BCS-054
1. (a) For the
following questions use 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 3456789 and 3455155 as floating
point numbers using chopping in normalised form.
(ii) What are the advantages of representing these
numbers in normalised form?
(iii) Compare the
error in representation when chopping is used to error in representation when
rounding is used for these numbers.
(iv) Subtract the
two numbers and find the error in result.
(v) Divide the
first number by second. The result should be in normalised form.
(vi) Explain the
concept of overflow and underflow using any two numbers. (3 marks)
(b) Explain the concept of Ill-conditioned problem with the help
of an example other than given in the unit. (2marks)
(c) Find the Maclaurin series for calculating sin x using the
first four terms of this series. Also find the bounds of truncation error for
such cases. (3 marks)
(d) Obtain approximate value of (2.8)-1 using first four terms of
Taylor’s series expansion. (2 marks)
2.(a) Solve the
system of equations
4x + 2y + z = 14
2x + 4y + 3z = 18
2x + 3y – 2z = 06
using
Gauss elimination method with partial pivoting. Show all the steps.
(b) Perform
four iterations (rounded to four decimal places) using
(i) Gauss - Jacobi Method and
(ii) Gauss-Seidel method,
for
the following system of equations.
With initial estimates as (0,0,0)T. The
exact solution is (2,1, 1)T. Which method gives better approximation
to the exact solution? (5 marks)
3. Determine
the smallest positive root of the following equation :
f(x) ≡ x2
- cos (x) = 0 to three significant
digits using
(a) Regula-falsi method
(b) Newton Raphson method
(c) Bisection method
(d) Secant method (10 marks)
4.(a) Find
Lagrange’s interpolating polynomial for the following data. Hence obtain the
value of f(3). (5marks)
x 0 2 5 6
f(x) 7 21 51 61
(b) Using the inverse Lagrange’s
interpolation, find the value of x when y=3 for the following data:(5 marks)
x 42 65 95 106
y=f(x) -1 1 4 6
5.(a) By
decennial census, the population of a town was given below. [ 3+2+3= 8 Marks]
|
Year
(x) :
|
1976
|
1986
|
1996
|
2006
|
2016
|
|
Population
(y) :
(in
lakhs)
|
37
|
43
|
53
|
58
|
72
|
(i) Using Stirling's central difference formula,
estimate the population for the year 2001.
(ii) Using Newton’s forward formula, estimate the
population for the year 1984.
(iii) Using Newton’s backward formula, estimate the
population for the year 2010.
(3+2+3 = 8 marks)
(b) Derive
the relationship between the operators E and δ. (2 marks)
6.(a) Find the
values of the first and second derivatives of y = x3/2 at x = 23
from the following table. Use forward difference method. Also, find Truncation
Error (TE) and actual errors. (5 marks)
|
x :
|
20
|
25
|
30
|
35
|
|
y :
|
89.4427
|
125.0000
|
164.3168
|
207.0628
|
(b) Find the values of the first and second derivatives of y = x3/2 at x = 23 from the following table using Lagrange’s interpolation
formula. Compare the results with part (a) above. (5 marks)
|
x :
|
20
|
25
|
30
|
35
|
|
y :
|
89.4427
|
125.0000
|
164.3168
|
207.0628
|
7. Compute
the value of the integral
By taking 8 equal subintervals using (a) Trapezoidal
Rule and then (b) Simpson's 1/3 Rule. Compare the result with the actual value.
(10 marks)
8.(a) Solve the
Initial Value Problem, using Euler’s Method y¢ = 1+xy, y(0) = 1.
Find
y(1.0) taking (i) h = 0.2 and then (ii) h = 0.1 (4 marks)
(b) Solve
the following Initial Value Problem using
(i) R-K method of O(h2) and (ii) R-K
method of O(h4)
y'
= x2+ y2 and
y(0) = 0.
Find
y(0.4) taking h = 0.2, where y' = dy/dx
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