Homework #3
i. Number of iterations that each function needed to reach an accuracy of 0.00001%.
| solver | initial guess(es) | ea | number of iterations |
|---|---|---|---|
| falsepos | 0,45 | 3.3785e-06 | 11 |
| bisect | 0,45 | 5.6529e-06 | 28 |
| newtraph | 0 | 9.9216e-06 | 687 |
| mod_secant | 0 | 1.1874e-09 | 3 |
Root=2.965
ii.Compare the convergence of the 4 methods. Plot the approximate error vs the number of iterations that the solver has calculated.
(couldnt get image to upload here, so i made it into an "issue" instead)
iii. Discription of files used
[root,fx,ea,iter]=falsepos(@(x) projectile(15,x),0,45,0.00001,150) [root,fx,ea,iter]=bisect(@(x) projectile(15,x),0,45,0.00001,150) [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),0.001,0,0.00001,150)
run convergence.m
2
Divergence
| iteration | x_i | approx error |
|---|---|---|
| 0 | 2 | n/a |
| 1 | 2 | 12.50 |
| 2 | 2.285 | 9.57 |
| 3 | 2.527 | 7.82 |
| 4 | 2.742 | 6.65 |
| 5 | 2.937 | 5.79 |
Convergence
| iteration | x_i | approx error |
|---|---|---|
| 0 | 0.2 | n/a |
| 1 | 0.2 | 1.250e03 |
| 2 | -0.17 | 1.653e05 |
| 3 | 1.052e-05 | 4.512e11 |
| 4 | -2.332e-15 | 4.512e11 |
| 5 | 0 | 4.512e11 |
HW#4
A) collar potential energy function:
function [ PE_t ] = collar_potential_energy(x_C,Theta ) %computes the total potential energy of a collar connected to a spring and sliding on a rod %
%initialize variables g=9.18; m=0.5; K= 30; DL= 0.5 - sqrt(0.5^2+(0.5-x_C)^2)
%The potential energy due to gravity PE_g= abs(m x_Cgsin(theta));
%potential energy due to the spring PE_s=abs(1/2*K *(DL)^2)
%Total potential energy PE_t= PE_g+PE_s end SEE ACTUAL FUCNTION
B) Finding Min with goldmin: Solving for x and fx: f =@(x_C) collar_potential_energy(x_C,theta) [x,fx,ea,iter]=goldmin(f,0,0.5,0.0000001,500); x fx
C)For loop for min PE position: theta(1) =0; for i=1:100 f =@(x_C) collar_potential_energy(x_C,theta(i)) [x(i),fx(i),ea(i),iter(i)]=goldmin(f,0,0.5,1e-6,1000); theta(i+1) = theta(i) +1; end
D) PLOT SEE ISSUE 2 (still havent figured out how to put a png into the repository)