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TEST2 is better than TEST1
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4 changes: 4 additions & 0 deletions HW1/README.md
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vii. Paste this into the Homework #1 Google form reponse.
(https://goo.gl/forms/jINjsioRLwQOiYtl2)[https://goo.gl/forms/jINjsioRLwQOiYtl2]

![Step 9](g9.png)

viii. Click on the gear for "Settings" then "Collaborators" on the left menu. Add
`rcc02007` (Ryan C. Cooper) as a collaborator.
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6 changes: 3 additions & 3 deletions README.md
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Expand Up @@ -19,11 +19,11 @@ matlab/octave functions and programming best practices.

**Instructor**: Prof. Ryan C. Cooper (ryan.c.cooper@uconn.edu)

**Office hours**: Fridays 10am-12pm in Engineering II room 315
**Office hours**: Mon 2:30-4:30pm and Thur 11am-1pm in Engineering II room 315

## Teaching Assistants:
- Graduate: **TBD**
- Office hours: 2 hours / week in office **TBD**
- Graduate: Peiyu Zhang <peiyu.zhang@uconn.edu>
- Office hours: 2 hours / week

**Prerequisite:** CE 3110, MATH 2410Q

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# Solution to Form #1


```octave
[1,2,3]*[1;2;3]
```

ans =

14




```octave
[1,2,3]*[1;2;3]=?
```


```octave
```

# The first source of error is roundoff error
## Just storing a number in a computer requires rounding


```octave
fprintf('realmax = %1.20e\n',realmax)
fprintf('realmin = %1.20e\n',realmin)
fprintf('maximum relative error = %1.20e\n',eps)
```

realmax = 1.79769313486231570815e+308
realmin = 2.22507385850720138309e-308
maximum relative error = 2.22044604925031308085e-16



```octave
s=1;
for i=1:1000
s=s+eps/10;
end
s==1
```

ans = 1


# Freefall Model (revisited)
## Octave solution (will run same on Matlab)

Set default values in Octave for linewidth and text size


```octave
%plot --format svg
```


```octave
set (0, "defaultaxesfontname", "Helvetica")
set (0, "defaultaxesfontsize", 18)
set (0, "defaulttextfontname", "Helvetica")
set (0, "defaulttextfontsize", 18)
set (0, "defaultlinelinewidth", 4)
```

Define time from 0 to 12 seconds with `N` timesteps
function defined as `freefall`


```octave
function [v_analytical,v_terminal,t]=freefall(N)
t=linspace(0,12,N)';
c=0.25; m=60; g=9.81; v_terminal=sqrt(m*g/c);
v_analytical = v_terminal*tanh(g*t/v_terminal);
v_numerical=zeros(length(t),1);
delta_time =diff(t);
for i=1:length(t)-1
v_numerical(i+1)=v_numerical(i)+(g-c/m*v_numerical(i)^2)*delta_time(i);
end
% Print values near 0,2,4,6,8,10,12 seconds
indices = round(linspace(1,length(t),7));
fprintf('time (s)|vel analytical (m/s)|vel numerical (m/s)\n')
fprintf('-----------------------------------------------\n')
M=[t(indices),v_analytical(indices),v_numerical(indices)];
fprintf('%7.1f | %18.2f | %15.2f\n',M(:,1:3)');
plot(t,v_analytical,'-',t,v_numerical,'o-')
end
```


```octave
[v_analytical,v_terminal,t]=freefall(120);
```

time (s)|vel analytical (m/s)|vel numerical (m/s)
-----------------------------------------------
0.0 | 0.00 | 0.00
2.0 | 18.76 | 18.82
4.0 | 32.64 | 32.80
6.1 | 40.79 | 40.97
8.0 | 44.80 | 44.94
10.0 | 46.84 | 46.93
12.0 | 47.77 | 47.82



![svg](output_13_1.svg)


# Types of error
## Freefall is example of "truncation error"
### Truncation error results from approximating exact mathematical procedure

We approximated the derivative as $\delta v/\delta t\approx\Delta v/\Delta t$

Can reduce error by decreasing step size -> $\Delta t$=`delta_time`

## Another example of truncation error is a Taylor series (or Maclaurin if centered at a=0)

Taylor series:
$f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^{2}+\frac{f'''(a)}{3!}(x-a)^{3}+...$

We can approximate the next value in a function by adding Taylor series terms:

|Approximation | formula |
|---|-------------------------|
|$0^{th}$-order | $f(x_{i+1})=f(x_{i})+R_{1}$ |
|$1^{st}$-order | $f(x_{i+1})=f(x_{i})+f'(x_{i})h+R_{2}$ |
|$2^{nd}$-order | $f(x_{i+1})=f(x_{i})+f'(x_{i})h+\frac{f''(x_{i})}{2!}h^{2}+R_{3}$|
|$n^{th}$-order | $f(x_{i+1})=f(x_{i})+f'(x_{i})h+\frac{f''(x_{i})}{2!}h^{2}+...\frac{f^{(n)}}{n!}h^{n}+R_{n}$|

Where $R_{n}=\frac{f^{(n+1)}(\xi)}{(n+1)!}h^{n+1}$ is the error associated with truncating the approximation at order $n$.

The $n^{th}$-order approximation estimates that the unknown function, $f(x)$, is equal to an $n^{th}$-order polynomial.

In the Freefall example, we estimated the function with a $1^{st}$-order approximation, so

$v(t_{i+1})=v(t_{i})+v'(t_{i})(t_{i+1}-t_{i})+R_{1}$

$v'(t_{i})=\frac{v(t_{i+1})-v(t_{i})}{t_{i+1}-t_{i}}-\frac{R_{1}}{t_{i+1}-t_{i}}$

$\frac{R_{1}}{t_{i+1}-t_{i}}=\frac{v''(\xi)}{2!}(t_{i+1}-t_{i})$

or the truncation error for a first-order Taylor series approximation is

$\frac{R_{1}}{t_{i+1}-t_{i}}=O(\Delta t)$


1. digital representation of a number is rarely exact

2. arithmetic (+,-,/,\*) causes roundoff error


```octave
fprintf('%1.20f\n',double(pi))
fprintf('%1.20f\n',single(pi))
```

3.14159265358979311600
3.14159274101257324219



```octave
```
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