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.

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