GTU Computer Engineering (Semester 4)

Numerical And Statistical Methods For Computer Engineering

May 2015

Numerical And Statistical Methods For Computer Engineering

May 2015

1 (a) (i)
Discuss briefly the various types of errors in performing numerical calculations.

4 M

1 (a) (ii)
Define ill-conditional and well conditional of linear equations.

3 M

1 (b)
The population of the town is given below. Estimate the population for the year 1895 and 1930 using suitable interpolation.

Year | 1891 | 1901 | 1911 | 1921 | 1931 |

Populationin thousand | 46 | 66 | 81 | 93 | 101 |

7 M

2 (a)
Derive Newton- Raphson method in brief.

7 M

Answer any one question from Q2 (b) & Q2 (c)

2 (b)
Find positive root of an equation x

^{3}+x^{2}-1=0 by iteration method correct to four decimal places.
7 M

2 (c)
Find smallest positive root of an equation x-e

^{-x}=0 using Regula Falsi method correct to four significant digits.
7 M

Answer any two question from Q3 (a), (b) & Q3 (c), (d)

3 (a)
By Gauss Seidel method solve the following system

2x+y+6z=9

8x+3y+2z=13

x+5y+z=7

2x+y+6z=9

8x+3y+2z=13

x+5y+z=7

7 M

3 (b)
Fit a second degree polynomial using least square method to data given below:

X | 0 | 1 | 2 | 3 | 4 |

Y | 1 | 1.8 | 1.3 | 2.5 | 6.3 |

7 M

3 (c)
Solve the following equations using Gauss Elimination

x+y+2z=4

3x+y-3z=-4

2x-3y-5z=-5

x+y+2z=4

3x+y-3z=-4

2x-3y-5z=-5

7 M

3 (d)
Obtain the cubic splines for the first two subinterval to following data:

X | 1 | 2 | 3 | 4 |

Y | 1 | 2 | 5 | 11 |

7 M

Answer any two question from Q4 (a), (b) & Q4 (c), (b)

4 (a) (i)
Write an algorithm for Simpson's 3/8 rule to integrate the tabulated function.

4 M

4 (a) (ii)
Evaluate \(\int^1_0 \dfrac {1}{1+x^2} dx \) using Trapezoidal rule.

3 M

4 (b)
Solve initial value problem \(\dfrac {dy}{dx} = x \sqrt{y}, \ y(1)=1 \) and hence find y(1.5) by taking h=0.1 using Euler's method.

7 M

4 (c) (i)
Write an algorithm for Lagrange's interpolation method to find functional value.

4 M

4 (c) (ii)
Construct Divided difference table for the data given below

X | -4 | -1 | 0 | 2 | 5 |

f(x) | 1245 | 33 | 5 | 9 | 1335 |

3 M

4 (d)
Solve boundary value problem \(\dfrac {d^2y}{dx^2} = \dfrac {dy}{dx}, y(0)=0 \ and \ y(1)=1.17 \)

7 M

Answer any two question from Q5(a), (b) & Q5 (c), (d)

5 (a)
Develop a C program of Runge-Kutta second order method to solve ordinary differential equation.

7 M

5 (b)
Obtain the two regression lines from the following data and hence find the correlation coefficient.

X | 6 | 2 | 10 | 4 | 8 |

Y | 9 | 11 | 5 | 8 | 7 |

7 M

5 (c)
Develop a C program to fit regression line x on y through set of points using method of least squares.

7 M

5 (d)
Assume a four yearly cycle and calculate trend by method of moving averages from the following data relating to the production in pen drives in india.

Year | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 |

Production(million kgs) | 464 | 515 | 518 | 467 | 502 | 540 | 557 | 571 | 586 | 612 |

7 M

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