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74 changes: 74 additions & 0 deletions HW3/README.md
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# Homework #3
## due 2/15/17 by 11:59pm


1. Create a new github repository called 'roots_and_optimization'.

a. Add rcc02007 and pez16103 as collaborators.

b. Clone the repository to your computer.

c. Copy your `projectile.m` function into the 'roots_and_optimization' folder.
*Disable the plotting routine for the solvers*

d. Use the four solvers `falsepos.m`, `incsearch.m`, `newtraph.m` and `mod_secant.m`
to solve for the angle needed to reach h=1.72 m, with an initial speed of 1.5 m/s.

e. The `newtraph.m` function needs a derivative, calculate the derivative of your
function with respect to theta, `dprojectile_dtheta.m`. This function should
output d(h)/d(theta).


f. In your `README.md` file, document the following under the heading `#
Homework #3`:

i. Compare the number of iterations that each function needed to reach an
accuracy of 0.00001%. Include a table in your README.md with:

```
| solver | initial guess(es) | ea | number of iterations|
| --- | --- | --- | --- |
|falsepos | | | |
|incsearch | | | |
|newtraph | | | |
|mod_secant | | | |
```

ii. Compare the convergence of the 4 methods. Plot the approximate error vs the
number of iterations that the solver has calculated. Save the plot as
`convergence.png` and display the plot in your `README.md` with:

`![Plot of convergence for four numerical solvers.](convergence.png)`

iii. In the `README.md` provide a description of the files used to create the
table and the convergence plot.

2. The Newton-Raphson method and the modified secant method do not always converge to a
solution. One simple example is the function f(x) = x*exp(-x^2). The root is at 0, but
using the numerical solvers, `newtraph.m` and `mod_secant.m`, there are certain initial
guesses that do not converge.

a. Calculate the first 5 iterations for the Newton-Raphson method with an initial
guess of x_i=2.

b. Add the results to a table in the `README.md` with:

```
### divergence of Newton-Raphson method

| iteration | x_i | approx error |
| --- | --- | --- |
| 0 | 2 | n/a |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
```

c. Repeat steps a-b for an initial guess of 0.2. (But change the heading from
'divergence' to 'convergence')

3. Commit your changes to your repository. Sync your local repository with github. Then
copy and paste the "clone URL" into the following Google Form [Homework
#3](https://goo.gl/forms/UJBGwp0fQcSxImkq2)
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544 changes: 480 additions & 64 deletions lecture_07/lecture_07.ipynb
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160 changes: 110 additions & 50 deletions lecture_07/lecture_07.md
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Expand Up @@ -18,20 +18,22 @@ $f'(x_{i})=\frac{f(x_{i})-0}{x_{i}-x_{i+1}}$

$x_{i+1}=x_{i}-\frac{f(x_{i})}{f'(x_{i})}$

Use Newton-Raphson to find solution when $e^{x}=x$
Use Newton-Raphson to find solution when $e^{-x}=x$


```octave
f= @(x) exp(-x)-x;
df= @(x) -exp(-x)-1;
x_i= 0;
x_r = x_i-f(x_i)/df(x_i)
error_approx = abs((x_r-x_i)/x_r)
x_r=x_i;
x_i=x_r;
```

error_approx = 1
x_r = 0.50000
error_approx = 1



Expand All @@ -41,8 +43,8 @@ error_approx = abs((x_r-x_i)/x_r)
x_i=x_r;
```

x_r = 0.50000
error_approx = 1
x_r = 0.56631
error_approx = 0.11709



Expand All @@ -52,8 +54,8 @@ error_approx = abs((x_r-x_i)/x_r)
x_i=x_r;
```

x_r = 0.56631
error_approx = 0.11709
x_r = 0.56714
error_approx = 0.0014673



Expand All @@ -64,7 +66,7 @@ x_i=x_r;
```

x_r = 0.56714
error_approx = 0.0014673
error_approx = 2.2106e-07


In the bungee jumper example, we created a function f(m) that when f(m)=0, then the mass had been chosen such that at t=4 s, the velocity is 36 m/s.
Expand All @@ -90,10 +92,12 @@ df_m = @(m) 1/2*sqrt(g./m/c_d).*tanh(sqrt(g*c_d./m)*t)-g/2./m*sech(sqrt(g*c_d./m


```octave
newtraph(f_m,df_m,140,0.00001)
[root,ea,iter]=newtraph(f_m,df_m,140,0.00001)
```

ans = 142.74
root = 142.74
ea = 8.0930e-06
iter = 48


## Secant Methods
Expand All @@ -119,32 +123,46 @@ $x_{i+1}=x_{i}-\frac{f(x_{i})(\delta x_{i})}{f(x_{i}+\delta x_{i})-f(x_{i})}$


```octave
mod_secant(f_m,1e-6,50,0.00001)
[root,ea,iter]=mod_secant(f_m,1,50,0.00001)
```

ans = 142.74
root = 142.74
ea = 3.0615e-07
iter = 7



