Numerical Method 2075( Old Course)
Attempt all questions:
1. What is error? Discuss various types of errors. Estimate a real root of the following nonlinear equation using bisection method or Newton's method correct up to two decimal places sin x - x2 - x + 3 = 0 (3 + 5)
2. How polynomial interpolation differs with Cubic Spline interpolation? Explain. Find the best fit curve of quadratic polynomial using least square approximation from following data.
3. (a) For the function 
estimate f'(2.1) and f''(2.7) [take h = 0.05] (3)
(b) Evaluate 
using trapezoidal rule taking h = 0.1 and h = 0.2. Also improve your result using Romberg integration.(4)
4. Decompose the given matrix [A] into LU form. Then using decomposed matrices L and U solve the system of linear equation Ax = b for given [b].(4 + 4)
OR
How can you calculate the inverse of a matrix? Explain. Solve the system of linear equations Ax = b for given [A] and [b] using Gauss elimination method. (use partial pivoting) (3 + 5)
5. (a) How can you solve higher order differential equation? Explain. (3)
(b) Solve the following differential equation within 1 ≤ x ≤ 2 using Runge-Kutta 4th order method. (5)
6. (a) Derive a difference equation to solve Laplace equation.(3)
(b) The steady-state two-dimensional heat-flow in a metal plate is defined by 
. Given the boundary conditions as shown in figure below. find the temperatures at interior points A, B, C, and D.(5)
7. Write an algorithm to solve non-linear equation using secant method. On the basis of your algorithm write a C-program that reads two initial guess from keyboard and displays the following information if the solution is obtained:(5 + 7)
a) Calculated root of the equation
b) Required number of iterations
c) Functional value at calculated root
If solution is not obtained within 200 iterations your program should be terminated by displaying the following message:
NO SOLUTION WITHIN 200 ITERATIONS
TRY AGAIN WITH NEW INITIAL VALUES