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# Homework #3 | |
## Hand calculations due 10/18 by 11:59 pm | |
The following problems should be worked out by hand. You can check your work with | |
Matlab/Octave. Upload a pdf of the completed calculations into a github repository called | |
'03_review_1-10'. Then, submit your repo link to | |
[https://goo.gl/forms/h6OMBURAZ6bJB2xG2](https://goo.gl/forms/h6OMBURAZ6bJB2xG2) | |
1. Use the Newton-Raphson method to approximate when f(x)=0. Start with an initial guess | |
of $x_{0}=0$. | |
$f(x)=e^{-x}-x^{3}$ | |
a. Compute the first 3 iterations and calculate the approximate error for each. | |
b. Compare the exact derivative to the derivative used in the modified secant for | |
$\delta x=0.1$ and $\delta x=0.001$ at $x_{0}$. | |
2. A simple computer is being assembled with 5-bits of storage for each integer. | |
a. How many different integers can be stored with 5 bits? | |
b. If we want the maximum number of positive and negative integers, what is the | |
largest and smallest integer we can store with 5 bits? | |
3. Convert the following binary numbers to base-10 in two ways, 1- the exact conversion, and | |
2- the conversion if only 4 digits are saved after addition/subtraction | |
a. 1.001 | |
b. 100.1 | |
c. 1.001 + 100.1 | |
d. 1000 - 0.0001 | |
4. In Problem 3c-d what kind of error is introduced by limiting the number of digits | |
stored? | |
5. Solve the following problems with matrix A: | |
$A=\left[ \begin{array}{ccc} | |
4 & 6 & 2 \\ | |
0 & 2 & 6 \\ | |
1 & 2 & 1\end{array} \right]$ | |
a. Compute the LU-decomposition | |
b. Solve for x if $Ax=b$ and $b=\left[\begin{array}{c} 1\\2\\1\end{array}\right]$ | |
6. Solve the following problems with matrix A: | |
$A=\left[ \begin{array}{ccc} | |
4 & -2 & 1\\ | |
-2 & 4 & -2\\ | |
1 & -2 & 2\end{array} \right]$ | |
a. Compute the Cholesky factorization of A | |
$C_{ii}=\sqrt{a_{ii}-\sum_{k=1}^{i-1}C_{ki}^{2}}$ | |
$C_{ij}=\frac{a_{ij}-\sum_{k=1}^{i-1}C_{ki}C_{kj}}{C_{ii}}$. | |
b. Find the determinant of A, |A|. | |
c. Find the inverse of A, $A^{-1}$ | |
7.Determine the lower (L) and upper (U) triangular matrices with LU-decomposition for the | |
following matrices, A. Then, solve for x when Ax=b: | |
a. $A=\left[ \begin{array}{cc} | |
1 & 3 \\ | |
2 & 1 \end{array} \right] | |
b= | |
\left[\begin{array}{c} | |
1 \\ | |
1\end{array}\right]$ | |
a. $A=\left[ \begin{array}{cc} | |
1 & 1 \\ | |
2 & 3 \end{array} \right] | |
b= | |
\left[\begin{array}{c} | |
3 \\ | |
4\end{array}\right]$ | |
a. $A=\left[ \begin{array}{cc} | |
1 & 1 \\ | |
2 & -2 \end{array} \right] | |
b= | |
\left[\begin{array}{c} | |
4 \\ | |
2\end{array}\right]$ | |
b. $A=\left[ \begin{array}{ccc} | |
1 & 3 & 1 \\ | |
-4 & -9 & 2 \\ | |
0 & 3 & 5\end{array} \right] | |
b= | |
\left[\begin{array}{c} | |
2 \\ | |
0 \\ | |
0\end{array}\right]$ | |
c. $A=\left[ \begin{array}{ccc} | |
1 & 2 & 3 \\ | |
-4 & -3 & 2 \\ | |
0 & 3 & 5\end{array} \right] | |
b= | |
\left[\begin{array}{c} | |
1 \\ | |
-1 \\ | |
-3\end{array}\right]$ | |
d. $A=\left[ \begin{array}{ccc} | |
1 & 3 & -5 \\ | |
1 & 4 & -8 \\ | |
-3 & -7 & 9\end{array} \right] | |
b= | |
\left[\begin{array}{c} | |
1 \\ | |
-1 \\ | |
-3\end{array}\right]$ | |
d. $A=\left[ \begin{array}{ccc} | |
0 & 2 & -1 \\ | |
2 & 5 & 2 \\ | |
1 & -1 & 2\end{array} \right] | |
b= | |
\left[\begin{array}{c} | |
2 \\ | |
3 \\ | |
5\end{array}\right]$ | |
9. Calculate the determinant of A from 1a-g. | |
10. Determine the Cholesky factorization, C, of the following matrices, where | |
$C_{ii}=\sqrt{a_{ii}-\sum_{k=1}^{i-1}C_{ki}^{2}}$ | |
$C_{ij}=\frac{a_{ij}-\sum_{k=1}^{i-1}C_{ki}C_{kj}}{C_{ii}}$. | |
a. A=$\left[ \begin{array}{cc} | |
3 & 2 \\ | |
2 & 1 \end{array} \right]$ | |
a. A=$\left[ \begin{array}{cc} | |
10 & 5 \\ | |
5 & 20 \end{array} \right]$ | |
a. A=$\left[ \begin{array}{ccc} | |
10 & -10 & 20 \\ | |
-10 & 20 & 10 \\ | |
20 & 10 & 30 \end{array} \right]$ | |
a. A=$\left[ \begin{array}{cccc} | |
21 & -1 & 0 & 0 \\ | |
-1 & 21 & -1 & 0 \\ | |
0 & -1 & 21 & -1 \\ | |
0 & 0 & -1 & 1 \end{array} \right]$ | |
11. Verify that $C^{T}C=A$ for 3a-d | |