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GTU Computer Engineering (Semester 4)
Numerical And Statistical Methods For Computer Engineering
May 2015
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

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 Population in 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 x3+x2-1=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
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
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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