How to implement cubic spline interpolation in 3 dimensions? - python

I am trying to implement cubic spline interpolation in 3 dimensions, however I am unsure how to modify the code I have currently written to implement the z-axis. The purpose of this code will be to calculate a trajectory between a starting point and an end point, which passes through several intermediate points. Any assistance would be greatly appreciated!
import sys
import numpy as np
import matplotlib.pyplot as plt
X = np.array([1, 5, 8, 12, 16, 20, 25, 30, 38], np.float)
Y = np.array([20, 14, 10, 7, 3, 8, 17, 5, 3], np.float)
num_points = 1000
H_x = np.diff(X)
H_y = np.diff(Y)
H_n = N - 1
Alfa = 1 / H_x[1 : H_n - 1]
Gamma = 1 / H_x[1 : H_n - 1]
Beta = 2 * (1 / H_x[:H_n - 1] + 1 / H_x[1:])
dF = H_y / H_x
Delta = 3 * (dF[1:] / H_x[1:] + dF[:H_n-1] / H_x[:H_n-1])
TDM = np.diag(Alfa, k=-1) + np.diag(Beta, 0) + np.diag(Gamma, +1)
B = np.linalg.solve(TDM, Delta)
B = np.hstack([0, B, 0])
C = (3*dF - B[1:] - 2 * B[:H_n]) / H_x
D = (B[:H_n] + B[1:] - 2 * dF) / (H_x ** 2)
x_step = (X[N-1] - X[0]) / num_points
x_points = []
x_base = X[0]
for i in range(num_points):
x_points.append(x_base+x_step*i)
y_points = []
for x_point in x_points:
for i in range(N-1):
if ((x_point >= X[i]) and (x_point <= X[i+1])):
y_point = Y[i] + B[i] * (x_point - X[i]) + C[i] * ((x_point - X[i]) ** 2) + D[i] * ((x_point - X[i]) ** 3)
y_points.append(y_point)
spline, nodes = plt.plot(x_points, y_points, "-g", X, Y, "o")
plt.axis([X[0]-3, X[N-1]+3, np.min(y_points)-3, np.max(y_points)+3])
plt.title(u'P(x)')
plt.xlabel(u'X')
plt.ylabel(u'Y')
plt.grid()
plt.savefig('cubic_spline.png', format = 'png')
plt.show()

Related

Simple Linear Regression - what am I doing wrong?

I am new to ML and tried to build a Linear Regression Model by myself. Object is to predict the fahrenheit values for celcius values.
This is my code:
celsius_q = np.array([-40, -10, 0, 8, 15, 22, 38], dtype= float)
fahrenheit_a = np.array([-40, 14, 32, 46, 59, 72, 100], dtype = float)
inputs = celsius_q
output_expected = fahrenheit_a
# y = m * x + b
m = 100
b = 0
m_gradient = 0
b_gradient = 0
learning_rate = 0.00001
#Forwardpropagation
for i in range(10000):
for i in range(len(inputs)):
m_gradient += (m + (b * inputs[i] - output_expected[i]))
b_gradient += inputs[i] * (m + (b * inputs[i]) - output_expected[i])
m_new = m - learning_rate * (2/len(inputs)) * m_gradient
b_new = b - learning_rate * (2/len(inputs)) * b_gradient
The code generates wrong weights for m and b, no matter how much I change the learning_rate and the epochs. The weights for minimal loss function has to be:
b = 1.8
m = 32
What am I doing wrong?

The update of m and b needs to happen every step but this is not going to be enough. You also need to slightly increase your learning rate, say twice:
import numpy as np
celsius_q = np.array([-40, -10, 0, 8, 15, 22, 38], dtype=float)
fahrenheit_a = np.array([-40, 14, 32, 46, 59, 72, 100], dtype=float)
inputs = celsius_q
output_expected = fahrenheit_a
# y = m * x + b
m_new = m = 100.0
b_new = b = 0.0
m_gradient = 0.0
b_gradient = 0.0
learning_rate = 0.0002
# Forwardpropagation
for i in range(10000):
m_gradient, b_gradient = 0, 0
for i in range(len(inputs)):
m_gradient += (m_new + (b_new * inputs[i] - output_expected[i]))
b_gradient += inputs[i] * (m_new + (b_new * inputs[i]) - output_expected[i])
m_new -= learning_rate * m_gradient
b_new -= learning_rate * b_gradient
print(m_new, b_new)
Getting:
31.952623523538897 1.7979482813813066
which is close to the expected 32 and 1.8.

You should continually update your parameters, in every step. Something like:
for i in range(10000):
m_gradient, b_gradient = 0, 0
for i in range(len(inputs)):
m_gradient += (m + (b * inputs[i] - output_expected[i]))
b_gradient += inputs[i] * (m + (b * inputs[i]) - output_expected[i])
m -= learning_rate * m_gradient
b -= learning_rate * b_gradient
(But I didn't check your math.)

How to create a plot functions orientated in a specific way (portal frame - moment diagram)?

