SolveTask One

We will start our exploration of the connections between sequences, series, power series, and computing by examining and executing a very basic

computer program. In doing so$,$ we will explore the basics of "Floating Point Numbers," which underlies how our computers store numbers.

The following code uses a "while" loop, which starts at the command "while" and ends at the command "end". A "while" loop operates much like a

for loop. After each command "sum$=$sum$+$diff", the computer evaluates the condition in the "while" line. If the condition is met, the computer runs

the loop again, and the computer stops if the condition is not met. In this case, as long as "sum the computer will run the command

"sum$=$sum$+$diff". Documentation on a "while" loop can be found here in the Mathworks Help Center $($MathWorks$).B$

The following code also uses the "vpa" function, which approximates numbers to various levels of precision. For instance, vpa$($pi$)$ gives $32$

significant digits of pi$.$ vpa$($pi$,50)$ gives $50$ digits of pi$,$ and vpa$($pi$,3)$ gives $3$ digits of pi$.$ Documentation on "vpa" can be found here in the

Mathworks Help Center $($Mathworks$),$ B

Task: Make the following posts. First, answer Questions for Code $1,$ Questions for Code $2,$ and then work towards a Final Group Post as

described below.

For your first post, read the following two codes $($i$.$e$.,$ Code $1$ and Code $2)$ and answer the following questions:

Code $1$

$x0=\mathrm{.}91$

$n=9$

diff$=\frac{1\mathrm{.}000-x0}{n}$

sum$=x0$

while sum$<mn\; id="EditorElement\_8906415">1</mn>$

sum$=$sum$+$diff

end

disp$(sum)$

This code can also be found in this Code $1$ document. $($DOCX$)$ darr

Questions for Code $1$

Before you run the code, explain what the code is doing and what you expect the final value of "sum" to be$.$

Run the code. Was the output what you expected?

Run the code at $x0=\mathrm{.}5,$ was the output as you expected?

Run the code at $x0=\mathrm{.}1,$ was the output as you expected?

Explain any discrepancies between what you expect the code to do and what the code actually does. See Code $2$ below if you need more

insight into these discrepancies.

This code can also be found in this Code $2$ document. $($DOCX$)$:$,$darr

Questions for Code $2$

Before you run the code, explain what the code is doing and what you expect the value of $"$a$-$b$"$ to be$.$

Explain any discrepancies between what you expect the code to do and what the code actually does.

Matlab stores $\mathrm{.}1$ as a "Floating$-$Point Number." Use the result of $"$a$-$b$"$ to explain what happens with $x0=\mathrm{.}1$ in Code $1.$

Fixing Code $1$

By the end of Module $7,$ your group should agree on changes to Code $1$ that would make it give a mathematically accurate output for $x0=\mathrm{.}1,$ and

for other potentially problematic $0$ values. Your suggested changes should also include an explanation of the errors in terms of sequences and

series and the resulting errors associated with partial summations.