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function gradient_descent(f, ∇f, x0, ϵ) | |
e = ϵ | |
n = 1 | |
t = 1 | |
x_n = x0 | |
while abs(e) >= ϵ | |
f_n = f(x_n) | |
grad = ∇f(x_n) # vetor do gradiente | |
if sqrt(sum(grad).^2) == 0 # se o gradiente for 0 você chegou ao seu destino | |
break | |
end | |
# vetor unitário na direção do gradiente | |
u_n = grad./sqrt(sum(grad.^2)) | |
# achar o valor de t que minimiza a função | |
t = quadratic_interp(f, x_n, u_n) | |
x_n1 = x_n - t*u_n # novo X | |
f_n1 = f(x_n1) | |
e = sqrt(sum(((f_n1 - f_n)./f_n).^2)) | |
x_n = x_n1 | |
n = n + 1 | |
end | |
return n | |
end | |
function quadratic_interp(f, x_n, u_n) | |
b = 1 | |
c = 2b | |
if f(x_n - u_n) > f(x_n) | |
while f(x_n) < f(x_n - b*u_n) | |
b = b/2 | |
c = 2b | |
end | |
else | |
while f(x_n - b*u_n) >= f(x_n - c*u_n) | |
b = 2*b | |
c = 2b | |
if c > 1e10 | |
exit("help") | |
end | |
end | |
end | |
# display(b) | |
# os três pontos da parábola | |
p1 = [0, f(x_n)] | |
p2 = [b, f(x_n - b*u_n)] | |
p3 = [c, f(x_n - c*u_n)] | |
# coeficientes Ax^2 + Bx + C | |
# https://stackoverflow.com/questions/717762/how-to-calculate-the-vertex-of-a-parabola-given-three-points | |
denom = (p1[1] - p2[2])*(p1[1] - p3[1])*(p2[1] - p3[1]) | |
A = (p3[1] * (p2[2] - p1[2]) + p2[1] * (p1[2] - p3[2]) + p1[1] * (p3[2] - p2[2])) | |
B = (p3[1]^2 * (p1[2] - p2[2]) + p2[1]^2 * (p3[2] - p1[2]) + p1[1]^2 * (p2[2] - p3[2])) | |
C = (p2[1] * p3[1] * (p2[1] - p3[1]) * p1[2] + p3[1] * p1[1] * (p3[1] - p1[1]) * p2[2] + p1[1] * p2[1] * (p1[1] - p2[1]) * p3[2]) | |
t = -B / 2A # x do vértice | |
f_t = (C - B^2 / 4A) / denom # y do vértice | |
if f(x_n - b*u_n) < f_t # reavalia em b | |
t = b | |
end | |
return t | |
end | |
using Plots | |
f1(X) = X[1]^2 + X[2]^2 # x^2 + y^2 | |
∇f1(X) = [2*X[1], 2*X[2]] # [2x, 2y] | |
n1(x, y, e) = gradient_descent(f1, ∇f1, [x,y], e) | |
x = range(-5, 5, length=100) | |
y = range(-5, 5, length=100) | |
contour(x, y, (x, y) -> n1(x, y, 0.1), | |
aspect_ratio = 1, fill = true, c=:matter, title=L"\epsilon = 0.1") | |
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