I´m a mechanical engineering student and I'm trying to create a script for moment diagram in Python. What is missing in my code is how to orientate the moment functions in order to be aligned like the portal frame.
Mpilar1 is the moment function for the fisrt column (lef to right).
Masna1 is the moment function for the fisrt beam (lef to right).
Masna2 is the moment function for the second beam (lef to right).
Mpilar2 is the moment function for the second column (lef to right).
Code:
import math as mt
import numpy as np
import warnings
import matplotlib.pyplot as plt
warnings.filterwarnings("ignore", category=np.VisibleDeprecationWarning)
#Definir parâmetros do pavilhão
v = 20 #(Vão em metros)
h = 6 #("Altura do pilar em metros:")
ht = 8 #("Altura total metros:")
alfa = (mt.atan((int(ht)-int(h))/(int(v)/2)))*180/((mt.pi))
print("Ângulo da vertente:", round(alfa, 1), "º")
lasna = ((v/2) ** 2 + (ht-h) ** 2) ** 0.5
print("Comprimento de cada asna: ", round(lasna, 2), "m")
h1 = np.arange(0, h+1, 1)
ha1 = np.arange(0, lasna, 0.1)
def draw_line():
x_number_list = [0, 0, (v/2), v, v]
y_number_list = [0, h, ht, h, 0]
plt.plot(x_number_list, y_number_list, linewidth=3)
plt.title("Pórtico", fontsize=15)
plt.xlabel("Vão (m)", fontsize=10)
plt.ylabel("Altura (m)", fontsize=10)
plt.tick_params(axis='both', labelsize=9)
plt.show()
if __name__ == '__main__':
draw_line()
Mpilar1 = 1500 * h1 ** 2 + 350 * h1
Masna1 = 300 * ha1 ** 2 + 15 * ha1
Masna2 = 200 * ha1 ** 2 + 15 * ha1
Mpilar2 = 1400 * h1 ** 2 + 10 * h1
plt.plot(h1, Mpilar1)
plt.plot(ha1, Masna1)
plt.plot(ha1, Masna2)
plt.plot(h1, Mpilar2)

You have to use transformation matrices over the points representing your curves. In particular, you'd have to use a roto-translation matrix to rotate and translate a curve to the correct position and orientation, and you might have to apply a mirror matrix to get the moments aligned according to your convention.
Please note that my structural engineering days are loooong gone, so I don't remember the convention to properly orient the moments. That's left to you as an exercise.
import math as mt
import numpy as np
import warnings
import matplotlib.pyplot as plt
warnings.filterwarnings("ignore", category=np.VisibleDeprecationWarning)
#Definir parâmetros do pavilhão
v = 20 #(Vão em metros)
h = 6 #("Altura do pilar em metros:")
ht = 8 #("Altura total metros:")
alfa_rad = mt.atan((int(ht)-int(h))/(int(v)/2))
alfa_deg = alfa_rad*180/mt.pi
print("Ângulo da vertente:", round(alfa_deg, 1), "º")
lasna = ((v/2) ** 2 + (ht-h) ** 2) ** 0.5
print("Comprimento de cada asna: ", round(lasna, 2), "m")
h1 = np.arange(0, h+1, 1)
ha1 = np.arange(0, lasna, 0.1)
# Roto-translation matrix:
# Rotates the points by an angle theta and translates
# them by x in the horizontal direction, and y in the
# vertical direction
R = lambda x, y, theta: np.array([
[np.cos(theta), np.sin(theta), x],
[-np.sin(theta), np.cos(theta), y],
[0, 0, 1],
])
# mirror matrix about the x-axis
Mx = np.array([
[1, 0, 0], [0, -1, 0], [0, 0, 1]
])
# mirror matrix about the y-axis
My = np.array([
[-1, 0, 0], [0, 1, 0], [0, 0, 1]
])
Mpilar1 = 1500 * h1 ** 2 + 350 * h1
Masna1 = 300 * ha1 ** 2 + 15 * ha1
Masna2 = 200 * ha1 ** 2 + 15 * ha1
Mpilar2 = 1400 * h1 ** 2 + 10 * h1
def draw_line():
plt.figure()
x_number_list = [0, 0, (v/2), v, v]
y_number_list = [0, h, ht, h, 0]
plt.plot(x_number_list, y_number_list, linewidth=3)
# left column
points1 = np.stack([h1, Mpilar1 / max(Mpilar1), np.ones_like(h1)])
points1 = np.matmul(R(0, 0, -np.pi/2), points1)
plt.plot(points1[0, :], points1[1, :], label="Mpilar1")
# right column
points2 = np.stack([h1, Mpilar2 / max(Mpilar2), np.ones_like(h1)])
points2 = np.matmul(R(20, 0, -np.pi/2), points2)
plt.plot(points2[0, :], points2[1, :], label="Mpilar2")
# left asna
points3 = np.stack([ha1, Masna1 / max(Masna1), np.ones_like(ha1)])
points3 = np.matmul(R(0, 6, -alfa_rad), points3)
plt.plot(points3[0, :], points3[1, :], label="Masna1")
# right asna
points4 = np.stack([ha1, Masna2 / max(Masna2), np.ones_like(ha1)])
points4 = np.matmul(np.matmul(R(20, 6, alfa_rad), My), points4)
plt.plot(points4[0, :], points4[1, :], label="Masna2")
plt.title("Pórtico", fontsize=15)
plt.xlabel("Vão (m)", fontsize=10)
plt.ylabel("Altura (m)", fontsize=10)
plt.tick_params(axis='both', labelsize=9)
plt.legend()
plt.show()
draw_line()
There are a few things to note in the above code:
let's consider points1 = np.stack([h1, Mpilar1 / max(Mpilar1), np.ones_like(h1)]). It creates a 3xn matrix of coordinates. The first row is the x-coordinates, h1. The second row is the y-coordinates of the moment, Mpilar1 / max(Mpilar1) (note that I have adimentionalized it in order to fit the chart). The third row is 1, and it is a trick to be able to apply a translation matrix. In the plot commands, we will only use the first and second rows (the x and y coordinates).
points4 = np.matmul(np.matmul(R(20, 6, alfa_rad), My), points4) here I first mirrored the points about the y-axis, then I applied a rotation and translation. You will have to play in order to properly orient the moment!

