Code: C-09 / T-09 Subject: NUMERICAL COMPUTING
Time: 3
Hours 
Max. Marks: 100
NOTE: There are 9 Questions in all.
· Question 1 is compulsory and carries 20 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.
· Out of the remaining EIGHT Questions answer any FIVE Questions. Each question carries 16 marks.
· Any required data not explicitly given, may be suitably assumed and stated.
Q.1 Choose the correct or best alternative in the following: (2x10)
a. Using Taylor series, an
approximation to 
is written as 
. Then, the bound of the
truncation error is given by
(A) 
. (B)
.
(C) 
. (D)
.
b. The rate of convergence of the secant method for finding a simple
root of the equation 
, is
(A) 1.22. (B) 2.
(C) 1.62. (D) 1.
c. Gauss-Jacobi method is applied to solve the system of equations
with 
.
The iteration diverges since the spectral radius of the iteration matrix is equal to
(A) 5. (B)
.
(C) 
. (D)
.
d. The matrix 
is to be reduced to the tri-diagonal
form by Givens method. Then, the angle of the orthogonal rotation is given by
(A) 
. (B) 
.
(C) 
. (D)
.
e. The third divided difference of the polynomial 
based on the
points 
and
is
(A) 1. (B) 0.
(C) 6. (D) 3
f. The value of 
, where 
is the forward difference, is given
by
(A) 
(B) 
(C) 
(D)
g. The error in a given numerical differentiation formula is given by 
. If this
derivative (denoted by g) is evaluated with step lengths 
and 
, then the Richardson’s extrapolation gives the better estimate of the value of this derivative as
(A) 
. (B) 
.
(C) 
. (D) 
.
h. The data
represents a
certain function 
. Then, the Simpson’s rule, using the
entire data with suitable step length, gives the value of the integral
, as
(A) 
. (B)
.
(C) 
. (D)
.
i. The integral 
is evaluated by the Gauss-Laguerre
one-point formula. The value of the integral is
(A) 0.65. (B) 0.75.
(C) 0.5 (D) 0.92.
j. The approximate value of y (1.2) using the Euler’s method with h =
0.1 for the IVP
is
(A) 2.25 (B) 2.486
(C) 3 (D) 3.246
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. The error in the secant method
, for finding a
simple root of 
is
obtained as 
,
where 
.
Find the order of the method and the error constant. (6)
b. Show that the equation 
has only one
negative root. Find this root correct to three decimal places, using the
Newton-Raphson method. Take suitable initial aproxmiation. (6)
c. The system of equations 
has a solution
near 
.
Formulate the Newton’s method for its solution (no iterations are to be
computed) (4)
Q.3 a. The following system of equations are given
Using Gauss elimination, find
the conditions on 
so that the system is solvable or
unsolvable. (8)
b. The
system of equations
is to be
solved by the Jacobi iteration method. Find the values of k > 0, for
which the iteration converges.(8)
Q.4 a. Transform the matrix 
to tridiagonal
form by the Given’s method.
Form the Sturm sequence and locate the eigen values in intervals of length 1
unit. (10)
b. Find the smallest eigen value in magnitude and the corresponding eigen vector
of the matrix.
using five iterations of the inverse power method. (6)
Q.5 a. Use Gauss-Chebyshev two-point
method to evaluate 
. (5)
b. Find the unique polynomial 
that fits the
data 
. (6)
c. We want to construct a table of values with uniform mesh size h for the
function 
on [ 0, 1 ]. Find
the step size h, if quadratic interpolation is
proposed to be used
with an error less than or equal to 
. (5)
Q.6 a. An experimentalist studies a decaying process and decides to fit an
approximation to it as 
. Derive the
normal equations using the
method of least squares. Then, fit this approximation to the following data
(4+4)
b. A numerical differentiation formula is written as
Find the values of the parameters so that the formula is of as high order as
possible. Find the leading term of the truncation error. Use this formula to
compute 
from the following
data
(6+2)
Q.7 a. Consider the formula
where the truncation
and RE is the
round off error.
Let 
be the round off errors in 
respectively and
. Determine the optimal
step length h satisfying the
criterion 
.
(5)
b. Evaluate the integral 
using Simpson’s rule with
2 and 4 intervals.
Extrapolate using Romberg integration. (7)
c. An integration formula is written as
Find the values of a and b such that it is of order as high as possible. (4)
Q.8 a. Evaluate the integral 
using the
Gauss-Legendre two point
formula. If the exact solution
is 
,
find the magnitude of
the error in the solution. (8)
b. Derive the Gauss-Laguerre two point formula in the form
. (8)
Q.9 a. Write the expression for the bound of the error in the Euler method to solve
the initial value problem
. The initial
value problem
, is being solved by the Euler
method in the
interval [0, 0.2]. Find the largest step length that can be used such that the
error
in the method is 
.
(6)
b. Use the classical Runge-Kutta method of fourth order to find the numerical
solution at 
for the initial
value problem 
.
Assume the step length 
.
(5)
c. For the initial value
problem 
,
estimate 
using
the
Adams-Moulton multistep method
with
.
Determine the required starting value using the Euler’s
method. (5)