Numerical and Statistical Methods for Computer — Summer 2018

Define terms: Relative errors, Percentage errors, Rounding errors.

Explain Bisection method geometrically and using it find the roots of equation1 x 3  x  1 1  for three iterations.

Calculate the coefficient of correlation for the following distribution. x 5 9 15 19 24 28 32 y 7 9 14 21 23 29 30 f 6 9 13 20 16 11 equation equation

Using Budan’s method show that f (x)  x4  7x2  6x 1  0 has no or even number of roots in [-3, 0].

Find the interpolating polynomial for (0,2), (1,3), (2,12) and (5,147).

Fit a second degree parabola to the following data: x 0 1 2 3 y 1 1.8 1.3 2.5 6.3

Fit cubic Splines for first to interval from the following data: x 3 4.5 7 9 f ( x )07 2.5 10 2.5 0.5

Evaluate the integral 0 e x d x03 by Simpson’s 1/3 rule. (h=1)

Prove that 1. (1 )(1 ) 1, 2.    

Use Secant and Regula Falsi methods to determine the root of equation cos x  xex  0 taking the initial approximation as x  0 & x  1 at the end of five iterations. Compare01 the both results.

Evaluate 0 1 1 x d x using Trapezoidal rule taking h  1 / 503 .

Show that    E  1  E  1   1  E  1 .

Solve x4 8x3  39x2  62x  50 by Bairstow method upto third iteration with p  q  0 .00

By Newton’s Divided differences formula , find f (6): x 1 2 7 F(x) 1 5 5

Wages earned in Rupees per day by the labourers are given the table : Wages in 10-20 20-30 30-40 40-50 50-60 Rs. No. of 5 8 13 10 Labourers Find the median of the distribution.

Solve the following linear system of equation by Gauss elimination method with partial pivoting:2 x3 x1 x 1 x1 2 x2  1 0  51  3 x  x 2 x 2 2 x7 x  x x3 3 3  x6 x  x 4 x 4 4 ,1 ,3 . ,07

dy

Use Euler’s method, find the approximate solution of  x  y, y(0)  0 in four steps dx with h  2 .

Find the mode for the following distribution: Class 0-10 10-20 20-30 30-40 40-50 50-60 60-70 F 4 7 8 12 25 18

Apply Gauss-Seidel iteration method to solve the following system of equations:1   0 x x x1 x1 1 1   x x2 x0   2 x2 x  0 x x3 x3  3 1 x 2 x x x 4    ,1 5 , , .07

Find P ( A  B ) , given that P ( A ) 1 , P ( B ) 1 & P ( A  B ) 1 .03

Evaluate0 s i n4 c o s d       by Simpson’s3 th rule, taking h6  04 .

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

Find 4y , from the following data: x 1 3 5 7 9 y 3 8 13 15 18

From the following distribution , find the mean and variance: x 1 2 3 4 f ( x )04 0.1 0.1 0.2 0.3 0.

Use fourth order Runge-Kutta method to find y(0.2) with h  0 .1 0.3 , given that dy 10  x2  y2, y(0) 1. dx