The solution works, but in this case I can´t have the correct moment diagram.
Code:
import math as mt
import numpy as np
import warnings
import matplotlib.pyplot as plt
warnings.filterwarnings("ignore", category=np.VisibleDeprecationWarning)
#Definir parâmetros do pavilhão
v = 20 #(Vão em metros)
h = 6 #("Altura do pilar em metros:")
ht = 8 #("Altura total metros:")
nm = 7 #("Número de madres por asna:")
npo = 5 #("Número de pórticos:")
dp = 5 #("Distância entre pórticos em metros:")
alfa_rad = mt.atan((int(ht)-int(h))/(int(v)/2))
alfa_deg = alfa_rad*180/mt.pi
alfa = (mt.atan((int(ht)-int(h))/(int(v)/2)))*180/((mt.pi))
print("Ângulo da vertente:", round(alfa, 1), "º")
lasna = ((v/2) ** 2 + (ht-h) ** 2) ** 0.5
print("Comprimento de cada asna: ", round(lasna, 2), "m")
dm = lasna / nm
print(("Distância entre madres da cobertura em metros:"), round(dm, 2), "m")
lp = npo * dp
print("Comprimento total do pavilhão:", round(lp, 1), "m")
def draw_line():
# x axis value list.
x_number_list = [0, 0, (v/2), v, v]
# y axis value list.
y_number_list = [0, h, ht, h, 0]
# Plot the number in the list and set the line thickness.
plt.plot(x_number_list, y_number_list, linewidth=3)
# Set the line chart title and the text font size.
plt.title("Pórtico", fontsize=15)
# Set x axis label.
plt.xlabel("Vão (m)", fontsize=10)
# Set y axis label.
plt.ylabel("Altura (m)", fontsize=10)
# Set the x, y axis tick marks text size.
plt.tick_params(axis='both', labelsize=9)
# Display the plot in the matplotlib's viewer.
plt.show()
if __name__ == '__main__':
draw_line()
#EXEMPLO - IPE 100
pppilar = 8.1 #variavel do peso próprio do pilar
arpilar = 10.32 * 10 ** -4 #variavél da área de secção do pilar - m^2
inypilar = 171 * 10 ** -8 #variavél da inércia pilar segundo y - m^4
ppasna = 8.1 #variavel do peso próprio da asna
arasna = 10.32 * 10 ** -4 #variavél da área de secção do pilar - m^2
inyasna = 171 * 10 ** -8 #variavél da inércia pilar - m^4
pesomadre = 4.84 #(Peso linear das madres de cobertura em kg/m)
pesopainel = 11.2 #(Peso painel revestimento kg/m2)
rotulado = 1
base = rotulado
#Definir ações
ppcobertura = (pesopainel * dp + ((pesomadre*dp)/dm) + ppasna) * 9.81 / 1000 #kN/m
#print("Peso próprio da cobertura:", round(ppcobertura, 1), "kN/m")
#Sobrecarga
qk = 0.4 #Sobrecarga em kN/m2
sb = qk * dp #Sobrecarga em kN/m
#Neve
Sk = 0.5 #Neve em kN/m2
ne = Sk * dp #Neve em kN/m
#Vento
Qp1 = 0.5 #Vento pilar 1 em kN/m2
vnp1 = Qp1 * dp #Vento pilar 1 em kN/m
Qp2 = 0.5 #Vento pilar 2 em kN/m2
vnp2 = Qp2 * dp #Vento pilar 2 em kN/m
Qa1 = 0.5 #Vento asna 1 em kN/m2
vna1 = Qa1 * dp #Vento asna 1 em kN/m
Qa2 = 0.5 #Vento asna 2 em kN/m2
vna2 = Qa2 * dp #Vento asna 2 em kN/m
#Decompor as ações em normal e tagencial
ppcoberturan = ppcobertura * mt.cos(alfa*mt.pi/180)
ppcoberturat = ppcobertura * mt.sin(alfa*mt.pi/180)
sbn = sb * mt.cos(alfa*mt.pi/180) * mt.cos(alfa*mt.pi/180)
sbt = sb * mt.cos(alfa*mt.pi/180) * mt.sin(alfa*mt.pi/180)
nen = ne * mt.cos(alfa*mt.pi/180) * mt.cos(alfa*mt.pi/180)
net = ne * mt.cos(alfa*mt.pi/180) * mt.sin(alfa*mt.pi/180)
#Coeficientes de majoração
psipp = 1.35
#print("\u03A8 peso próprio:", psipp)
psi0sb = 0
#print("\u03A8 0 sobrecarga:", psi0sb)
psi1sb = 0
#print("\u03A8 1 sobrecarga:", psi1sb)
psi2sb = 0
#print("\u03A8 2 sobrecarga:", psi2sb)
ne1 = 1
ne2 = 2
nete = ne1
if nete == ne1:
psi0ne = 0.5
#print("\u03A8 0 neve:", psi0ne)
psi1ne = 0.2
#print("\u03A8 1 neve:", psi1ne)
psi2ne = 0
#print("\u03A8 2 neve:", psi2ne)
psi0vn = 0.6
#print("\u03A8 0 vento:", psi0vn)
