BCSL-058 Computer oriented Numerical techniques Lab Solved Assignment 2018-2019

Course Code : BCSL-058
Course Title : Computer oriented Numerical techniques
Lab
Assignment Number : BCA(5)/L-058/Assignment/2018-19
Maximum Marks : 50
Weightage : 25%
Last Dates for Submission : 15th October, 2018 (For July, 2018 Session)
15th April, 2019 (For January, 2019 Session)

This assignment has eight problems of 40 marks, each of 5 marks. All problems
are compulsory. 10 marks are for viva voce. Please go through the guidelines
regarding assignments given in the programme guide for the format of
presentation. Note: The programs are to be written in C/C++ and/or in MS-Excel/Any spread
sheet.

Question 1:
Write a program that implements pivot condensation Gaussian
elimination method for solving n linear equations in n variables, that
calls procedure
(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
2x + y+ 3z= 6
4x+5y+2z =11
(5 Marks)
Question 2:
Write a program that uses Gauss-Jacobi iterative 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)
Question 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 this method should be explicitly mentioned. Use the
method to find smallest positive root of the equation
x
4 + 2×3
– 5×2 + 10x – 19=0.

(5 Marks)
Question 4:
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) = 2x at x =0.5.
(5 Marks)
17
The nodes are at points x 0 = 0, x 1 = 1, x 2 = 2.
Question 5:
Write a program to interpolate using Newton’s Forward difference
formula having only three points. Solve Problem No. 4 using Newton’s
Interpolating polynomial using Forward difference (instead of
Lagrangian Polynomial).
(5 Marks)
Question 6:
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 )
1/2 at x=0.5

(5 Marks)
Question 7:
Write a program that approximates the value of a definite integral
( )
b
z
f x dx 
using Trapezoidal Rule, with M sample points. Find an
approximate value of the integral of 2x
2
using the program with 8
intervals over the interval [0, 4].
(5 Marks)
Question 8:
Write a program that approximates the solution of the initial value
problem:
y f t y 
 ( , )
with
0
y a y ( ) 
over
[ , ] a b
using Euler’s method.
Using the program approximate the solution of the initial value problem:
y
1 = -2ty2
with y(0) =1