Code: C-09 / T-09 Subject: NUMERICAL COMPUTING
Time: 3 Hours Max. Marks: 100
NOTE: There are 11 Questions in all.
· Question 1 is compulsory and carries 16 marks.
· Answer any THREE Questions each from Part I and Part II. Each of these questions carries 14 marks.
· Any required data not explicitly given, may be suitably assumed and stated.
· Keep four decimal digits in your arithmetic calculations.
Q.1 Choose the correct or best alternative in the following: (2x8)
a. A root of the equation 
is obtained using
the Newton-Raphson method with the initial approximation x0 = 1.2.
The root after two iterations is obtained as
(A) 1.4879 (B) 1.4974
(C) 1.5243 (D) 1.7897
b.
The system of equations 
is solved using the
Gauss-Jacobi iteration method. The iteration matrix of the method is
(A) 
(B)
(C) 
(D)
c. The lowest degree interpolating polynomial which fits the data
|
x |
0 |
1 |
2 |
3 |
4 |
|
f(x) |
-24 |
-18 |
6 |
36 |
60 |
is
(A) 4 (B) 3
(C) 2 (D) 1
d. For the numerical differentiation method
the truncation error is written as 
. The value of p is
(A) 1 (B) 2
(C) 3 (D) 4
e. The
least squares polynomial approximation of degree 1 to the function 
on 
is
(A)
(B)
(C) 
(D)
f. The value of the
integral 
using
Simpson’s five-point formula is
(A) 0.8762 (B) 0.8548
(C) 0.8508 (D) 0.8476
g. The method given
by, 
where
is the
weight function, will give exact results when f(x) is a polynomial of degree
less than or equal to
(A) 1 (B) 2
(C) 3 (D) 4
h. The
Taylor’s series method of order two with step size h = 0.2 is used to solve the
initial value problem 
. The approximate value of y (1.4)
is
(A) 5.8625 (B) 5.8525
(C) 5.7525 (D) 5.2525
PART I
Answer any THREE Questions. Each question carries 14 marks.
Q.2 a. Determine
the values of p and q so that the order of the method 
for computing 
, where a is a
positive real constant, becomes as high as possible. Also find the order and
the asymptotic error constant for the method. (7)
b. Perform two iterations of the method
to
find a root of the equation 
. Take the initial approximation as 
. (7)
Q.3 a. The system of
equations 
has a solution near x = 1.9 and y = 0.9. Perform two iterations of the Newton’s method to obtain the solution. (7)
b. Obtain
the Choleski factorisation of the form 
, where 
is a lower triangular matrix and 
. Hence,
determine the matrix 
. (7)
Q.4 a. Set
up the Gauss-Seidel iteration scheme in matrix form to solve the linear system
of equations 
. Determine
its rate of convergence,
if the iteration scheme converges. (7)
b. Reduce the matrix 
to tridiagonal
form using Given’s method. Find the characteristic equation of the matrix
using Strum sequence. (Hence or otherwise) Find intervals of unit length each
containing one eigenvalue. (Use exact arithmetic) (7)
Q.5 a. Explain the Power
method to find the largest eigenvalue in magnitude of a given square matrix.
Perform three iterations of this method to obtain the largest eigenvalue in
magnitude of the matrix 
. Take
the
initial approximate vector on 
. (7)
b. Obtain the inverse of
the matrix 
using
Gauss-Jordan method. (7)
Q.6 a. Find all the eigenvalues and the corresponding
eigenvectors of the matrix 
using Jacobi method. (use exact
arithmetic). (8)
b. Solve the linear system of equations
using Gauss-elimination method with partial piroting. (6)
PART II
Answer any THREE Questions. Each question carries 14 marks.
Q.7 a. Write
the error term in the Lagrange interpolation 
using (n+1) data points 
and 
are Lagrange fundamental
polynomials. Find an expression for 
. (8)
b. Show that (i) 
(ii) 
where
and 
are respectively
the forward, central and average difference operators. (6)
Q.8 a. Construct the divided difference table for the data
|
x |
-3 |
-1 |
0 |
1 |
3 |
5 |
|
f(x) |
-23 |
1 |
1 |
1 |
25 |
121 |
Using divided differences interpolation, obtain the interpolating polynomial which fits this data. (7)
b. Obtain the least squares
polynomial approximation of degree 2 for the function 
on the interval [-1, 1]. (7)
Q.9 a. Find
the truncation error and the order of the method 
. Determine the optimal value of h
using the criterion 
where 
>0 is the maximum round off
error in function values and 
is the maximum value of 
in the given
interval. (7)
b. Using the method, 
and the
corresponding Richardson extrapolation obtain the best value of 
with the help of
the following data:
|
x |
-1 |
1 |
2 |
3 |
4 |
5 |
7 |
|
f(x) |
1 |
0.3333 |
0.25 |
0.2 |
0.1667 |
0.1429 |
0.1111 |
(7)
Q.10 a. Determine
the constants 
so
that the method 
where x is the weight function, is of highest possible order. Find the error term of the method. (8)
b. Evaluate the integral 
using
Gauss-Legendre two-point integration method. (6)
Q.11 a. Using
Taylor’s series method of order two with step size h = 0.1, obtain an
approximate value of y (1.2) for the initial value problem 
. (7)
b. Using Runge-Kutta fourth order method with step size h = 0.2, obtain an approximate value of y (2.4) for the initial value problem
.
(7)