psi1vn = 0.2
#print("\u03A8 1 vento:", psi1vn)
psi2vn = 0
#print("\u03A8 2 vento:", psi2vn)
#Combinação das ações para a cobertura - ELU - asna 1 - normal
comb_sbn = (psipp * ppcoberturan + sbn * 1.5 + (1.5 * psi0ne * nen + 1.5 * psi0vn * vna1)) * 1000 #N/m
comb_vnn = (psipp * ppcoberturan + vna1 * 1.5 + (1.5 * psi0ne * nen + 1.5 * sbn * psi0sb)) * 1000 #N/m
comb_nen =( psipp * ppcoberturan + nen * 1.5 + (1.5 * psi0vn * vna1 + 1.5 * sbn * psi0sb)) * 1000 #N/m
if (comb_sbn >= comb_vnn) and (comb_sbn >= comb_nen):
comb_a1n = comb_sbn
elif (comb_vnn >= comb_sbn) and (comb_vnn >= comb_nen):
comb_a1n = comb_vnn
else:
comb_a1n = comb_nen
#Combinação das ações para a cobertura - ELU - asna 1 - tangencial
comb_sbt = (psipp * ppcoberturat + sbt * 1.5 + (1.5 * psi0ne * net + 1.5 * psi0vn * 0)) * 1000 #N/m
comb_vnt = (psipp * ppcoberturat + 0 * 1.5 + (1.5 * psi0ne * net + 1.5 * sbt * psi0sb)) * 1000 #N/m
comb_net = (psipp * ppcoberturat + net * 1.5 + (1.5 * psi0vn * 0 + 1.5 * sbt * psi0sb)) * 1000 #N/m
if (comb_sbn >= comb_vnn) and (comb_sbn >= comb_nen):
comb_a1t = comb_sbt
elif (comb_vnn >= comb_sbn) and (comb_vnn >= comb_nen):
comb_a1t = comb_vnt
else:
comb_a1t = comb_net
#Combinação das ações para a cobertura - ELU - asna 2 - normal
comb_sb2n = (psipp * ppcoberturan + sbn * 1.5 + (1.5 * psi0ne * nen + 1.5 * psi0vn * vna2)) * 1000 #N/m
comb_vn2n = (psipp * ppcoberturan + vna2 * 1.5 + (1.5 * psi0ne * nen + 1.5 * sbn * psi0sb)) * 1000 #N/m
comb_ne2n = (psipp * ppcoberturan + nen * 1.5 + (1.5 * psi0vn * vna2 + 1.5 * sbn * psi0sb)) * 1000 #N/m
if (comb_sb2n >= comb_vn2n) and (comb_sb2n >= comb_ne2n):
comb_a2n = comb_sb2n
elif (comb_vn2n >= comb_sb2n) and (comb_vn2n >= comb_ne2n):
comb_a2n = comb_vn2n
else:
comb_a2n = comb_ne2n
#Combinação das ações para a cobertura - ELU - asna 2 - tangencial
comb_sbt2 = (psipp * ppcoberturat + sbt * 1.5 + (1.5 * psi0ne * net + 1.5 * psi0vn * 0)) * 1000 #N/m
comb_vnt2 = (psipp * ppcoberturat + 0 * 1.5 + (1.5 * psi0ne * net + 1.5 * sbt * psi0sb)) * 1000 #N/m
comb_net2 = (psipp * ppcoberturat + net * 1.5 + (1.5 * psi0vn * 0 + 1.5 * sbt * psi0sb)) * 1000 #N/m
if (comb_sb2n >= comb_vn2n) and (comb_sb2n >= comb_ne2n):
comb_a2t = comb_sbt2
elif (comb_vn2n >= comb_sb2n) and (comb_vn2n >= comb_ne2n):
comb_a2t = comb_vnt2
else:
comb_a2t = comb_net2
#Elementos finitos - Reações e deslocamentos
E = 210 * 10 ** 9 #módulo de elasticidade do aço em Pa
#Elemento 1 - asna1
a1 = E * arpilar / h
b1 = 12 * E * inypilar / h**3
c1 = 6 * E * inypilar / h**2
d1 = 4 * E * inypilar / h
e1 = 2 * E * inypilar / h
alfa1 = 90 * mt.pi / 180
l1 = mt.cos(alfa1)
m1 = mt.sin(alfa1)
t1 = np.matrix([[l1, m1, 0, 0, 0, 0],
[-m1, l1, 0, 0, 0, 0],
[0, 0 , 1, 0, 0, 0],
[0, 0, 0, l1, m1, 0],
[0, 0, 0, -m1, l1, 0],
[0 , 0, 0, 0, 0, 1]])
k1local = np.matrix([[a1, 0, 0, -a1, 0, 0],
[0, b1, c1, 0, -b1, c1],
[0, c1, d1, 0, -c1, e1],
[-a1, 0, 0, a1, 0, 0],
[0, -b1, -c1, 0, b1, -c1],
[0, c1, e1, 0, -c1, d1]])
invt1 = np.matrix.transpose(t1)
k1global = np.linalg.multi_dot([invt1, k1local, t1])
#Elmento 2 - asna 1
a2 = E * arasna / lasna
b2 = 12 * E *inyasna / lasna**3
c2 = 6 * E * inyasna / lasna**2
d2 = 4 * E * inyasna / lasna
e2 = 2 * E *inyasna / lasna
alfa2 = ((alfa) * mt.pi) / 180
l2 = mt.cos(alfa2)
m2 = mt.sin(alfa2)
t2 = np.matrix([[l2, m2, 0, 0, 0, 0],
[-m2, l2, 0, 0, 0, 0],
[0, 0 , 1, 0, 0, 0],
[0, 0, 0, l2, m2, 0],
[0, 0, 0, -m2, l2, 0],
[0 , 0, 0, 0, 0, 1]])
k2local = np.matrix([[a2, 0, 0, -a2, 0, 0],
[0, b2, c2, 0, -b2, c2],
[0, c2, d2, 0, -c2, e2],
[-a2, 0, 0, a2, 0, 0],
