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# 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 |