2007 Anna University B.E : NUMERICAL COMPUTING Question paper

Course: B.E

University: Anna University

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 to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.
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
Q.1 Choose the correct or best alternative in the following: (2x8)
a. Newton - Raphson method is to be applied to find the value of . Then, the formula can be written as
(A) (B)
(C) (D)
b. The divided difference f[x0,x1,x2] is equal to
(A) (B)
(C) (D)
Where x0,x1,x2 are equispaced points with spacing h and is the
forward difference operator.
c. Attempt is made to solve the system of equations Ax=b, 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.
d. The interpolating polynomial that fits the data
x 1 2 3 4
f(x) -1 -1 1 5
is
(A) x2 + 3x + 1. (B) x2 - 3x + 1.
(C) 2x2 - 6x + 3. (D) x2 -5x + 1.
e. The integral is evaluated by Simpson's rule using 3 points. Then, the value of I is equal to
(A) (B) 23 / 60
(C) 37 / 60. (D) 47 / 60.
f. The least squares straight line approximation to the data
x 1 2 3 4
f(x) -1 1 3 5
is given by
(A) 2x - 3 (B) 2x + 3
(C) x + 4 (D) 2x - 4
1
g. A numerical differentiation formula for finding is given by

Then, the value of a for which the method is of highest order is given by
(A) -2. (B) 0.
(C) -1. (D) 2.
h. The integration formula is to be used. The value of a for which the method is of highest order, is given by
(A) 1. (B) 2/3.
(C) 1/3. (D) 1/2.
PART I
Answer any THREE Questions. Each question carries 14 marks.
Q.2 a. The equation x2 - 2x -3 cosx = 0 is given. (6)

(i) Locate the smallest root in magnitude, in an interval of length one unit.
(ii) Hence, find this root correct to 3 decimals using the secant method.
b. A method for determining , where N is a positive real number, is written as . Determine the values of the parameters a, b, c such that the order of the method is as high as possible. (8)

Q.3 a. The system of equations has a solution near x = 1.5, y = -1.0 Set up the Newton's method for solution and iterate once. (7)
b. Using the Cholesky method, solve the system of equations (7)

Q.4 a. Solve the system of equations (7)

using the Gauss elimination method with partial pivoting.
b. Find the inverse of the matrix using the Gauss - Jordan method. (7)
2
Q.5 a. Perform 4 iterations of the Gauss - Seidel method for finding the solution of the linear system of equations

Assume the initial approximation as and x3 = 0.5. Find the iteration matrix and hence determine the rate of convergence of the method. (8)

b. Find all the eigenvalues of the matrix , using the Jacobi method. (Use exact arithmetic) (6)
Q.6 a. Find the smallest eigenvalue in magnitude and the corresponding eigenvector of the matrix , using the inverse power method. Assume the initial approximation to the eigen vector as . (8)
b. Transform the matrix to tri-diagonal form using the Given's method. Set up the Sturm sequence and find the smallest eigenvalue in magnitude. (6)
PART II
Answer any THREE Questions. Each question carries 14 marks.
Q.7 a. A table of values is to be constructed for the function f(x)=(1+x)5 on [1,2]. If the linear interpolation is to be used on this table of values, find the largest step size that can be used so that the error is bounded by 5x10-4. (7)

b. Obtain the unique polynomial P(x) of degree 3 or less corresponding
to a function f(x), where f(0)=1, f'(0) = 2, f(1)=5, f'(1)=4. (7)

Q.8 a. If f(x)=u(x)v(x), find the divided difference f[x0,x1] in terms of u(x0), v(x0) and the divided differences u[x0,x1], v[x0,x1]. (5)

b. If are the forward and backward differences respectively,
show that . (4)
3
c. Find the interpolating polynomial which fits the data (5)
x 0.1 0.2 0.3 0.4 0.5 0.6
f(x) 0.93 0.92 0.97 1.08 1.25 1.48

Q.9 a. Use the method of least squares to fit a function of the form to the following data (7)
x 1 2 3 4 5
y 5 3.5 3 2.7 2.5
b. The following table of values is given
x 0.2 0.3 0.4 0.5 0.6
y(x) 1.8054 1.5769 1.2834 0.9483 0.5981
Find all the possible approximations for y''(0.4) using the differentiation formula.

Perform Richardson's extrapolation to obtain a better estimate. (7)

Q.10 a. The generalised trapezoidal rule

where p is a constant and is given. Find the value of the constant p. Deduce the composite rule for evaluating the integral
(8)
b. Evaluate the integral using the Gauss-Legendre two-point and three point integration rules. (6)

Q.11 a. Find approximations to with the Taylor's series method of second order and step length h = 0.2, where y(x) is the solution of the initial value problem (8)

b. The initial value problem is given. Find an approximation to y(1.2), when h = 0.2, using the Runge - Kutta method

for the solution of differential equation (6)

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