[0, -b2, -c2, 0, b2, -c2],
[0, c2, e2, 0, -c2, d2]])
invt2 = np.matrix.transpose(t2)
k2global = np.linalg.multi_dot([invt2, k2local, t2])
#Elmento 3 - asna 2
a3 = E * arasna / lasna
b3 = 12 * E *inyasna / lasna**3
c3 = 6 * E * inyasna / lasna**2
d3 = 4 * E * inyasna / lasna
e3 = 2 * E *inyasna / lasna
alfa3 = -alfa2
l3 = mt.cos(alfa3)
m3 = mt.sin(alfa3)
t3 = np.matrix([[l3, m3, 0, 0, 0, 0],
[-m3, l3, 0, 0, 0, 0],
[0, 0 , 1, 0, 0, 0],
[0, 0, 0, l3, m3, 0],
[0, 0, 0, -m3, l3, 0],
[0 , 0, 0, 0, 0, 1]])
k3local = np.matrix([[a3, 0, 0, -a3, 0, 0],
[0, b3, c3, 0, -b3, c3],
[0, c3, d3, 0, -c3, e3],
[-a3, 0, 0, a3, 0, 0],
[0, -b3, -c3, 0, b3, -c3],
[0, c3, e3, 0, -c3, d3]])
invt3 = np.matrix.transpose(t3)
k3global = np.linalg.multi_dot([invt3, k3local, t3])
#Elmento 4 - pilar 2
a4 = E * arpilar / h
b4 = 12 * E *inypilar / h**3
c4 = 6 * E * inypilar / h**2
d4 = 4 * E * inypilar / h
e4 = 2 * E *inypilar / h
alfa4 = -(90 * mt.pi/180)
l4 = mt.cos(alfa4)
m4 = mt.sin(alfa4)
t4 = np.matrix([[l4, m4, 0, 0, 0, 0],
[-m4, l4, 0, 0, 0, 0],
[0, 0 , 1, 0, 0, 0],
[0, 0, 0, l4, m4, 0],
[0, 0, 0, -m4, l4, 0],
[0 , 0, 0, 0, 0, 1]])
k4local = np.matrix([[a4, 0, 0, -a4, 0, 0],
[0, b4, c4, 0, -b4, c4],
[0, c4, d4, 0, -c4, e4],
[-a4, 0, 0, a4, 0, 0],
[0, -b4, -c4, 0, b4, -c4],
[0, c4, e4, 0, -c4, d4]])
invt4 = np.matrix.transpose(t4)
k4global = np.linalg.multi_dot([invt4, k4local, t4])
k = [k1global, k2global, k3global, k4global]
kportico = np.zeros([15,15])
for i,m in enumerate(k):
kportico[i*3:i*3+6,i*3:i*3+6] += m
#[K] * {U} = {F} - ELU
F12x = (comb_a1n * lasna / 2) * mt.sin((alfa*mt.pi) / 180)
F12y = (comb_a1n * lasna / 2) * mt.cos((alfa*mt.pi) / 180)
F22x = (comb_a1t * lasna / 2) * mt.cos((alfa*mt.pi) / 180)
F22y = (comb_a1t* lasna / 2) * mt.sin((alfa*mt.pi) / 180)
F13x = F12x
F13y = F12y
F23x = F22x
F23y = F22y
F33x = (comb_a2n * lasna / 2) * mt.sin((alfa*mt.pi) / 180)
F33y = (comb_a2n * lasna / 2) * mt.cos((alfa*mt.pi) / 180)
F43x = (comb_a2t * lasna / 2) * mt.cos((alfa*mt.pi) / 180)
F43y = (comb_a2t * lasna / 2) * mt.sin((alfa*mt.pi) / 180)
F14x = F33x
F14y = F33y
F24x = F43x
F24y = F43y
F1x = (vnp1 * 1000 * h) / 2
F1y = 0
M1 = -(vnp1 * 1000 * h ** 2) / 12
F2x = int((vnp1 * 1000 * h / 2) + F12x - F22x)
F2y = - F12y - F22y
M2 = ((vnp1 * 1000 * h ** 2) / 12) - ((comb_a1n * lasna ** 2) / 12)
F3x = F13x - F23x - F33x + F43x
F3y = -F13y - F23y - F33y - F43y
M3 = ((comb_a1n * lasna ** 2) / 12) - ((comb_a2n * lasna ** 2) / 12)
F4x = (vnp2 * h * 1000 / 2) + F24x - F14x
F4y = - F14y - F24y
M4 = - ((vnp2 * 1000 * h ** 2) / 12) + ((comb_a2n * lasna ** 2) / 12)
F5x = (vnp2 * 1000 * h) / 2
F5y = 0
M5 = (vnp2 * 1000 * h ** 2) / 12
f = np.array([[F1x],
[F1y],
[M1],
[F2x],
[F2y],
[M2],
[F3x],
[F3y],
[M3],
[F4x],
[F4y],
[M4],
[F5x],
[F5y],
[M5]])
fel1 = np.array([[(vnp1 * 1000 * h) / 2],
[0],
[-(vnp1 * 1000 * h ** 2) / 12],
[(vnp1 * 1000 * h) / 2],
[0],
[(vnp1 * 1000 * h ** 2) / 12]])
fel2 = np.array([[(comb_a1n * lasna / 2) * mt.sin(alfa2) - (comb_a1t * lasna / 2) * mt.cos(alfa2)],
[- (comb_a1n * lasna / 2) * mt.cos(alfa2) - (comb_a1t * lasna / 2) * mt.sin(alfa2)],
[-(comb_a1n * lasna ** 2) / 12],
[(comb_a1n * lasna / 2) * mt.sin(alfa2) - (comb_a1t * lasna / 2) * mt.cos(alfa2)],
[- (comb_a1n * lasna / 2) * mt.cos(alfa2) - (comb_a1t * lasna / 2) * mt.sin(alfa2)],
[(comb_a1n * lasna ** 2) / 12]])
fel3 = np.array([[( - comb_a2n * lasna / 2) * mt.sin(-alfa3) + (comb_a2t * lasna / 2) * mt.cos(-alfa3)],
[( - comb_a2n * lasna / 2) * mt.cos(-alfa3) - (comb_a2t * lasna / 2) * mt.sin(-alfa3)],
