# Computer Based Numerical and Statistical Techniques

Note : (1) Attempt all questions.

(2) All questions can equal marks.

1. Attempt any four parts of the following :

(a) Discuss two important computer arithmetic systems. Illustrate with examples that associatiative laws of floating point arithmetic do not hold in numerical computation, ,-2 .r4 x6

(b) Derive the series : cos r – 1 + — – — . 2 ! 4 ! 6 ! Compute the number of terms required to estimate cos(rt/4) so that the result is correct to atleast two significant digits.

(e) In a triangle A ARC, a = 6 cm, c = 15 cm, Z B = 90°. Write a program in ‘C’ to find the absolute error in the computed value of A, if possible errors in a and c are 1/5% and 1/7% respectively. JJ-9967] llilll |f:ll IIIMI II 1

(d) Develop an iteration formula to find a real root of the equation x sin x + cos x = 0 Find it in the vicinity of xl)=K.

(e) Use the iteration method to find a real root of the equation : 3x- x/t + sin x = 0 correct to five decimal places.

(f)  Use Muller’s method to obtain a root of the equation : cos x-x ex =0 in the interval (0. 1).

2 . Attempt any four parts of the following :

(a) Prove : A-Y=-AY

(b) Estimate the missing term in the table :

 .V 0 1 2 4 /M 1 “•> 9 9 81

(c) Apply Stirling’s formula to find a polynomial of degree three which takes the following values of

x, .r :

 x 2 4 6 8 10 V -2 1 3 8 20

(d) Write an algorithm of any central difference interpolation formula.

(e) Apply Langrange’s formula to find a cubic polynomial which approximates the data :

 X -1 Z. -> y{x) -12 -8 •i 5

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(f)  A function f(x) satisfies the conditions : /(0)= 1 , /'(0)=i, /'(1) = 0, /'(l) = 0. Use Hermite interpolation to approximate f(x) by a polynomial. Also evaluate the maximum value of f(x) in [0, l].

3. Attempt any two parts of the following :

(a) State the importance of numerical differentiation.

Find /'(0.6) and /”(0.6) from the following table :

 ,v 0.4 0.5 0.6 0.7 0.8 A*) 1.5836 1.7974 2.0442 2.3275 2.651

(b) State the need and scope of numerical integration. Use Simpson’s rule to estimate the integral.

(2 .- J ex c/x with a stepsize 0.5,

(c) The area A inside the closed curve. y2 + .v2 = cos v a   t ‘ ‘ o is given by A – 4jo |cosx- x2 j 2 c/x where a is the positive root of the equation cos x = x~ . Compute the area with an absolute error less than 0.05.

4. Attempt any.two parts of the following :

(a) Apply Runge-Kutta fourth order method to find y{0.1), v(0.2) and >'(0.3) for the initial value

problem. — = xy/r y\ y (0) = 1 . Also, find y (0.4 j using Adam’s method.

(b) Solve the initial value problem :

y’ – x + sin (rcy); v(l) = 0, 1 < x < 2 by Milne’s predictor-corrector method

(c) Discuss the stability of Euler’s method applied to the initial value problem, y’ – Xy, J'(-Vq) = j’o .

5. Attempt any two parts of the following :

(a) State various methods for curve-fitting. Obtain the cubic splines approximation for the function given by the following table :

 X 0 1 2 3 /(*) 1 o 5 11

with the end conditions Mq = 0 =

( b) State objectives of control charts. A drilling machine bores holes with a mean diameter of 0.5230 cm and a standard deviation of 0.0032 cm. Calculate the 2-sigma and 3-sigma upper and lower control limits for means of sample of 4.

(c) Define lines of regression. Find the lines of regression for the given data :

 X 50 100 150 200 250 300 350 y 30 65 90 130 150 190 200

Also find the coefficient of correlation for the above data.