These extra practice problems are for students who wish to have an extra set to practice. These problems do not have a posted solution so students should reflect on ways to determine whether their answer is correct. Feel free to discuss with peers to solve these problems.

Solve the following equations using Naïve Gauss Elimination.

$$\begin{align} -0.55x_1+9.95x_2+1.41x_3+2.94x_4 &= 64.207 \\ -4.74x_2-5.62x_3-8.77x_4&=-61.751 \\ -5.31x_1-9.88x_2+4.36x_3-6.12x_4 &=-78.818 \\ 8.33x_1-4.12x_2-4.74x_3+4.88x_4 &=-2.62 \end{align}$$

Solve the following equations using Gauss Elimination, which includes partial pivoting and scaling.

$$\left[ \begin{matrix} 1&6&2&5&15&8&14\\-5&3&5&0&6&-4&7\\2&-1&1&13&8&1&9\\11&-5&14&6&7&6&9\\3&4&3&2&11&14&7\\1&-1&6&1&6&9&5\\8&0&10&-4&2&2&8 \end{matrix} \right| \left. \begin{matrix} 283.9\\57.7\\207.9\\220.3\\226.4\\127.4\\95 \end{matrix} \right]$$

Decompose the following matrix [A] and then solve for different [B] matrices. Check your decomposition by ensuring [L][U] = [A].

$$\begin{align}\left[ A \right]= \left[ \begin{matrix} -2.8&1&2&4&0.1\\-2.4&-2.1&2.4&-3.6&2.4\\-1.5&0.3&-4.9&-1.2&4.4\\1.4&-4.3&1.6&4.3&0.1\\-4.4&3.9&4.1&4.5&4.6 \end{matrix} \right] &&\left[ B_1 \right]= \left[ \begin{matrix} -84.3\\12.3\\21.4\\-10\\-141.1\end{matrix} \right]& &\left[ B_2 \right]= \left[ \begin{matrix} -69.2\\17.1\\72.3\\-14.8\\-65.5\end{matrix} \right] \end{align}$$

Find the roots from the following equation when y = 2 and y = 3 using bisection method. Plug the roots back into the original equation to determine if it is correct.

$$\begin{align}\left[A\right]= \left[ \begin{matrix} -14&12&0&0&0&0&0&0&0&0 \\ 14&-11&15&0&0&0&0&0&0&0 \\ 0&2&7&-7&0&0&0&0&0&0 \\ 0&0&9&-1&8&0&0&0&0&0 \\ 0&0&0&-5&5&9&0&0&0&0 \\ 0&0&0&0&11&9&2&0&0&0 \\ 0&0&0&0&0&12&4&10&0&0 \\ 0&0&0&0&0&0&14&14&10&0 \\ 0&0&0&0&0&0&0&13&-14&14 \\ 0&0&0&0&0&0&0&0&3&4 \\ \end{matrix} \right] & \left[B_1\right]= \left[ \begin{matrix}-162.88 \\ 285.77\\-65.18\\127.9\\91.29\\271.66\\379.46\\498.02\\151.45\\69.46\end{matrix}\right] \left[B_2\right]= \left[ \begin{matrix}-126.2\\375.32\\55.5\\264.5\\99.09\\272.57\\202.24\\335.22\\113.46\\78.84\end{matrix}\right]\end{align} $$

Rearrange the following equations and use Gauss-Seidel to find an answer. Make sure that the equations are diagonally dominant to ensure convergence.

$$\begin{align} 3x_1+6x_2+9x_3+4x_4&=68.3 \\ 8x_1+3x_2+4x_3-1x_4&=10.5 \\ 1x_1+4x_2+4x_3+9x_4&=46.8 \\ 2x_1+10x_2+5x_3+3x_4&=65.5 \\ \end{align}$$

Solve the following set of non-linear equations using Newton- Raphson method.

$$\begin{align} -0.55x_1+9.95x_2+1.41x_3+2.94x_4 &= 64.207 \\ -4.74x_2-5.62x_3-8.77x_4&=-61.751 \\ -5.31x_1-9.88x_2+4.36x_3-6.12x_4 &=-78.818 \\ 8.33x_1-4.12x_2-4.74x_3+4.88x_4 &=-2.62 \end{align}$$

Good initial guesses for Newton-Raphson’s method are x1 = 1, x2 = 3.