STUPIDSID
GTU Computer Engineering (Semester 4)
Numerical And Statistical Methods For Computer Engineering
May 2016
Numerical And Statistical Methods For Computer Engineering
May 2016
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
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
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.
\( \int ^{1.8}_0 f(x)dx, \) dividing the interval into 9 equal parts.
1 M
1(i)
Define ill-conditioned system.
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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).
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
2I1 -I2 +3I3 = 8
-I1 +2I2 +I3 = 4
3I1 +I2 -4I3 = 0
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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
10x1 +x2 +x3 = 12
2x1 +2x2 +10x3 = 14
2x1 +10x2 +x3 = 13
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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.
Apply Simpson's three-eight rule to determine the distance moved by the train in 30 seconds.
| 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 | 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 |
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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.
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).
| 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).
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