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.

1 M

1(j)
Can you apply False position method to obtain a root of the equation f(x) = xe

^{x}- 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) = x

^{3}+ x = 0 after the first iteration of Newton Raphson method with initial guess x_{0}= 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) ' xe

^{x}= 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 x

2)Apply Budan's theorem to the equation x

_{0}= 0.5.2)Apply Budan's theorem to the equation x

^{4}- 7x^{2}+ 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 x

^{2}+ px + q from the polynomial x^{4}+ x^{3}+ 2x^{2}+ 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.

2I

-I

3I

2I

_{1}-I_{2}+3I_{3}= 8-I

_{1}+2I_{2}+I_{3}= 43I

_{1}+I_{2}-4I_{3}= 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.

10x

2x

2x

10x

_{1}+x_{2}+x_{3}= 122x

_{1}+2x_{2}+10x_{3}= 142x

_{1}+10x_{2}+x_{3}= 13
7 M

3(d)
Write an algorithm for Simpson's 1/3

^{rd}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 |

7 M

4(d)
Using Euler's method compute y (0.3) for the initial value problem y' = y

^{2}- x^{2}, 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.

I Judge |
3 | 5 | 8 | 4 | 7 | 10 | 2 | 1 | 6 | 9 |

II 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).

7 M

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