[-(comb_a2n * lasna ** 2) / 12],
[( - comb_a2n * lasna / 2) * mt.sin(-alfa3) + (comb_a2t * lasna / 2) * mt.cos(-alfa3)],
[( - comb_a2n * lasna / 2) * mt.cos(-alfa3) - (comb_a2t * lasna / 2) * mt.sin(-alfa3)],
[(comb_a2n * lasna ** 2) / 12]])
fel4 = np.array([[(vnp2 * 1000 * h) / 2],
[0],
[-(vnp2 * 1000 * h ** 2) / 12],
[(vnp2 * 1000 * h) / 2],
[0],
[(vnp2 * 1000 * h ** 2) / 12]])
if base == rotulado:
u = np.dot(np.linalg.pinv(kportico[np.ix_([2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14])]), f[np.ix_([2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14])])
utotal = np.array([[0],
[0],
[u[0]],
[u[1]],
[u[2]],
[u[3]],
[u[4]],
[u[5]],
[u[6]],
[u[7]],
[u[8]],
[u[9]],
[0],
[0],
[u[10]]])
r = np.dot(kportico, utotal) - f
rp = np.array([[r[0]],
[r[1]],
[r[12]],
[r[13]]])
rv = np.array([["R1x ="],
["R1y ="],
["R5x ="],
["R5y ="]])
uni = np.array([["N"],
["N"],
["N"],
["N"]])
size = len(rv)
print(" ")
print(" ")
if len(rv) == len(rp) and len(rv) == len(uni):
for x in range(size):
print(rv[x],rp[x],uni[x])
Fpilar1 = np.dot(k1global, utotal[0:6]) - fel1 #ESFORÇOS NÓS PILAR 1
Fasna1 = np.dot(k2global, utotal[3:9]) - fel2 #ESFORÇOS NÓS ASNA 1
Fasna2 = np.dot(k3global, utotal[6:12]) - fel3 #ESFORÇOS NÓS ASNA 2
Fpilar2 = np.dot(k4global, utotal[9:15]) - fel4 #ESFORÇOS NÓS PILAR 2
#Diagrama de esforço transverso e momentos fletores
fig, ax = plt.subplots()
h1 = np.arange(0, h+1, 1)
ha1 = np.arange(0, lasna, 0.1)
ha2 = np.arange(0, lasna, 0.1)
hp2 = np.arange(0, h+1, 1)
Mpilar1 = ((-vnp1 * 1000 * h1 ** 2 / 2) - float(Fpilar1[0]) * h1 - float(Fpilar1[2])) / 1000
Masna1 = (((- comb_a1n * ha1 ** 2 ) / 2) + (float(Fasna1[1]) * mt.cos(alfa2) - float(Fasna1[0]) * mt.sin(alfa2)) * ha1 - float(Fasna1[2])) / 1000
Masna2 = (((- comb_a2n * ha2 ** 2 ) / 2) + (float(Fasna2[1]) * mt.cos(alfa2) + float(Fasna2[0]) * mt.sin(alfa2)) * ha2 - float(Fasna2[2])) / 1000
Mpilar2 = ((vnp2 * 1000 * hp2 ** 2 / 2) + float(Fpilar2[0]) * hp2 - float(Fpilar2[2])) / 1000
# Roto-translation matrix:
# Rotates the points by an angle theta and translates
# them by x in the horizontal direction, and y in the
# vertical direction
R = lambda x, y, theta: np.array([
[np.cos(theta), np.sin(theta), x],
[-np.sin(theta), np.cos(theta), y],
[0, 0, 1],
])
# mirror matrix about the x-axis
Mx = np.array([
[1, 0, 0], [0, -1, 0], [0, 0, 1]
])
# mirror matrix about the y-axis
My = np.array([
[-1, 0, 0], [0, 1, 0], [0, 0, 1]
])
def draw_line():
plt.figure()
x_number_list = [0, 0, (v/2), v, v]
y_number_list = [0, h, ht, h, 0]
plt.plot(x_number_list, y_number_list, linewidth=3)
# left column
points1 = np.stack([h1, Mpilar1 / max(Mpilar1), np.ones_like(h1)])
points1 = np.matmul(R(0, 0, -np.pi/2), points1)
plt.plot(points1[0, :], points1[1, :], label="Mpilar1")
# right column
points2 = np.stack([h1, Mpilar2 / max(Mpilar2), np.ones_like(h1)])
points2 = np.matmul(R(20, 0, -np.pi/2), points2)
plt.plot(points2[0, :], points2[1, :], label="Mpilar2")
# left asna
points3 = np.stack([ha1, Masna1 / max(Masna1), np.ones_like(ha1)])
points3 = np.matmul(R(0, 6, -alfa_rad), points3)
plt.plot(points3[0, :], points3[1, :], label="Masna1")
# right asna
points4 = np.stack([ha1, Masna2 / max(Masna2), np.ones_like(ha1)])
points4 = np.matmul(np.matmul(R(20, 6, alfa_rad), My), points4)
plt.plot(points4[0, :], points4[1, :], label="Masna2")
plt.title("Pórtico", fontsize=15)
plt.xlabel("Vão (m)", fontsize=10)
plt.ylabel("Altura (m)", fontsize=10)
plt.tick_params(axis='both', labelsize=9)
plt.legend()
plt.show()
draw_line()

