Code: C-09 / T-09 Subject: NUMERICAL COMPUTING
Time: 3 Hours June 2006 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. The approximation 
to 
is correct to n significant digits.
The value of n is
(A) 7. (B) 6.
(C) 5. (D) 4.
b. The equation f (x) = 0 has a simple root in the interval (1, 2). This root is to be determined correct to two decimal places using bisection method. The required number of iterations is
(A) 5. (B) 6.
(C) 7. (D) 8.
c. The system of equations
has
(A) no solution.
(B) two parameter family of solutions.
(C) one parameter family of solutions.
(D) unique solution.
d. Gauss-Seidel iteration method is used to solve the system of equations
The rate of convergence of this method is
(A) 0.7659. (B) –0.7659.
(C) 1.7635. (D) –1.7635.
e. The value of 
where 
is the forward difference operator
and E is the shift operator, with differencing h =1 is
(A) 
. (B)
.
(C) 6 x. (D) 6
f. The data
represents a third degree polynomial. It is known that f (3) is in error. The correct value of f (3) is
(A) 37. (B) 39.
(C) 41. (D) 43.
g. The least squares polynomial approximation of degree 1 to 
on
[0, 1] is 
.
The least squares error is
(A) 
. (B)
.
(C) 
. (D)
.
h. The truncation error in the method
is of 
.
The value of p is
(A) 1. (B) 2.
(C) 3. (D) 4.
i. The minimum number of sub-intervals that will be required, so that
the error in evaluating 
by trapezoidal rule is 
, is
(A) 8. (B) 9.
(C) 12. (D) 13.
j. The value of y(1.4) for the initial value problem 
using
Euler method with h=0.2 is
(A) 4.6. (B) 22.5552.
(C) 25.1552. (D) 26.2432.
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. The method 
is used to find a multiple root of
multiplicity two of the equation f (x)=0. Determine the rate of convergence of
the method. Also obtain the asymptotic error constant. (10)
b. Perform two iterations of the method
to
determine a root of 
with 
. (6)
Q.3 a. Set up the Newton-Raphson method in matrix form to solve the system of equations
Perform two iterations of the
method with the initial approximation 
. (8)
b. Solve the system of equations
using Gauss-Jordan method with partial pivoting, if necessary. (8)
Q.4 a. Using 
decomposition method, find the
inverse of the matrix 
Take
.
(8)
b. Perform four iterations of the power
method to obtain the largest eigenvalue in magnitude of the matrix 
. Take the initial approximation
vector as 
.
(8)
Q.5 a. Using Jacobi method, find all
the eigenvalues and the corresponding eigenvectors of the matrix 
. (Use exact arithmetic) (10)
b. Obtain the least squares approximation of
the form 
to the data
(6)
Q.6 a. Jacobi iteration method is used to solve the system of equations
Determine the rate of convergence of the method. Starting with the initial vector
perform
two iterations. (8)
b. Prove the following operator relations
(i) 
.
(ii) 
.
(iii) 
.
The operators have their standard meaning. (2+3+3)
Q.7 a. Determine the largest step size h that can be used in the tabulation of a function
f(x), 
, at equally spaced nodal points so
that the truncation error
of the quadratic
interpolation is less than 
in magnitude. Find h, when
and 
. (8)
b. For the method
determine the optimal value of h, so that
=minimum where 
is
the maximum round off error in function values and 
is the maximum value of 
in the given interval.
(8)
Q.8 a. Evaluate 
using Simpson’s 
rule with 3 and 5 nodal
points. Find the improved value of I using Romberg integration. (8)
b. Determine the constants A, B and C in the method
so that the method is of highest possible order. Obtain the truncation error and the
order of the
method. Use this method to obtain the value of 
. (8)
Q.9 a. Find the truncation error and the order of the Heun’s method for solving the initial
value problem 
.
(8)
b. Using classical Runge-Kutta method of order four with h=0.1, obtain the
approximate value of y (1.2) for the initial value problem
.
(8)