Difference between revisions of "Practice questions"
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\begin{align*} | \begin{align*} | ||
A &= \left[ \begin{array}{rrr} 2 & -2 & 4 \\ 1 & -3 & 4 \\ 3 & -1 & 5 \end{array} \right] | A &= \left[ \begin{array}{rrr} 2 & -2 & 4 \\ 1 & -3 & 4 \\ 3 & -1 & 5 \end{array} \right] | ||
\end{align*} | |||
\] | |||
== Tutorial 5, 2009 == | |||
In addition to elimination methods, we have seen that iterative methods can be used to solve systems of linear equations. In this question, you are to solve the following linear algebraic equations, using (a) the Jacobi method, (b) the Gauss-Seidel method and (c) the relaxed Gauss-Seidel method, with \(\omega=0.5\). | |||
In each case, start from the initial guess \(x_1^{(0)}=x_2^{(0)}=x_3^{(0)}=0\) and perform 3 full iterations. Can anything be said regarding the convergence/divergence of these methods for that equation system? | |||
\[ | |||
\begin{align*} | |||
\left\{\begin{array}{rcl} | |||
2 x_1 -2x_2 +4 x_3 & = & 0 \\ | |||
x_1 -3 x_2 + 4x_3 & = & -1 \\ | |||
3x_1 - x_2 +5x_3 &= & 0 | |||
\end{array}\right. | |||
\end{align*} | \end{align*} | ||
\] | \] | ||
Revision as of 13:09, 12 October 2010
Various practice questions will be posted here as the semester progresses. These questions are from previous exams, assignments and tutorials. No solutions will be posted.
Tutorial 3, 2009
- Convert into decimal representation: (a) \((10011101)_2\); (b) \((0.001101)_2\)
- Convert into binary representation: (a) \((45.625)_{10}\); (b) \((0.1)_{10}\)
- Consider the following system of linear algebraic equations.
- Use Gauss elimination (without pivoting) to solve these equations for \((x_1,x_2,x_3)\).
- Validate your solution by comparing it to the one obtained with computer software.
\[ \begin{align*} \left\{\begin{array}{rcl} 2 x_1 -2x_2 +4 x_3 & = & 0 \\ x_1 -3 x_2 + 4x_3 & = & -1 \\ 3x_1 - x_2 +5x_3 &= & 0 \end{array}\right. \end{align*} \]
Tutorial 4, 2009
- We have already seen several elimination techniques for linear equation solving. In this question, you are to solve the following linear algebraic equations, using LU decomposition
\[\begin{align*} \left\{\begin{array}{rcl} 2 x_1 -2x_2 +4 x_3 & = & 0 \\ x_1 -3 x_2 + 4x_3 & = & -1 \\ 3x_1 - x_2 +5x_3 &= & 0 \end{array}\right. \end{align*} \]
- We have also seen that the LU decomposition technique can be used to calculate matrix inverses. Using the results obtained in the question above, compute the inverse of the following matrix,
\[ \begin{align*} A &= \left[ \begin{array}{rrr} 2 & -2 & 4 \\ 1 & -3 & 4 \\ 3 & -1 & 5 \end{array} \right] \end{align*} \]
Tutorial 5, 2009
In addition to elimination methods, we have seen that iterative methods can be used to solve systems of linear equations. In this question, you are to solve the following linear algebraic equations, using (a) the Jacobi method, (b) the Gauss-Seidel method and (c) the relaxed Gauss-Seidel method, with \(\omega=0.5\).
In each case, start from the initial guess \(x_1^{(0)}=x_2^{(0)}=x_3^{(0)}=0\) and perform 3 full iterations. Can anything be said regarding the convergence/divergence of these methods for that equation system? \[ \begin{align*} \left\{\begin{array}{rcl} 2 x_1 -2x_2 +4 x_3 & = & 0 \\ x_1 -3 x_2 + 4x_3 & = & -1 \\ 3x_1 - x_2 +5x_3 &= & 0 \end{array}\right. \end{align*} \]