Gradient Descent Problem with smallest/simplest data on planet Earth

I want to implement the Gradient Descent Algorithm on this simple data but I am facing problems. It would be great if someone points me in the right direction. The answer should be 7 for x=6 but I'm not getting there.
X = [1, 2, 3, 4]
Y = [2, 3, 4, 5]
m_gradient = 0
b_gradient = 0
m, b = 0, 0
learning_rate = 0.1
N = len(Y)
for p in range(100):
for idx in range(len(Y)):
x = X[idx]
y = Y[idx]
hyp = (m * x) + b
m_gradient += -(2/N) * x * (y - hyp)
b_gradient += -(2/N) * (y - hyp)
m = m - (m_gradient * learning_rate)
b = b - (b_gradient * learning_rate)
print(b+m*6)

You are calculating the gradients incorrectly for all but the first iteration. You need to set both gradients to 0 in the outer for loop.
X = [1, 2, 3, 4]
Y = [2, 3, 4, 5]
m_gradient = 0
b_gradient = 0
m, b = 0, 0
learning_rate = 0.1
N = len(Y)
for p in range(100):
for idx in range(len(Y)):
x = X[idx]
y = Y[idx]
hyp = (m * x) + b
m_gradient += -(2/N) * x * (y - hyp)
b_gradient += -(2/N) * (y - hyp)
m = m - (m_gradient * learning_rate)
b = b - (b_gradient * learning_rate)
m_gradient, b_gradient = 0, 0
print(b+m*6)
For example consider b_gradient. Before first iteration b_gradient = 0 and is calculated as 0 + -0.5*(y0 - (m*x0 +b)) + -0.5(y1 - (m*x1 +b)) + -0.5(y2 - (m*x2 + b)) + -0.5(y3 - (m*x3 + b)), where x0 and y0 are X[0] and Y[0], respectively.
After the first iteration the value of b_gradient is -7, this is correct.
The problem starts with the second iteration. Instead of calculating b_gradient as the sum of (-0.5(yn - (m*xn + b)) for 0 <= n <= 3, you calculated it as the previous value of b_gradient plus the sum of (-0.5(yn - (m*xn + b)) for 0 <= n <= 3.
After the second iteration the value of b_gradient is -2.6, this is incorrect. The correct value is 4.4, note that 4.4 - 7 = -2.6.

