Numerical Method 2078

Section A

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

1. How can Horner's rule be used to evaluate the f(x) and f(x) of a polynomial at given point? Explain. Write an algorithm and program to calculate a real root of a polynomial using Horner's rule.

2. What is matrix factorization? How can it be used to solve system of linear equations? Factorize the given matrix A and solve the system of equations Ax=b for given b using L and U matrices.

     and 

3. What is higher order differential equation? How can you solve the higher order differential equation? Explain. Solve the following differential equation for , taking h=0.25.

     with y(1) = 1 and y'(1) = 2

Section B

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

4. How the half-interval method can estimate a root of non-linear equation? Find a real root of following equation using half-interval method correct up to two decimal places.

                x² - e^-x - x = 1

5. Calculate a real root of the given equation using fixed point iteration correct up to 3 significant figures.

        2x³ - 2x = 5

6. What is Newton's interpolation? Obtain the divided difference table from the following data set and estimate the f(x) at x=2 and x=5.

7. What is linear regression? Fit the linear function to the following data.

8. What are the problems with polynomial interpolation for large number of data set? How such problems are addressed? Explain with example.

9. Evaluate the following integration using Romberg integration.

10. Solve the following set of linear equations using Gauss Jordan method.

            x₂ + 2x₃ + 3x₄ = 9

            7x₁ + 6x₂ + 5x₃ + 4x₄ = 33

            8x₁ + 9x₂ + x₄ = 27

            2x₁ + 5x₂ + 4x₃ + 3x₄ = 23

11. Solve the following differential equation for , taking h = 0.25 using Heun's method.

        y'(x) + x²y = 3x, with y(1) = 1

12. Consider a metallic plate of size 90 cm by 90 cm. The two adjacent sides of the plate are maintained at temperature of 100⁰C and remaining two adjacent sides are held at 200⁰C. Calculate the steady state temperature at interior points assuming a grid size of 30 cm by 30 cm.