Exercise 4.22

Consider two systems of equations:

$\displaystyle \left\{ \begin{array}{r} 10x_1+7x_2+8x_3+7x_4=32\\ 7x_1+5x_2+6x_3+5x_4=23\\ 7x_1+6x_2+10x_3+9x_4=33\\ 7x_1+5x_2+9x_3+10x_4=31 \end{array} \right.$    


$\displaystyle \left\{ \begin{array}{r} 2y_1+y_2+5y_3+y_4=9\\ y_1+y_2-3y_3-y_4=-5\\ 3y_1+6y_2-2y_3+y_4=8\\ 2y_1+2y_2+2y_3-3y_4=3 \end{array} \right.$    

Solve both systems of equations
By changing the right-hand-side of both systems of equations, we can obtain an impression of the sensitivity of the matrices for small changes.
Solve the first system of equations with the righthandsides $ (32.1, 22.9, 32.9, 31.1)$ and $ (32.01, 22.99, 32.99, 31.01)$ . Solve the second system of equations with right hand-sides $ (9.1, -5.1, 7.9, 3.1)$ and $ (9.01, -5.01, 7.99, 3.01)$ . What is the effect of these minor changes?
By slightly changing the coefficients of the equations and studying the effect of these changes to the solutions, one can get an impression of the sensitivity for small data errors.
Change the elements of the original matrices by adding a matrix 0.1*rand(4) to both of them, such that for each element, a random number between 0 and 0.1 is added. Solve the systems of equations with the original right-hand sides.What is the effect of the changes?
Systems of equations that are sensitive for small data changes have ill-conditioned matrices of coefficients. Which of the two systems has the most well-conditioned matrix of coefficients?

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Esteur 2010-03-22