README.md

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

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