GTU Computer Engineering (Semester 4)

Numerical And Statistical Methods For Computer Engineering

December 2015

Numerical And Statistical Methods For Computer Engineering

December 2015

1(a)1
Using method of successive approximation solve the equation \[2x-\log _{10}x=7\] correct to four decimal places.

3 M

1(b)1
Discuss briefly the different types of errors encountered in performing numerical calculations.

3 M

1(b)2
Using method of False-position, compute the real root of the equation \[x\log_{10}x-1.2=0\] correct to four decimals.

4 M

1(b)2
Use Newton-Raphson method to find smallest positive root of f(x) = x

^{3}-5x+1=0 correct to four decimals.
4 M

2(a)
Solve this system of linear equations using Jacobi's method in three iterations first check the co-efficient matrix of the following systems is diagonally dominant or not?

20x+y-2z=17

2x-3y+20z=25

3x+20y-z=-18

20x+y-2z=17

2x-3y+20z=25

3x+20y-z=-18

7 M

Solved any one question from Q.2(b) & Q.2(c)

2(b)1
State Budan's theorem
And hence show that

p(x)=x

p(x)=x

^{5}-x^{4}-3x^{3}+2x+5 has one root in [-2, -1].
3 M

2(b)2
Apply Budan's theorem to find the no. of roots of the equation

x

x

^{5}+x^{4}-4xx^{3}-3xx^{2}+3x+1 in the interval [ - 2, -1], [0,1] & [1, 2].
4 M

2(c)
Perform two iterations of the Bairstow method to extract a quadratic factor from the polynomial

p(x)=x

p(x)=x

^{3}+x^{2}-x+2=0.
7 M

Solved any one question from Q.3 & Q.4

3(a)
State the Direct & iterative methods to solve system of linear equations. Using Gauss-Seidel method, solve

2x

-x

-x

2x

_{1}-x_{2}=7-x

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

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

3(b)1
Define ill-conditional linear systems condition number of the matrix
of equations. Determine the condition number of the matrix \[A=\begin{bmatrix}
1 & 4 & 9\\
4 & 9 & 16\\
9 & 16 & 25
\end{bmatrix}\] .

3 M

3(b)2
From the following data find the value of x when y f ( x ) = 0.390 .

x | 20 | 25 | 30 |

y=f(x) | 0.342 |
0.423 |
0.500 |

4 M

4(a)
Obtain the cubic Spline approximation for the function defined by the data.

Hence find an estimate of f (2.5)

x | 0 | 1 | 2 | 3 |

f(x) | 1 |
2 |
33 | 244 |

x | 0 | 1 | 2 | 3 |

f(x) | 1 |
2 |
33 | 244 |

Hence find an estimate of f (2.5)

7 M

4(b)i
Fit a straight line for the data.

y | 12 | 15 | 21 | 25 |

x | 50 |
70 |
100 | 120 |

3 M

4(b)ii
The following table gives distance (in nautical miles) of the visible
horizon for the given heights (in feet) above Earth's surface. Find the
values of y when x = 390 feet.

Height(x) | 100 | 150 | 200 | 250 | 300 | 350 | 400 |

Distance(y) | 10.63 | 13.03 | 15.04 | 16.81 | 18.42 | 19.90 | 21.47 |

4 M

Solved any one question from Q.5 & Q.6

5(a)1
Use Euler's method to find an approximation value of y at x = 0.1 for the initial value problem \[\dfrac{dy}{dx}=x-y^2;y(0)=1\] .

3 M

5(a)2
Find the least squares approximations of second degree for the following data

x | -2 | -1 | 0 | 1 | 2 |

y=f(x) | 15 | 1 | 1 | 3 | 19 |

4 M

5(b)
Solve the initial value problem \[\dfrac{dy}{dx}=-2xy^2;y(0)=1\] with h=0.2 for y(0, 2) using Runge-Kutta fourth order method.

7 M

6(a)1
Evaluate \[\int ^5_1\log_{10}x dx\] taking 8 subintervals by Trapezoidal rule.

3 M

6(a)2
Evaluate \[\int ^1_0\dfrac{dx}{1+x}\] using Simpson 3/8 rule

4 M

6(b)
State different predictor-corrector method. For the initial value problem \[\dfrac{dy}{dx}=y+x^2;y(0)=1\] , use Milne's prediction-corrector method to find y(0.8) by taking h=0.2 from following data

x | 0 | 0.2 | 0.4 | 0.6 |

y | 1 | 1.2242 | 1.5155 | 1.9063 |

7 M

Solved any one question from Q.7 & Q.8

7(a)
From the following data calculate moments about (i) Assumed mean 25
(ii) Actual mean (iii) zero.

Variable | 0-10 | 10-20 | 20-30 | 30-40 |

Frequency | 1 | 3 | 4 | 2 |

7 M

7(b)
Explain co-relation, co-relation Types, co-relation co-efficient. Also state the methods to find correlation between two variables. Find the correlation co- efficient between the serum diastolic blood pressure & serum cholesterol levels of 10 randomly selected persons.

Persons | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Cholesterol | 307 | 259 | 341 | 317 | 274 | 416 | 267 | 320 | 274 | 336 |

Diastolic B.P. |
80 | 75 | 90 | 74 | 75 | 110 | 70 | 85 | 88 | 78 |

7 M

8(a)
The quantities of water (in liter) supplied by municipal corporation on ten consecutive days in certain area are shown below:

218.2, 199.7, 207.3, 185.4, 213.7, 184.7, 179.5, 194.4, 224.3, 203.5.

Evaluate the mean & the first four central moments of the water (in liter) of that area.

218.2, 199.7, 207.3, 185.4, 213.7, 184.7, 179.5, 194.4, 224.3, 203.5.

Evaluate the mean & the first four central moments of the water (in liter) of that area.

7 M

8(b)
State the formula for two regression equations. Also give algorithm for the following data find the line of regression of y on x

x | 1.53 | 1.78 | 2.60 | 2.95 | 3.42 |

y | 33.5 | 36.3 | 40.0 | 45.8 | 53.5 |

7 M

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