README.md

# 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   |85,100  | 0.0116 | 50 |
|incsearch  | 85,90 | 8.7659e-05 | 16 |
|newtraph   | 100-(h(80)*(80-100))/(h(80)-h(100)),100 |  8.0459e-05 | 6 |
|mod_secant | @projectile,@dprojectile_theta, 90 | 8.0023e-5 | 4 |
```

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. 

### divergence of Newton-Raphson method

| iteration | x_i | approx error |
| --- | --- | --- |
| 0 | 2 | n/a |
| 1 |  0.3678 |     |
| 2 | 0.0366  |     |
| 3 | 3.70e-4  |     |
| 4 | 4.50e-7  |     |
| 5 | 6.94e-11 |     |
 
# Homework #4

# Part A

function PE = collar_potential_energy(x,theta).

 theta_d = theta*180/pi;
 
 x_C = 0.5+x; 
 
             %spring length when the spring is unstretched.
 
             %positive x means the collar move away from point o.
             
              %negative x means the collar move towards the point o.
              
 m = 0.5; 

 %mass of the collar.
 
 g = 9.81; 
 
 %unit m/s^2 .
 
 K = 30; 
  
  %spring stifness unit of N/m.
 
 PE_g = m*g*x_C*sin(theta_d); 

                           %Potential energy equation due to gravity.
 
                           %which 'theta' is the angle between the collar.
                           
                           %and the horizontal ground.
                           
 DL = 0.5-sqrt(0.5^2+(0.5-x_C)^2);
 
 PE_s = 0.5*K*(DL)^2;  

%Potential energy equation due to the spring.
 
 PE_tol = PE_g + PE_s 
 
 %Total energy equation.

 end

# Part b

function PE = xcollar_potential_energy(x)

theta_d = 0*180/pi;

x_C = 0.5+x; 

             %spring length when the spring is unstretched
            
             %x is the distance that collar moves
            
             %positive x means the collar move away from point o
            
             %negative x means the collar move towards the point o

m = 0.5; %mass of the collar

g = 9.81; %unit m/s^2 

K = 30; %spring stifness unit of N/m

PE_g = m*g*x_C*sin(theta_d); 
                          
              %Potential energy equation due to gravity,
                         
              %which 'theta' is the angle between the collar 
                         
              %and the horizontal ground.

DL = 0.5-sqrt(0.5^2+(0.5-x_C)^2);

PE_s = 0.5*K*(DL)^2;  %Potential energy equation due to the spring

PE_tol = PE_g + PE_s %Total energy equation.


end

Using goldmin function, it gives us the minimum potential energy is 0, at x_c = 0

#Part C

g = 9.81; %unit m/s^2 

m = 0.5;

K = 30; %spring stifness unit of N/m


for x = -1:0.1:1
   
    x_C = 0.5+x
    
    for theta = 0:90
   
    theta_d = theta*180/pi;
    
    PE_g = m*g*x_C*sin(theta_d);
    
    end
  
    DL = 0.5-sqrt(0.5^2+(0.5-x_C)^2);
  
    PE_s = 0.5*K*(DL)^2;
   
    PE_tol = PE_g + PE_s
    
end
    
 From for loop function, the minimum potential energy happens at x_c = 0, the total potential energy = 0.6434