Code: A-20

AMIETE – ET (OLD SCHEME)

Code: AE07 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 the best alternative in the following: (210)

a. If we take each correct to six digits in decimal system then the value of is correct to

(A) 3 digits (B) 4 digits

(C) 5 digits (D) 6 digits

b. Identify the number of True statements among the following:

(i) An iterative method is said to be of order p, if there exists a non-zero constant C and p is the largest positive real number such that

is satisfied where is the error at the k-th iteration

(ii) The rate of convergence of Secant method is p=1

(iii) The Regula-Falsi method has linear rate of convergence

(A) 1 (B) 2

(C) 3 (D) None of the above

c. Suppose the coefficient matrix A of a given system of equations is decomposed in to

A=LU

where L and U are the lower and upper triangular matrices respectively. If we choose the diagonal elements of L to be equal to the value 1 then the method is called

(A) Gauss-Jordan method (B) Doolittle’s method

(C) Crout’s method (D) None of the above

d. For the following values given

x ( in degrees ) 10 20 30

f (x) 1.1585 1.2817 1.3660

using quadratic interpolating polynomial f(.) that fits the data, find

(A) 1.0729 (B) 1.1925

(C) 1.2246 (D) None of the above

e. The following table of values is given:

x -1 1 2 3 4 5 7

f(x) 1 1 16 81 256 625 2401

Using the formula and the Richardson extrapolation, find

(A) 108 (B) 115

(C) 127 (D) None of the above

f. Identify the correct statements from the following

(i) The problem of Least Squares approximation is a minimization problem

(ii) The Legendre polynomials defined on [-1,1] are orthogonal polynomials

(iii) The Chebyshev polynomials are defined on [-1,1] by,

(A) (i) & (ii) (B) (i) & (iii)

(C) (ii) & (iii) (D) (i), (ii) & (iii)

g. Simpson’s three-eighth rule of numerical integration is exact for polynomials of degree up to

(A) 1 (B) 2

(C) 3 (D) any finite degree

h. The value of the integral

using 1-point Gauss-Chebyshev formula will be

(A) 2.1276 (B) 2.5672

(C) 2.9831 (D) None of the above

i. The order of convergence of Newton-Raphson method is

(A) 1 (B) 2

(C) 3 (D) 4

j. The value of y corresponding to x=0.1 for the differential equation

Using Euler’s method.

(A) 1.10 (B) 1.36

(C) 1.94 (D) 2.19

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2 a. Show that the Newton-Raphson method for finding the root of the equation f(x)=0 has second order convergence. (8)

b. Write a C program to find a simple root of the equation of f(x)=0 by the Regula-Falsi method. The inputs are : (i) x0, x1 (the initial interval in which the root lies), (ii) maximum number of iterations, (iii) the error tolerance ‘tol’. The outputs are: (i) approximate root (ii) number of iterations taken. If the input value of ‘n’ is not sufficient then your program should give an error message: “Iterations not sufficient”. Also write a function to evaluate f(x) where . (8)

Q.3 a. Obtain a second degree polynomial approximation to

using the Taylor series expansion about x=0. Use the expansion to approximate f(0.05) and find a bound of the truncation error (8)

b. Perform three iterations of the Newton-Raphson method to solve the system of equations

by taking the initial approximation as (8)

Q.4 a. Solve the following system of equations using Gauss elimination with partial

pivoting (8)

b. Using the Gauss-Seidel method, solve the system of equations

starting from (0,0,0) up to 5 iterations. (8)

Q.5 a. Differentiate the following:

(i) Call by values and Call by reference in C program

(ii) Structures & Unions (4+4)

b. A polynomial fits the points (1,4), (3,7), (4,8) and (6,11). Using Newton’s divided difference formula interpolate the value of y at x=2. (8)

Q.6 a. Find the least-squares approximation of second degree for the discrete data

x -2 -1 0 1 2

f(x) 15 1 1 3 19 (8)

b. Determine the step size that can be used in the tabulation of f(x)=sin x in the interval at equally spaced nodal points so that the truncation error of the quadratic interpolation is less than . (8)