Q1. Write a program that implements Gaussian elimination method with pivot condensation for solving n linear equations in n variables, that calls procedures
(i) lower-triangularisation and
(ii) back substitutions
(codes of procedures are also to be written).
Use the program for solving the following system of linear equations:
5x – 7y + 8z = 11
4x – 9y + 3z = 2
9x +2y + 5z = 25
Q2. Write a program that uses Gauss-Seidel iterative method to solve system of linear equations. Use the method to solve the system of linear equations given in Q. No. 1 above.
Q3. Write a program that approximates a root of the equation f (x) = 0 in an interval [a, b] using Newton-Raphson method. The necessary
assumptions for application of this method should be explicitly mentioned. Use the method to find smallest positive root of the equation
2×4- 5×2 + 10x – 32=0.
Q4. Write a program that uses Lagrangian polynomials for interpolation. You must use only three nodes such that the interpolating polynomial is at most quadratic. Using this program find approximate value of f (x) = 5x at x =1.5. The nodes are at points x 0 = 0, x 1 = 1, x 2 = 2.
Q5. Write a program to interpolate using Newton’s Backward difference formula using only three points. Solve Problem asked in Question 4 using Newton’s Interpolating polynomial using Backward difference (instead of Lagrangian Polynomial).
Q6. Write a program that approximates the derivative of a given (differentiable) function f (x) at x = x0, using forward difference formula
taking only 3 points having value of x as 0, 1 and 2 respectively. Using the program find the derivative of function f(x)=( x ) 5/2 at x=0.5
Q7. Write a program that approximates the value of a definite integral
using Simpson’s 1/3rd rule, with M sample points. Find an approximate value of the integral of 2×2 + x + 5 using the program with 8 intervals over the interval [0, 4].