## Description

BCS-054 Solved Assignment IGNOU BCA 5th Semester For July 2018 and January 2019 Session

**Course Code : BCS-054**

**Course Title : Computer Oriented Numerical Techniques**

**Assignment Number : BCA(5)/054/Assignment/2018-19**

**Maximum Marks : 100**

**Weightage : 25%**

**Last Dates for Submission : 15th October, 2018(For July, 2018 Session)**

**15th April, 2019(For January, 2019 Session)**

*This assignment has eight questions of total 80 marks. Answer all the questions. 20 marks are for viva voce. You may use illustrations and diagrams to enhance explanations. Please go through the guidelines regarding assignments given in the Programme Guide for the format of presentation. Illustrations/ examples, where-ever required, should be different from those given in the course material. You must use only simple calculator to perform the calculations.*

**Question 1:**

(a)Consider that you are using an eight-decimal digit floating point representation as given in your Block 1, Unit 1, Section 1.3.1 page 29. Perform the following operations:

(i) Represent 0.000035234545 and 789264123 as floating point numbers using Rounding in normalised form.

(ii) Given the above two numbers what is the absolute and relative error in their representation.

(iii) Subtract the smaller number from the bigger numbers. What is the error in the resulting number?

(iv) Divide the first number by the second number. Convert the result into normalized form in the given format.

(v) Take the first number as 0.000035234545 and assume any second number to demonstrate the concepts of overflow or underflow for the given representation. (You may assume any second number to demonstrate overflow or underflow).

(vi) Define the concept of machine-epsilon.

(b) Consider the following two equations:

5x + 8y = 340

2.45 x + 4.10* y = 172

Does the problem of solving the above two equations can be categorised as Ill-conditioned? Justify your answer.

(c) Find the Maclaurin series for calculating e2x. Use first four terms of this series to calculate the value of e2x for any value of x. Also find the bounds of truncation error for such cases.

(d) Obtain Approximate the value of (0.999)-1 using first four terms of Taylor’s series expansion.

**Question 2:**

(a) Solve the system of equations

3x + 4y + 5z = 6

2x – 6y + 3z = -13

5x – 7y + 2z = -11

using Gauss elimination method with partial pivoting. Show all the steps.

(b) Perform four iterations (rounded to four decimal places) using

(i) Jacobi Method and

(ii) Gauss-Seidel method

for the following system of equations.

3 1 -4 x 0

2 -3 -4 y = 1

-3 3 7 z 2

With x = (0, 0, 0)T. The exact solution is (3, -1, 2)T. Which method gives better approximation to the exact solution?

**Question 3:**

Determine the smallest positive root of the following equation:

f(x) = 2×3 + 3×2- 9x – 10 = 0

The root should be correct up to 2 decimal places, using

(a) Regula-falsi method (b) Newton-Raphson method (c) Bisection method (d) Secant method

**Question 4:**

(a) Find Lagrange’s interpolating polynomial that fits the following data. Hence obtain the value of f(3.5).

x 1 3 6 10

f(x) 5 12 22 35

(b) Using the Lagrange’s inverse interpolation method, find the value of x when y is 6.

x 2 23 34 55

y=f(x) 1 3 5 8

**Question 5:**

(a) The GDP of a country is given in the following table:

Year (x) : 1995 2000 2005 2010 2015

GDP (y) (in Billion Rupees): 347 610 920 1402 1745

(i) Using Stirling’s central difference formula estimate the GDP for the year 2007

(ii) Using Newton’s forward difference formula estimate the GDP for the year 1998.

(iii) Using Newton’s backward difference formula estimate the GDP for the year 2013.

(b) Derive an expression of E operators in terms of δ.

**Question 6:**

(a) Find the values of the first and second derivatives of y = x2 for x=2.75 using the following table. Use forward difference method. Also, find Truncation Error (TE) and actual errors.

(b) Find the values of the first and second derivatives of y = x2 for x=2.75 from the following table using Lagrange’s interpolation formula. Compare the results with (a) part above.

**Question 7:**

Compute the value of the integral

4

∫ (x3+ x2 + 7) dx

0

By taking 8 equal subintervals using (a) Trapezoidal Rule and then

(b) Simpson’s 1/3 Rule. Compare the result with the actual value.

**Question 8:**

(a)

Solve the Initial Value Problem, using Euler’s Method for the differential Equation:

y = 1+xy, given that y(0) = 1.

Find y(1.0) taking (i) h = 0.2 and then (ii) h = 0.1

(b) Solve the following Initial Value Problem using (i)R-K method of O(h2) and (ii) R-K method of O(h4)

y’ = xy + x and y(0) = 1.

Find y(0.4) taking h = 0.2, where y’ means dy/dx.

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