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 root of the equation is obtained by using the
Newton-Raphson method with initial approximation
. The root obtained after one
iteration is
(A) 3.5151. (B) 3.5306.
(C) 4.5151. (D) 4.5306.
b. The rate of convergence of the secant method to find a simple root
of is m and that of the Regula-Falsi
method is n. Then
(A) m < n. (B) m > n.
(C) m = n. (D) m = 2n.
c. The rate of convergence of the Gauss-Jacobi method for solving the system of equations
is
(A) 0.98. (B) 0.85.
(C) 0.56. (D) 0.49.
d. Given’s method is used to reduce the matrix to tri-diagonal
form. The angle of orthogonal transformation is
(A) . (B)
.
(C) . (D)
.
e. The interpolating polynomial which fits the data
is
(A) . (B)
.
(C) . (D)
.
f. The operator is equivalent to the operator
(A) . (B)
.
(C) . (D)
.
g. The least squares polynomial approximation of degree 1 to the
function on
is a + bx. The value of b is
(A) . (B)
.
(C) . (D)
.
h. The truncation error of the method
is
(A) . (B)
.
(C) . (D)
.
i. The value of the integral using
Gauss-Legendre two-point method is
(A) . (B)
.
(C) . (D)
.
j. The approximate value of y(1.2) for the initial value problem
obtained by using the Taylor series second order method with h=0.2 is
(A) 2.84. (B) 2.65.
(C) 1.84. (D) 1.65.
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. Perform three iterations of
the Regula-Falsi method to obtain the approximate value of which lies between 3 and 3.2. To
how many decimal digits is this value accurate? (8)
b. Find the rate of convergence and the asymptotic error constant for the method
to
obtain a simple root of the equation . (8)
Q.3 a. Perform two iterations of the Newton’s method to solve the system of equations
Take
the initial approximation as . (9)
b. Using LU decomposition method solve the system of equations
(7)
Q.4 a. Obtain the inverse of the matrix
Using
Gauss-Jordan method. Hence solve the system of equations where
.
(9)
b. Show that the Gauss-Jacobi iteration method for solving the system of equations
is divergent. (7)
Q.5 a. Find all the eigenvalues and the corresponding eigenvectors of the matrix
Using Jacobi’s method. (8)
b. Find the least squares
approximation of the form to
on the interval
.
(8)
Q.6 a. Prove the following operator relations
(i)
(ii) (4)
b. Using Lagrange interpolation and the data
obtain the interpolating polynomial which fits this data. Hence obtain an
approximate value of . (6)
c. Obtain
second degree Taylor series polynomial approximation to the function about x = 2. Find the maximum
absolute error if this approximation is used in the interval
. (6)
Q.7 a. Obtain the maximum
absolute truncation error and the maximum
absolute round
off error in
the method
. Assuming that
is the maximum
round
off error in evaluating
f(x) and ,
where
,
determine the optimal
step size h so that . (8)
b. Derive the Richardson’s extrapolation scheme for the method
Using this method and the corresponding Richardson’s extrapolation, obtain the
best value of using the data
(8)
Q.8 a.
Evaluate the
integral using Simpson’s
rule of integration with 3 and 5 nodal points. Find the best value using
Romberg integration. (8)
b. Obtain the values of the constants and
so that the method
is of highest possible order. Determine the order and error term of the method.
Use this method to find
the value of .
(8)
Q.9 a. Obtain the order and truncation error of the method
for solving the initial value problem .
(8)
b. Using the classical fourth order Runge-Kutta method with h = 0.2, obtain the
approximate value of y
(0.4) for the initial value problem. (8)