Numerical and Statistical Methods for Computer — Winter 2018

Discuss about mathematical modeling.

Discuss various types of errors used for numerical calculations.04

Obtain cubic spline approximation for the function defined by the data given below for the first two subintervals. Take1 M0  M3  0 . x 0 1 2 f ( x )07

Write an algorithm for Simpson’s rule.

Using Simpson’s rule, find 0 .6 0 e  x 2 d x03 taking seven ordinates. Show the calculations up to four decimal places.

Define divided difference. Using Newton’s divided difference interpolation, find f ( 6 ) from the following table: x 1 2 7 f ( x )07

Define interpolation. Using Lagrange interpolation, fit a second degree polynomial passing through the points (0,0), ( 1 ,1 ) and ( 2 , 2 0 )07 .

State Budan’s theorem. Define diagonally dominant system with example.

Use Newton-Raphson method to find a positive root of x 3  x 2  1  0 correct up to four decimal places taking x0  103 .

What do you mean by diagonally dominant system? Solve the following system of linear equations using Gauss-Seidel method: 9 x  y  z  1 0 , 2 x  1 0 y  3 z  1 9 , 3 x  4 y  1 1 z  .

Explain geometrically the method of false position.

Using Euler’s method, find y ( 1 ) if d d y x  x  y and y ( 0 )  103 . Take n10.04

Perform one iteration of the Bairstow method to extract a quadratic factor from the polynomial x 4  x 3  2 x 2  x  1 with initial factor x 2  0 . 5 x  0 . 507 .

Write the steps for engineering problem solving.

Determine the condition number of the matrix2  04 .

State direct and iterative methods to solve system of linear equations. Solve the following system of linear equations using Gauss elimination method: x  y  z  9 , 2 x  3 y  4 z  1 3 , 3 x  4 y  5 z  4 .

Write the formula for Runge-Kutta fourth order method.

Fit a second degree polynomial to the following data using least square method. y -3 -2 -1 0 1 2 x03

Calculate the first four moments of the following distribution about the mean. x 0 1 2 3 4 5 6 7 f ( x )07

Develop a Cprogram to fit regression line of y on

x through given set of points using the least square method. The probability distribution of a commodity is given below. Demand 5 6 7 8 9 Probability 0.05 0.10 0.30 0.40 0.10 0.05 Find expected demand.

For the following data, obtain trend values using five years moving average. Year 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Value 3 7 14 8 10 11 14 12 16 20 25

Discuss the pitfalls of Gauss elimination.

Define the following terms with examples: 1. Ill-conditioned system 2. Significant figure

Obtain the correlation coefficient for the following data: x 100 98 78 85 110 93 80 y07