ARYABHATTA KNOWLEDGE UNIVERSITY
NUMERICAL METHOD AND COMPUTATIONAL TECHNIQUE
Sem. iv 2013 Time
3hr Full marks: 70
Attempt any five questions in which question no 1 is
compulsory
1.
Answer any seven from the following:
(a)
In C++ ,a mathematical expression 137% 10 yields
(i)
13
(ii)
13.7
(iii)
0
(iv)
7
(b)
Which of the following statements is true about
the break statement?
(i)
A break statement causes an exit only from the
innermost loop
(ii)
A break statement causes an exit only from the
innermost switch
(iii)
A break statement causes an exit only from all
loops and switches
(iv)
A break statement causes an exit from the
innermost loop or switches
(c)
What is the process of converting one predefined
type into another called?
(i)
Type casting
(ii)
Expression
(iii)
Type promotion
(iv)
Type conversion
(d)
The assignment statement a-=b; can also be
written as
(i)
a=b-1
(ii)
a=a-b
(iii)
a=a-(-b)
(iv)
a=b-a
(e)
the operator which is used to access the address
of a variable is
(i)
&
(ii)
#
(iii)
@
(iv)
*
(f)
What do you mean by a difference equation?
(g)
Define the order and degree of a difference
equation with examples.
(h)
What is the difference between an initial value
problem and a boundary value problem ?
(i)
What do you mean by the order of convergence of
an iterative method for finding the root of the equation f(x)=0 ?
(j)
When is the convergence of an iterative method
for solving the equation f(x)=0 said to be (i) linear and (ii) quadratic?
2.
(a) find the root of the equation xtanx=1.28,
that lies between 0 and 1, correct to two
places of decimals, by
bisection method
(b) Write a computer program using C/ C++
for the above equation using bisection method.
3.
(a) Find the Newton- raphson iterative formula
for the reciprocal of a number N and hence find the value of 1/23, correct to 5
decimals.
(b) Write a computer program using C/C++ to
find the smallest positive root /the largest negative root of the equation
f(x)=0 , by using the simple iterative method.
4.
Solve the following system of equations by Jacobi’s
iteration method and gauss-seidel’s method:
10x1+2x2 +x3=3
X1+ 10x2- x3=-22
-2X1+3x2+10 x3=22
5.
The population of a town Patna in the census is
as given in the data. Estimate the
population in the year 1996 using Newton’s forward interpolation and backward
interpolation formulae:
Year(x)
|
1961
|
1971
|
1981
|
1991
|
2001
|
Population (in 1000’s approx)
|
46
|
66
|
81
|
93
|
101
|
6.
(a) fit a curve of the form xy= a+bx2
to the following data by the method of least square :
X
|
1
|
2
|
4
|
6
|
8
|
Y
|
5.43
|
6.28
|
10.32
|
14.86
|
19.51
|
(b) Write
a computer program using C/C++ to fit a straight line of the form xy=a bx2
taking the above data, using the method of least square.
7.
The velocity of a particle at distance s from a
point on its linear path is given in the following data :
S(m)
|
0
|
2.5
|
5.0
|
7.5
|
10.0
|
12.5
|
15.0
|
17.5
|
20.5
|
V(m/sec)
|
16
|
19
|
21
|
22
|
20
|
17
|
13
|
11
|
9
|
Estimate the time taken by the particle to
traverse the distance of 20 meters , using Simpson’s one-third rule.
8.
(a) Find the value of y(1.1) ,using runge-kutta
method of the fourth order , given that dy/dx=y2 +xy ;y(1)=1
(b) Write a computer program using C/c++ to
solve the above differential equation at specified pivotal points, using runge
–kutta method of the fourth order.
9.
Solve the boundary value problem v’’(x)- xy(x)=0
for xi=0, 1/3 ,2/3 , given that y(0)+y’(0)=1 and y(1)=1.