BCSL-058
1. Write a
program that implements (with pivot condensation) Gaussian elimination method
for solving n linear equations
in n variables, that calls
procedures (5 Marks)
(i) Exchange of rows
(ii) lower-triangularisation and
(iii) back substitutions
(codes of procedures are also to be written).
Use
the program for solving the following system of linear equations:
x+ y+z = 3
5x+ 2y+7z=14
3x+y+ 5z =9
2. Write
a program that uses Gauss-Jacobi method
to solve system of linear equations. Use the method to solve the system
of linear equations given in Q. No. 1 above. (5 Marks)
3. Write a
program that approximates a root of the equation f(x) = 0 in an interval [a, b] using Bisection method. The necessary
assumptions for application of Bisection
method should be explicitly mentioned. Use the method to find one root
of the equation x4 + 5x - 3=0. (5 Marks)
4. Write a
program that uses Lagrangian polynomials, for which at most three nodes are
given (hence interpolating polynomial will be at most quadratic). Use the
program to find approximate value of f(x)
= x4 at x =1.5.
The nodes given may be assumed as x0
= 1, x1 = 2, x2 = 3.
5. Repeat
Problem No. 4 using Newton’s Interpolating polynomial (instead of Lagrangian
Polynomial). (5 Marks)
6. Write
a program that approximates the derivative of a given (differentiable) function
f (x) at x = x0,
using forward–difference formula. Using the program find the derivative of f(x) = ex at x=1. (5 Marks)
7. Write
a program that approximates the value of a definite integral 
using Trapezoidal Rule, with M sample points.
Find an approximate value of the integral of sin (2
) using the program with 6 intervals over the interval [1,
7]. (5 Marks)
using Trapezoidal Rule, with M sample points.
Find an approximate value of the integral of sin (2) using the program with 6 intervals over the interval [1,
7]. (5 Marks)
8. Write
a program that approximates the solution of the initial value problem: y¢ = f(t, y) with y(a) = y0 over [a, b] using
Euler’s method. Using the
program to approximate the solution of the initial value problem: y¢ = -2ty2 with y(0) =1.
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