Numerical and Statistical Methods for Computer — Summer 2020

Find the relative error if the number X 0.004997is

truncated to three decimal places. (ii) rounded off to three decimal places.

Find the negative root of x3 7x30by the bisection method correct up to three decimal places.

Using Gauss Jacobi method solve the following system of the equations: 8x y2z 13 x10y3z 17 3x2y12z 25

2 x Using trapezoidal rule to evaluate  dx, dividing the 2x20 interval into four equal parts.

By using Lagrange’s interpolation formula, find y(10). x 5 6 9 11 y 12 13 14 16

Using the Runge-Kutta method of fourth order, solve dy 10  x2  y2,y(0)1at x0.1,x0.2 taking h0.1 dx

Using Euler’s method find the approximate value of y at x1.5 dy yx taking h0.1. Given that  and y(1)2. dx xy

Using Newton Raphson method find the positive root of x4 x100correct up to three decimal places.

Fit a least square quadratic curve to the following data: x 1 2 3 y 1.7 1.8 2.3 3.2 Estimate y(2.4).

Find the regression coefficients b and b hence, find the yx yx correlation coefficient between x and y for the following data x 4 2 3 4 y 2 3 2 4 41

0.6 Using Simpson’s 1/3 rule, find  ex2dx, by taking n = 6.0

Using Newton’s divided difference formula, compute f(10.5) from the following data: x 10 11 13 17 f(x) 2.3026 2.3979 2.5649 2.8332

Solve x4 8x339x2 62x50by using Lin Bairstow method up to third iteration starting with p q 0.00

Find a real root of the equation xlog x1.2 by the regula falsi method.

The first four moments of distribution about x2are 1, 2.5, 5.5 and 16. Calculate the four moments about x and about zero.

dy Given that 2  y2 x2y2,y(0)1, y(0.1)1.06, dx y(0.2)1.12, y(0.3)1.21evaluate y(0.4)by Milne’s predictor- corrector method.

Find the arithmetic mean form the following data: Marks less 10 20 30 40 50 60 than No. of 10 30 60 110 150 180 students

(i) Obtain relation between  and E. (ii) Obtain relation between Dand E.

Obtain cubic spline for every subinterval from the following data x 0 1 2 f(x) 1 2 33 244

Two unbiased coins are tossed. Find expected value of number of heads.

1 sinx 1 By Simpson’s 3/8 rule, evaluate  dxtaking h . x 60

From the following table, estimate the number of students who obtained marks between 40 and 45. Marks 30-40 40-50 50-60 60-70 70-80 No. of 31 42 51 35 31 students

Using Budan’s theorem find the number of roots of the equation f(x)x4 4x33x210x80in the interval 1,0.

Find the positive solution of x2sinx0,correct up to three decimal places starting from x 2 and x 1.9. Using secant01 method.

Using Gauss Siedel method solve the following system of the equations: 3x0.1y0.2z 7.85 0.1x7y0.3z 19.3 0.3x0.2y10z 71.4