Numerical and Statistical Methods for Computer — Winter 2022

If the approximate solution of a problem is x0  3 5 . 2 5 with relative error of at the most 2 %. Find the range of values correct up to four decimal digits in which the exact value of the solution lie.

Find the positive root of x3 x110by the bisection method correct up to fourth approximation.

Using three iteration of Gauss Jacobi method solve the following system of the equations:2 x x x   3 y y0   y2  z0 z  z 1  7

Using Simpson’s 3/8 rule, evaluate 301 d  x x , taking n  603 .

By using Lagrange’s interpolation formula, find y ( 1 0 ) . x 5 6 9 11 y 12 13 14 16

Find the cubic splines and evaluate y ( 1 .5 ) and y(3). x: 1 2 3 y: 1 2 5 11

Fit a curve of the form yabx for the data using least square criteria and hence find the estimation for y when x = 8. x 1 2 3 4 5 6 y 87 97 113 129 202 195 193

Using Newton Raphson method find the square root of 10 correct up to three decimal places.

1 1 1 1 1 Prove that 1.  E2 E 2 2.  E2 E 2 2  

dy 2x Given that  y ,y(0)1. Using Runge–Kutta method of fourth dx y order, compute y for x = 0.2 and x = 0.4.

Use Trapezoidal rule to approximate the definite integral2 101 d  x x d x03 with n = 4.

Find Newton’s interpolating polynomial satisfying the data x 4 6 8 y 1 3 8 16

dy xy Use Euler’s method to solve the initial value problem  , on dx  0 , 3 07 with y(0)1, Compare the numerical solution with exact solution for the step size h = 0.25.

Find a real root of the equation x 3  x  1  0 by the regula falsi method.

Using Newton`s divided – difference interpolation, find f (1) and f(9) from the following table: x -1 0 2 5 y -2 -1 7 124 999

Given that d d y x  x 2  y 2 , y ( 0 )  1 , Compute y ( 0 .3 )07 by Milne’s predictor-corrector method h = 0.1.

Apply Budan’s theorem to find the number of the roots of the equation f ( x )  x 4  4 x 3  3 x 2  1 0 x  803 in the interval [-1, 0] and [0, 1].

The first moments of a distribution about the value 5 of variable are 2, 20, 40 and 50. Find the mean, variance1 and  .2

Find the roots of the equation x 4  9 x 3  3 6 x 2  5 1 x  2 7  0 to three decimal places to Lin – Bairstow method.

Develop Cprogram for secant method.

Compute the correlation coefficient between Xand Yusing the following data: x 2 4 5 6 8 11 y 18 12 10 8 7

Using Gauss Siedel method solve the following system of the equations: 30x2y3z 75 2x2y18z 30 x17y2z 48

In a lottery of 10,000 tickets, only one ticket bears a prize of Rs. 5,000. The price of a ticket is Rs. 100. Rakesh has on ticket of this lottery. Find his expectation.

For the following data find the two regression lines: x 1 2 3 4 5 6 7 8 9 y 10 12 16 28 25 36 41 49 40 50

dy Solve  x y,with y(0)1by Euler`s modified method for dx x  0 . 107 correct up to four decimal places by taking h = 0.05.