Anna University Chennai 2005 B.E NUMERICAL ANALYSIS

Code: A-07 Subject: NUMERICAL ANALYSIS & COMPUTER PROGRAMMING
Time: three Hours Max. Marks: 100
NOTE: There are 11 ques. in all.
ques. one is compulsory and carries 16 marks. ans to Q. 1. must be written in the space given for it in the ans book supplied and nowhere else.
ans any 3 ques. every from Part I and Part II. every of these ques. carries 14 marks.
Any needed data not explicitly given, may be suitably presumed and said.
Q.1 Choose the accurate or best option in the following: (2x8)
a. Using one-point Gauss-Chebyshev formula an approximate value of the integral is
(A) 0. (B) 2.40.
(C) 3.14. (D) 1.57.

b. The postfix notation for the subsequent infix notation is
(A) . (B) .
(C) . (D) .

c. The n-point Gauss Quadrature formula is exact for all polynomials of degree upto
(A) n. (B) 2n.
(C) two n - 1. (D) 2n +1.

d. If int s[5] is a 1 dimensional array of integers, which of the subsequent refers to the 3rd element in the array?
(A) (B)
(C) s + 3. (D) s + 2.

e. The interpolating polynomial that fits the data
f -1 0 2 4 5
f(x) 1 -1 1 11 19
is
(A) x2 - x - one . (B) x2 + x + one .
(C) x 2- x + 1 (D) x2 + x - 1
f. If is an eigenvalue of A, then the eigenvalue of A-1 is
(A) . (B)
(C) (D)

g. Given that k = four and m = 1, the value of k after execution of the statement k + = k > m is
(A) equal to 4. (B) equal to 8.
(C) equal to 9. (D) cannot be obtained because the
statement is invalid.

h. The rate of convergence of the Newton-Raphson method for finding a simple root of the formula f (x) = 0 is
(A) 1 (B) 3.
(C) 5. (D) 2.
PART I
ans any 3 ques.. every ques. carries 14 marks.
Q.2 Show that the subsequent 2 sequences have convergence of 2nd order with the identical limit
(i) (ii)
If xn is a close approximation to , show that the fault in the 1st formula is about one-third of that in the 2nd formula and deduce that the formula provide a sequence with third-order convergence. (14)

Q.3 a. Apply Cholesky's method to the matrix to find upper triangular matrix U such that A = UUT. (7)
b. Write a program in C to solve a system of linear equations using Jacobi's iteration method.

Q.4 a Derive Euler's method for solving the initial value issue . Use this method to find an approximate value of y(0.4) for the formula with h = 0.1.
b. Find the inverse of the matrix

using Gauss-Jordan method. (5)
c. Find the linear lowest squares polynomial approximation for the data (4)
xi -2 -1 0 1
fi 6 3 2 2

Q.5
define power method for finding the largest eigen value in magnitude of a square matrix. Perform 4 iterations of this method to obtain the largest eigen value in magnitude and the corresponding eigen vector of the matrix

Take the initial approximation to the eigenvector as . (14)

Q.6 a. Find a bound on the truncation fault in quadratic interpolation based on equispaced points with spacing h. Determine the largest step size h that can be used in tabulation of in [1,0] so that fault in quadratic interpolation is less than . (8)
b. Write a program in C that reads a variable x and computes the sum of series containing 25 terms.
PART II
ans any 3 ques.. every ques. carries 14 marks.
Q.7 a. Write a program in C using a recursive function for calculating values of Chebyshev polynomial by
(7)
b. For the provided data, use Newton divided difference interpolation to obtain f (2) and f (8) : (7)
x 4 5 7 10
f(x) 48 100 294 900
Q.8 a. Find a quadratic factor of the polynomial
starting with and using 1 iteration of the Bairstow's method. (7)

b. Use the classical Runge-Kutta 4th order method to find an approximate value of y(1.2) for the initial value issue with . (7)

Q.9 a. Use the 2 point Gauss-Legendre quadrature formula to evaluate (7)


b. Write a program in C that reads a text file and then computes the total number of words and phrases in it. (7)

Q.10 a. Calculate the nth divided difference of based on the points . (7)

b. Write the differences ranging from a structure and union in C. provide an example of every. (7)

Q.11 a. Determine p and q such that the order of the iteration method for calculating becomes as high as possible. obtain the order and the fault constant of this method. (8)

b. Gauss - Seidel iteration method is used to solve the system of equations

Find the iteration matrix. Show that the method converges. (6)

Earning:   Approval pending.