README.md

# Homework #3

#### i. Number of iterations that each function needed to reach an accuracy of 0.00001%.
##### In degrees (root = 2.9655):

| solver | initial guess(es) | ea | number of iterations|
| --- | --- | --- | --- |
|falsepos   | xl = 0, xu = 45 | 3.3785e-06 | 11 |
|bisect     | xl = 0, xu = 45 | 5.6529e-06 | 28 |
|newtraph   | xr = 0 | 9.9216e-06 | 687 |
|mod_secant | xr = 0 | 4.4638e-09 | 3 |
##### In radians (root = 0.0518):

| solver | initial guess(es) | ea | number of iterations|
| --- | --- | --- | --- |
|falsepos   | xl = 0, xu = pi/4 | 3.3785e-06 | 11 |
|bisect     | xl = 0, xu = pi/4 | 5.6529e-06 | 28 |
|newtraph   | xr = 0 | 5.6054e-09 | 3 |
|mod_secant | xr = 0 | 1.1874e-09 | 3 |

#### ii. Compare the convergence of the 4 methods.

![Plot of convergence for four numerical solvers. (degrees)](convergence.png)
![Plot of convergence for four numerical solvers. (radians)](convergence2.png)

#### iii. Creation code
To create degrees table:
```
[root,fx,ea,iter]=falsepos(@(x) projectile(15,x),0,45,0.00001,100)
[root,fx,ea,iter]=bisect(@(x) projectile(15,x),0,45,0.00001,100)
[root,ea,iter]=newtraph(@(x) projectile(15,x),@(x) dprojectile_dtheta(15,x),0,0.00001,1000)
[root,ea,iter]=mod_secant(@(x) projectile(15,x),0.001,0,0.00001,100)
```
To create radians table:
```
[root,fx,ea,iter]=falsepos(@(x) projectile2(15,x),0,pi/4,0.00001,100)
[root,fx,ea,iter]=bisect(@(x) projectile2(15,x),0,pi/4,0.00001,100)
[root,ea,iter]=newtraph(@(x) projectile2(15,x),@(x) dprojectile2_dtheta(15,x),0,0.00001,1000)
[root,ea,iter]=mod_secant(@(x) projectile2(15,x),0.001,0,0.00001,100)
```
To create the convergence plots:
```
Degrees: run convergence.m
Radians: run convergence2.m
```

### Divergence of Newton-Raphson method

| iteration | x_i | approx error |
| --- | --- | --- |
| 0 | 2 | n/a |
| 1 | 2 | 12.5000 |
| 2 | 2.2857 | 9.5703 |
| 3 | 2.5276 | 7.8262 |
| 4 | 2.7422 | 6.6492 |
| 5 | 2.9375 | 5.7945 |

### Convergence of Newton-Raphson method

| iteration | x_i | approx error |
| --- | --- | --- |
| 0 | 0.2 | n/a |
| 1 | 0.2 | 1.2500e+03 |
| 2 | -0.0174 | 1.6531e+05 |
| 3 | 1.0527e-05 | 4.5122e+11 |
| 4 | -2.3329e-15 | 4.5122e+11 |
| 5 | 0 | 4.5122e+11 |

# Homework #4

#### a. collar potential energy function
```
function [PE] = collar_potential_energy(x_C, theta)
% theta taken in terms of degrees
% x_C in meters
theta1 = theta*(pi/180);
PE_g = (0.5)*x_C*9.81*sin(theta1);
DL = 0.5 - sqrt((0.5)^2 + (0.5 - x_C)^2);
PE_s = (0.5)*30*(DL^2);
PE = PE_g + PE_s;
end
```

#### b. minimum PE at theta = 0
PE = 0, x_C = 0.5
```
function [PE] = min_PE
[x,fx,ea,iter]=goldmin(@(x) collar_potential_energy(x, 0),0,10,0.0001,100);
PE = collar_potential_energy(x, 0);
end
```

#### c. minimum PE position for theta 0 to 90
```
theta = linspace(0,90,91);
x_C = zeros(1,91);
for n = 0:90
    [x,fx,ea,iter]=goldmin(@(x) collar_potential_energy(x,n),0,20,0.00001,100);
    x_C(n+1) = x;
end
```

#### d. plot of part c
![Steady-state position of collar on rod at angle theta](position.png)