Code:
AC09/AT09 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. The exact
solution of the Initial Value Problem 
is
(A)
(B)
(C)
(D)
b. The
expression 
is
equal to
(A)
(B)
(C)
(D)
c. The
equation 
has a
multiple root x = 1 of multiplicity 2. The
other 2 roots will be
(A) both real but different (B) a pair of real roots
(C) a complex pair (D) one real and one complex
d. Which of the following is a Composite Simpson’s Rule?
(A)
(B)
(C)
(D)
e. The nth
divided difference 
of 
is given by
(A)
(B)
(C)
(D)
f. Attempt is
made to solve the system of equations 
where 
and 
by the Gauss–Jacobi iteration
method. Then, the iteration
(A) has rate of convergence 0.5634. (B) has rate of convergence 0.235.
(C) has rate of convergence 1.234. (D) diverges.
g. The least squares straight line approximation to the data
is given by
(A)
(B)
(C)
(D)
h. The
integration formula 
is to be used. The value of 
for which the
method is of highest order, is given by
(A) 1 (B) 2/3
(C) 1/3 (D) 1/2
i. The order
of Newton-Raphson method 
for finding out a multiple root
of multiplicity 3 of the equation 
, is
(A) 1 (B) 2
(C) 3 (D) 4
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. Obtain
a second degree polynomial approximation to 
using the Taylor series expansion
about 
.
Use the expansion to approximate 
and find a bound of the truncation
error. (6)
b. Use the classical Runge-Kutta
fourth order method to obtain an approximate value 
for the initial value
problem 
with
. (10)
Q.3 a. How
should the constant be chosen to ensure the fastest
possible convergence with the iteration formula
.
(8)
b.
Obtain a bound on the truncation error in linear interpolation based on the
nodal points 
.
Determine the maximum step-size 
that can be used in the tabulation
of 
in 
so that the
error in linear interpolation will be less than 
. (8)
Q.4 a. Solve the following system of equations using the Gauss-Seidel iteration method
with
the initial approximation as 
and 
. Perform two iterations. Find the
iteration matrix and hence find the rate of convergence of the method. (8)
a. Use the method of least
squares to fit a function of the form 
to the following data
(8)
Q.5 a.
Determine all the eigenvalues of the matrix 
, using Jacobi’s method. (Use exact
arithmetic). (8)
b. The
equation 
has
a simple root in the interval 
. The function 
is such that 
and 
for all 
in . Assuming that
the Newton-Raphson’s method converges for all initial approximations in , find the number
of iterations required to obtain the root correct to 
. (8)
Q.6 a. Find the Lagranges interpolation polynomial which fits the following data.
and
use the same to estimate the value of 
. (8)
b. Use
the two point Gauss-Legendre quadrature formula to evaluate 
. (8)
Q.7 a. The following table of values is given
Find
all the possible approximations for 
using the differentiation formula 
.
Perform Richardson’s extrapolation to obtain a better estimate. (8)
b. The
polynomial 
interpolates
the first four points in the table:
By
adding one additional term to 
, find a polynomial that
interpolates the whole of table. (8)
Q.8 a. Calculate
using
Trapezoidal rule with number of points as 3, 5 and 9. Improve the results using
Romberg Integration. (8)
b. Find
the inverse of the matrix 
using the Choleski factorization
method, taking 
,
where U is an upper triangular matrix. (8)
Q.9 a. Use
Taylor’s series method of order four to obtain the approximate value of 
for the initial
value problem 
.
Take
the step size 
. (8)
b. Perform four iterations of power method to find the largest eigen value in magnitude and the corresponding eigen vector of the matrix
Take
the initial approximation to the eigenvector as 
. (8)