Code: A-07 Subject: NUMERICAL ANALYSIS & COMPUTER PROGRAMMING
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.
A second
degree polynomial approximation to 
using Taylor Series expansion is
given by
(A)
. (B)
.
(C)
. (D)
None of above.
b. The order of Newton-Raphson method 
for finding a multiple root of
multiplicity 3 of the equation 
, is
(A) 1 (B) 2
(C) 3 (D) 4
c. The combination “\a” in C gives a
(A) vertical tab. (B) form feed.
(C) system beep. (D) carriage return.
d. For
a two point Gauss-Hermite integration rule 
, the values of 
are given by
(A)
.
(B)
.
(C) 
.
(D) 
.
e. If x = 5, the output of the statement sign = (x < 0) ? -1 : ((x == 0) ? 0 : 1); would be
(A) –1 (B) 0
(C) 1 (D) error
f. If a general
iteration method 
is of order p then
, by definition.
The value of r is
(A) p+1 (B) p
(C) p-1 (D) None of above
g. The left shift operator “<<” has the effect of
(A) multiplying the number by 2.
(B) dividing the number by 2.
(C) adding 2 to the number.
(D) subtracting 2 from the number.
h. The solution for the system of equations
is given by
(A)
. (B) 
.
(C) 
. (D) 
.
i. The format for unsigned integer in C printf statement is
(A) %i (B) %I
(C) %u (D) None of above
j. What
should be the condition on 
such that the method 
where 
, converges?
(A) 
. (B)
.
(C) 
. (D)
None of above.
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. The system of equations
is
to be solved. Write the Newton’s iterative procedure and iterate two times
starting with 
and
. (8)
b. The
equation 
has
a simple root in the interval 
. The function 
is such that 
and 
for all values of
in this
interval. Assuming that the Newton-Raphson method converges for all initial
approximations in , find the number of iterations
required to obtain the root correct to 
. (8)
Q.3 a. Show that the matrix
is positive definite and hence
decompose it as 
using
Cholesky method. (8)
b. Consider the linear system of equations
Discuss the convergence of Gauss Jacobi and Gauss Seidel methods. (8)
Q.4 a. Construct the divided difference table for the following data:
Use
Newton’s divided difference formula to find the Interpolating Polynomial and
hence find the value of 
. (6)
b. When does a function need an include directive? (5)
c. Write
a programme using a function comb() to calculate C(n,k) to print
Pascal’s Triangle down to row number 12. Each number in Pascal’s Triangle is
one of the combinations 
. If we count the rows and the
diagonal columns starting with 0, then the number C(n,k) is in row n
and column k. Pascal’s Triangle looks like
(5)
Q.5 a. Starting with the
set 
,
generate a set of orthogonal polynomials on 
with the weight function 
. Using these polynomials
find a Least squares approximation of the form 
for the function 
on 
. (8)
b. For the method
determine the optimal
value of 
using
the criteria
(i) 
(ii) 
minimum (8)
Q.6 a. Find the error term of the method 
as a power series in . Derive the corresponding
Richardson’s extrapolation scheme. Using this method and the Richardson’s
extrapolation, find the best value of 
when 
is given in tabular form as
(8)
b. Write a function that rotates a two-dimensional square array 900 “clock-wise”. Consider an array of integers. For example, an array
(8)
Q.7 a. Using
the method of undetermined coefficients, find the nodes and weights of the
quadrature formula 
(8)
b. Calculate 
using Trapezoidal
rule with number of points as 3, 5 and 9. Improve the results using Romberg
Integration. (8)
Q.8 a. The
following IVP is given 
Use second order Taylor series
method to get 
with
step length 
. (8)
b. Write a programme that plays the game of “rock, paper, scissors”. In this game, two players will simultaneously say either “rock” or “paper” or “scissors”. The winner is the one whose choice dominates the other. The rules are: paper dominates (wraps) rock, rock dominates (breaks) scissors and scissors dominates (cuts) paper. Use enumerated data type for the choices and the result. The result will be either a “tie” or “player 1 wins” or “player 2 wins”. (8)
Q.9 a. Use
the classical fourth order Runge-Kutta method to find the solution at 
for 
Assume the step length 
. (10)
b. Construct a truth table for each of the following boolean expressions, showing its truth value (0 or 1) for all its combinations of truth values of its operands p, q and r.
i) !p || q
ii) p && q || !p && !q
iii) (p || q) && !(p && q)
iv) !(p && q)
v) p && (q && r)
vi) p || (q && r) (6)