AMIETE – CS/IT (OLD SCHEME)

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 the best alternative in the following: (2x10)

a. If X is the true value of a quantity and is its approximate value, then the absolute error and relative errors are defined by

(A) (B)

(C) (D)

b. The root of the equation by bisection method after 2 iteration is

(A) 2.75 (B) 2.5

(C) 2.625 (D) 2.16

c. A matrix is said to be orthogonal if, (where stands for transpose of A).

(A) (B)

(C) (D)

d. If is the averaging or mean operator and is the central difference operator, then which is correct

(A) (B)

(C) (D)

e. The error in trapezoidal rule for finding numerical integration is of the order (where h is the interval width)

(A) h (B) h²

(C) h³ (D) h⁴

f. In Gauss-Jordan Elimination Method

(A) coefficient matrix is reduced to a diagonal matrix.

(B) coefficient matrix is reduced to a triangular matrix.

(C) coefficient matrix is reduced to a unit matrix.

(D) coefficient matrix is reduced to a null matrix.

g. Which interpolation method is used to find a tabulated value near the beginning of the table?

(A) Stirling’s formulae. (B) Langrage’s interpolation formulae.

(C) Newton’s forward formulae. (D) Newton’s backward formulae.

h. The rate of convergence of Newton Raphson method is

(A) cubic (B) quadratic

(C) linear (D) none of the above

i. The value of , correct to three decimal places by Simpson’s rd rule with h=0.5 is

(A) 0.7084 (B) 0.6945

(C) 0.3246 (D) 0.8956

j. Which of the method(s) is / are used for finding solutions of differential equations

(A) Euler method (B) Mid-point method

(C) Taylor’s series method (D) All of the above

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2 a. If a number 37.46235 is rounded off to four significant figures, then compute truncation error and percentage error. (3)

b. Find a root of the equation by using Regula-Falsi method for four iterations. (7)

c. Prove that the rate of convergence of Secant method is . (6)

Q.3 a. Solve the equations by using Gauss elimination method.

(7)

b. Solve the following equations by Jacobi’s iteration method

(9)

Q.4 a. Find all the eigen values and eigen vectors of the matrix

(12)

b. Write any four important properties of eigen values of a matrix. (4)

Q.5 a. From the following table, estimate f(9), using Langrage’s interpolation formulae. (7)

x	5	7	11	13	17
f(x)	150	392	1452	2366	5202

b. Find the polynomial of degree 2 such that f(0) = 1, f(1) = 3, f(3) = 55, using Newton divided difference interpolation. (9)

Q.6 a. Obtain the least square polynomial approximation of degree two for . (12)

b. Prove that (where is forward difference operator and interval of difference is 1). (4)

Q.7 a. Find from the following data (where ) (10)

x	3	5	11	27	34
f(n)	-13	23	899	17315	35606

b. Prove that, (where = forward difference operator & h = interval width) (6)

Q.8 a. From the following table, find out the area bounded by the curve and the x-axis from x = 7.47 to x = 7.52 (6)

x	7.47	7.48	7.49	7.50	7.51	7.52
f(x)	1.93	1.95	1.98	2.01	2.03	2.06

b. Evaluate the integral using Gauss-Legendre three point formulae. (10)

Q.9 a. Solve the initial value problem with h=0.2 on the interval [0, 0.4], by using fourth order Runge Kutta Method (10)

b. Find by Taylor’s series method the values of Y at x = 0.1 and x = 0.2 from . (6)