Numerical Methods Model Question

Group A

Attempt ANY TWO Question.    {10 ×2=20}

1. How Secant methods differs from Newton Raphson method? Derive the formula for Secant Method. Solve the equation cosx+2sinx-x²=0 using Secant method. Assume error precision is 0.01.     {2+4+4}

2. How interpolation differs from regression? Write down algorithm and program for Lagrange interpolation.     {2+4+4}

3. Explain the working of Jacobi Iteration method? Solve the following system of equations using the method. Assume error precision is 0.01. Compare Jacobi Iteration method with Gauss-Seidel method.     {4+4+2}

        5x-2y+3z=-1

        -3x+9y+z=2

        2x-y-7z=-3

Group B

Attempt ANY EIGHT Questions.    {5 ×8=40}

4. Define the terms true error and relative error? Write down algorithm for Horner’ method to evaluate polynomial and use the method to evaluate the polynomial 2x³ -3x²+5x-2 at x=3. {1+2+2}

5. Construct Newton’s backward difference table for the given data points and approximate the value of f(x) at x=45.

6. Fit the quadratic curve through the following data points and estimate the value of f(x) at x=2.

7. Derive formula for the Doolittle LU decomposition matrix factorization method.

8. How can we calculate derivatives of continuous functions? Write down algorithm and program for differentiating continuous function using two point forward difference formula.

9. Find following integral using composite trapezoidal rule using 2 segments (k=2) and 4 segments (k=4).

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

11. Solve the Poisson’s equation  with f = 2 on boundary by assuming square domain ,  and h=1.

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