```octave
Amt_numerical=mod_secant(@(A) car_payments(A,30000,0.05,5),1e-6,50,0.001)
car_payments(Amt,30000,0.05,5)
car_payments(400,30000,0.05,5,1)
```

error: 'plot_bool' undefined near line 12 column 6
error: called from
car_payments at line 12 column 3
mod_secant at line 22 column 8
error: 'Amt' undefined near line 1 column 14
error: evaluating argument list element number 1
ans = 1.1185e+04



![svg](lecture_07_files/lecture_07_15_1.svg)



```octave
Amt*12*5
Amt_numerical=mod_secant(@(A) car_payments(A,700000,0.0875,30,0),1e-6,50,0.001)
car_payments(Amt_numerical,700000,0.0875,30,1)
```

error: 'Amt' undefined near line 1 column 1
Amt_numerical = 5467.0
ans = 3.9755e-04



![svg](lecture_07_files/lecture_07_16_1.svg)



```octave
Amt_numerical*12*30
```

ans = 1.9681e+06


Amortization calculation makes the same calculation for the monthly payment amount, A, paying off the principle amount, P, over n pay periods with monthly interest rate, r.
Expand Down Expand Up @@ -257,12 +275,13 @@ ea_ms=zeros(1,N); % appr error Modified Secant
ea_fp=zeros(1,N); % appr error false point method
ea_bs=zeros(1,N); % appr error bisect method
for i=1:length(iterations)
[root_nr,ea_nr(i),iter_nr]=newtraph(f_m,df_m,200,0,iterations(i));
[root_nr,ea_nr(i),iter_nr]=newtraph(f_m,df_m,300,0,iterations(i));
[root_ms,ea_ms(i),iter_ms]=mod_secant(f_m,1e-6,300,0,iterations(i));
[root_fp,ea_fp(i),iter_fp]=falsepos(f_m,1,300,0,iterations(i));
[root_bs,ea_bs(i),iter_bs]=bisect(f_m,1,300,0,iterations(i));
end
setdefaults
semilogy(iterations,abs(ea_nr),iterations,abs(ea_ms),iterations,abs(ea_fp),iterations,abs(ea_bs))
legend('newton-raphson','mod-secant','false point','bisection')
```
Expand All @@ -281,31 +300,27 @@ legend('newton-raphson','mod-secant','false point','bisection')



![svg](lecture_07_files/lecture_07_22_1.svg)
![svg](lecture_07_files/lecture_07_24_1.svg)



```octave
ea_ms
ea_nr
```

ea_ms =
ea_nr =

Columns 1 through 6:
Columns 1 through 8:

2.3382e+03 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14
6.36591 0.06436 0.00052 0.00000 0.00000 0.00000 0.00000 0.00000

Columns 7 through 12:
Columns 9 through 16:

1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

Columns 13 through 18:
Columns 17 through 20:

1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14

Columns 19 and 20:

1.9171e-14 1.9171e-14
0.00000 0.00000 0.00000 0.00000



Expand Down Expand Up @@ -344,32 +359,28 @@ legend('newton-raphson','mod-secant','false point','bisection')



![svg](lecture_07_files/lecture_07_24_1.svg)
![svg](lecture_07_files/lecture_07_26_1.svg)



```octave
ea_bs
ea_nr
newtraph(f,df,0.5,0,12)
```

ea_bs =

Columns 1 through 6:
ea_nr =

9.5357e+03 -4.7554e-01 -2.1114e-01 6.0163e-02 -2.4387e-03 6.1052e-04
Columns 1 through 7:

Columns 7 through 12:
99.03195 11.11111 11.11111 11.11111 11.11111 11.11111 11.11111

2.2891e-04 -9.5367e-06 2.3842e-06 8.9407e-07 -2.2352e-07 9.3132e-09
Columns 8 through 14:

Columns 13 through 18:
11.11111 11.11111 11.11111 11.11109 11.11052 11.10624 10.99684

-2.3283e-09 -8.7311e-10 3.6380e-11 -9.0949e-12 -3.4106e-12 8.5265e-13
Columns 15 through 20:

Columns 19 and 20:

-3.5527e-14 8.8818e-15
8.76956 2.12993 0.00000 0.00000 0.00000 0.00000

ans = 16.208

Expand All @@ -383,6 +394,55 @@ df(300)



```octave
% our class function
f= @(x) tan(x)-(x-1).^2
mod_secant(f,1e-3,1)
```

f =

@(x) tan (x) - (x - 1) .^ 2

ans = 0.37375



```octave
f(ans)
```

ans = -3.5577e-13



```octave
tan(0.37375)
(0.37375-1)^2
```

ans = 0.39218
ans = 0.39219



```octave
f([0:10])
```

ans =

Columns 1 through 8:

-1.0000 1.5574 -3.1850 -4.1425 -7.8422 -19.3805 -25.2910 -35.1286

Columns 9 through 11:

-55.7997 -64.4523 -80.3516




```octave
```
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