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 = mgx_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 = 0.5K(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 = mgx_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 = 0.5K(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