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GTU Computer Engineering (Semester 4)
Numerical And Statistical Methods For Computer Engineering
May 2016
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

Short Questions
1(a) Define: Accuracy and Precision
1 M
1(b) If a=0.8461538461 is approximated by 0.84615 then find percentage relative error.
1 M
1(c) What is convergence rate of Bisection method and Newton Raphsonmethod.
1 M
1(d) Write any two pitfalls of Newton Raphson method.
1 M
1(e) The error caused by truncating an infinite series to a finite number of terms is called _______ and the error associated with chopping and rounding is called _______ .
1 M
1(f) Check the following system is diagonally dominant or not. Justify your answer.
10x ' 4y + z = 7; x + 5y ' 2z = 5; 8x ' 4y ' 3z = 6
1 M
1(g) Employ partial pivoting to the following system of equations:
4x + 2y ' z = -2; 5x + y + 2z = 4; 6x + y + z = 6
1 M
1(h) Write appropriate Simpson's integration formula to solve the integration
$$\int ^{1.8}_0 f(x)dx,$$ dividing the interval into 9 equal parts.
1 M
1(i) Define ill-conditioned system.
1 M
1(j) Can you apply False position method to obtain a root of the equation f(x) = xex - 2 = 0 in the interval (0, 0.5)? Justify your answer.
1 M
1(k) Find the arithmetic mean of the following frequency distribution.
 x: 1 2 3 4 f: 4 5 2 1
1 M
1(l) What is mode of the following frequency distribution?
 Data values x: 1 2 3 4 frequency f: 4 7 10 8
1 M
1(m) What is the approximate value of the $$\int ^2_1 f(x)dx,$$ using trapezoidal rule with h = 1, where f(1) = 2, f(2) = 4.
1 M
1(n) Find the approximate root of the equation f(x) = x3 + x = 0 after the first iteration of Newton Raphson method with initial guess x0 = 1.
1 M

2(a) Discuss the steps of an engineering problem solving.
3 M
2(b) Perform three iterations of Bisection method to obtain a root of the equation f(x) = cos(x) ' xex = 0 in the interval (0.5, 1).
4 M
Solve any one question from Q.2(c) & Q.2(d)
2(c) 1) Test the convergence condition for the equation $$x=\dfrac{1}{3}\left ( \cos (x)+1 \right )$$ in the interval $$\left ( 0,\dfrac{\pi}{2} \right )$$ and then solve the equation using successive approximation method correct up to three places of decimals taking initial guess as x0 = 0.5.
2)Apply Budan's theorem to the equation x4 - 7x2 + 6x ' 1 = 0 to draw the inference about the roots in the interval (-2, -1).
7 M
2(d) Perform one iteration of the Bairstow method to extract a quadratic factor x2 + px + q from the polynomial x4 + x3 + 2x2 + x + 1 = 0. Use the initial approximation r = 0.5, s = 0.5. Also, calculate the relative approximate error in r and s after first iteration.
7 M

Solve any three question from Q.3(a), Q.3(b), Q.3(c) & Q.3(d), Q.3(e), Q.3(f)
3(a) Write an algorithm to fit a straight line using least square method.
3 M
3(b) The following system of equations was generated by applying mess current law to the circuit. Use Gauss Elimination method to find the current in the circuit.
2I1     -I2     +3I3 = 8
-I1     +2I2     +I3 = 4
3I1     +I2     -4I3 = 0
4 M
3(c) State the Direct & iterative method to solve system of linear equations. Arrange following system of equations into diagonally dominant form and solve it using Gauss Seidel method.
10x1     +x2     +x3 = 12
2x1     +2x2     +10x3 = 14
2x1     +10x2     +x3 = 13
7 M
3(d) Write an algorithm for Simpson's 1/3rd rule to integrate the tabulated function.
3 M
3(e) A train is moving at the speed of 30m/sec. Suddenly brakes are applied. The speed of the train per second after t seconds is given by the following table.
 Time (t) 0 5 10 15 20 25 30 Speed (v) 30 24 19 16 13 11 10

Apply Simpson's three-eight rule to determine the distance moved by the train in 30 seconds.
4 M
3(f) Obtain cubic Splines approximation for the following data and hence compute f (1.5).
 X 1 2 3 f(x) -8 -1 18
7 M

Solve any three question from Q.4(a), Q.4(b), Q.4(c) & Q.4(d), Q.4(e), Q.4(f)
4(a) Use following data to evaluate f (2.5).
 x: 0 1 2 3 f(x): 1 2 1 10
3 M
4(b) Use following data to construct a Lagrange's polynomial of degree two.
 x: 0 0.6 1.2 f(x): 1 0.825336 0.362358
4 M
4(c) From the following data obtain the two regression lines and the correlation coefficients.
 x: 100 98 78 85 110 93 80 y: 85 90 70 72 95 81 74
7 M
4(d) Using Euler's method compute y (0.3) for the initial value problem y' = y2 - x2, y(0) = 1 taking the step size h = 0.1.
3 M
4(e) Use the Runge-Kutta 4 th order method with h=0.1 to find the approximate solution for y(1.1) for the initial value problem $$\dfrac{dy}{dx}=2xy, y(1)=1$$
4 M
4(f) Fit a polynomial of degree two using least square method for the following experimental data. Also estimate y(2.4)
 x: 1 2 3 4 5 y: 5 12 26 60 97
7 M

Solve any three question from Q.5(a), Q.5(b), Q.5(c) & Q.5(d), Q.5(e), Q.5(f)
5(a) Find standard deviation from the following data.
 Class 9-11 12-14 15-17 18-20 Frequency 2 3 4 1
3 M
5(b) Find correlation coefficient for the data given below.
 x: 4 5 9 14 18 22 24 y: 16 22 11 16 7 3 17
4 M
5(c) Use the finite difference approach with h=0.25 to solve the boundary value problem y'' = x + y, y (0) =1, y (1) =1.
7 M
5(d) In a college, IT department has arranged one competition for IT students to develop an efficient program to solve a problem. Ten students took part in the competition and ranked by two judges given in the following table. Find the degree of agreement between the two judges using Rank correlation coefficient.
 Ist Judge 3 5 8 4 7 10 2 1 6 9 IInd Judge 6 4 9 8 1 2 3 10 5 7
3 M
5(e) The following data represents the number of foreign visitors in a multinational company in every 10 days during last 2 months. Use the data to find median.
 x: 0-10 10-20 20-30 30-40 40-50 50-60 No. of visitors f: 12 18 27 20 17 06
4 M
5(f) The table below shows the demand for a new hard disk for each of the last 7 months.
 Month 1 2 3 4 5 6 7 Demand 23 29 33 40 41 43 49

1) Calculate a two month moving average for months two to seven.
2) What would be your forecast forthe demand in month eight?
3)Calculate Mean Square Error(MSE).
7 M

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