following equation.

X² + cos(x) - e ^-x -  2 = 0

2. What is matrix factorization? Factorize the given matrix A into LU using Dolittle algorithm and also solve Ax = b for given b using L and U matrices.

3. What is initial value problem and boundary value problem? Write an algorithm and program to solve the boundary value problem using shooting method.

Group B

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

4. Calculate a real negative root of following equation using Newton's method for polynomial.

x⁴ + 2x³ + 3x² + 4x = 5

5. What is least squares approximation of fitting a function? How does it differ with polynomial interpolation? Explain with suitable example.

6. Find the lowest degree polynomial, which passes through the following points:

Using this polynomial estimate f(x) at x = 0

7. Fit function of type y = a + bx for the following points using least square method.

8. Calculate the integral value of the function given below from x = 1.8 to x = 3.4 using Simpson's 1/3 rule.

9. Evaluate the following integration using Romberg integration.

10. Solve the following set of equations using Gauss Seidel method.

        x + 2y + 3z = 4

        6x - 4y + 5z = 10

        5x + 2y + 2z = 25

11. From the following differential equation estimate y(1) using RK 4^th order method.

12. Solve the Poison's equation over the square domain 0 ≤ x ≤ 1.5, 0 ≤ y ≤ 1.5 with f = 0 on the boundary and h = 0.5.