It seems you want coefficients for Linear Regression using Gradient Descent. Some more data points, a slightly smaller learning rate, training for more epochs by looking at the loss will help reduce error.
As input size gets larger the code below will give slightly off results. The above mentioned methods such as training for more epoch will give correct results for larger range of numbers.
Vectorized Version
import numpy as np
X = np.array([1, 2, 3, 4, 5, 6, 7])
Y = np.array([2, 3, 4, 5, 6, 7, 8])
w_gradient = 0
b_gradient = 0
w, b = 0.5, 0.5
learning_rate = .01
loss = 0
EPOCHS = 2000
N = len(Y)
for i in range(EPOCHS):
# Predict
Y_pred = (w * X) + b
# Loss
loss = np.square(Y_pred - Y).sum() / (2.0 * N)
if i % 100 == 0:
print(loss)
# Backprop
grad_y_pred = (2 / N) * (Y_pred - Y)
w_gradient = (grad_y_pred * X).sum()
b_gradient = (grad_y_pred).sum()
# Optimize
w -= (w_gradient * learning_rate)
b -= (b_gradient * learning_rate)
print("\n\n")
print("LEARNED:")
print(w, b)
print("\n")
print("TEST:")
print(np.round(b + w * (-2)))
print(np.round(b + w * 0))
print(np.round(b + w * 1))
print(np.round(b + w * 6))
print(np.round(b + w * 3000))
# Expected: 30001, but gives 30002.
# Training for 3000 epochs will give expected result.
# For simple demo with less training data and small input range 2000 in enough
print(np.round(b + w * 30000))
Output
LEARNED:
1.0000349103409163 0.9998271260509328
TEST:
-1.0
1.0
2.0
7.0
3001.0
30002.0
Loop Version
import numpy as np
X = np.array([1, 2, 3, 4, 5, 6, 7])
Y = np.array([2, 3, 4, 5, 6, 7, 8])
w_gradient = 0
b_gradient = 0
w, b = 0.5, 0.5
learning_rate = .01
loss = 0
EPOCHS = 2000
N = len(Y)
for i in range(EPOCHS):
w_gradient = 0
b_gradient = 0
loss = 0
for j in range(N):
# Predict
Y_pred = (w * X[j]) + b
# Loss
loss += np.square(Y_pred - Y[j]) / (2.0 * N)
# Backprop
grad_y_pred = (2 / N) * (Y_pred - Y[j])
w_gradient += (grad_y_pred * X[j])
b_gradient += (grad_y_pred)
# Optimize
w -= (w_gradient * learning_rate)
b -= (b_gradient * learning_rate)
# Print loss
if i % 100 == 0:
print(loss)
print("\n\n")
print("LEARNED:")
print(w, b)
print("\n")
print("TEST:")
print(np.round(b + w * (-2)))
print(np.round(b + w * 0))
print(np.round(b + w * 1))
print(np.round(b + w * 6))
print(np.round(b + w * 3000))
# Expected: 30001, but gives 30002.
# Training for 3000 epochs will give expected result.
# For simple demo with less training data and small input range 2000 in enough
print(np.round(b + w * 30000))
Output
LEARNED:
1.0000349103409163 0.9998271260509328
TEST:
-1.0
1.0
2.0
7.0
3001.0
30002.0

My program calculates only the last value

Here is my code:
import numpy as np
cx = np.array([0, 0, 3, 3])
cy = np.array([0, 3, 4, 0])
M = len(cx)
for j in range(M):
wx = 0
wy = 0
for i in range(M):
if i == j:
continue
x = cx[i] - cx[j]
y = cy[i] - cy[j]
wx += -x / np.sqrt(x ** 2 + y ** 2)
wy += -y / np.sqrt(x ** 2 + y ** 2)
Move = (
wx / np.sqrt(wx ** 2 + wy ** 2),
wy / np.sqrt(wx ** 2 + wy ** 2),
)
What is wrong with my code?
Your help will be highly appreciated

import numpy as np
cx = np.array([0, 0, 3, 3])
cy = np.array([0, 3, 4, 0])
M = len(cx)
for j in range(M):
wx = 0
wy = 0
for i in range(M):
if i == j:
continue
x = cx[i] - cx[j]
y = cy[i] - cy[j]
wx += -x / np.sqrt(x ** 2 + y ** 2)
wy += -y / np.sqrt(x ** 2 + y ** 2)
Move = (
wx / np.sqrt(wx ** 2 + wy ** 2),
wy / np.sqrt(wx ** 2 + wy ** 2),
)
print(Move)

Try this:
import numpy as np
cx = np.array([0, 0, 3, 3])
cy = np.array([0, 3, 4, 0])
M = len(cx)
for j in range(M):
wx = 0
wy = 0
for i in range(M):
if i == j:
continue
x = cx[i] - cx[j]
y = cy[i] - cy[j]
wx += -x / np.sqrt(x ** 2 + y ** 2)
wy += -y / np.sqrt(x ** 2 + y ** 2)
Move = (
wx / np.sqrt(wx ** 2 + wy ** 2),
wy / np.sqrt(wx ** 2 + wy ** 2),
)
print(Move)

IndexError: index 3 is out of bounds for axis 0 with size 3

So this is my code to plot a Bézier curve:
def bezier(a):
n = np.shape(a)[0]-1
# initialise arrays
B = np.zeros([101, 2])
terms = np.zeros([n+1, 2])
# create an array of values for t from 0 to 1 in 101 steps
t = np.linspace(0, 1, 101)
# loop through all t values
for i in range(0, 101):
#calculate terms inside sum in equation 13
for j in range(0, n + 1):
# YOUR CODE HERE
terms[j,:] = ((1 - t[i]) ** 3 * a[0,:] \
+ 3 * t[i] * (1-t[i]) ** 2 * a[1,:] \
+ 3 * t[i] ** 2 * (1-t[i]) * a[2,:]
+ t[i] ** 3 * a[3,:])
#sum terms to find Bezier curve
B[i, :] = sum(terms, 0)
# plot Bezier
pl.plot(B[:, 0], B[:, 1])
# plot control points
pl.plot(a[:, 0], a[:, 1],'ko')
# plot control polygon
pl.plot(a[:, 0], a[:, 1],'k')
return B
And when I try to pass it some control points:
a = np.array([[0, 0], [0.5, 1], [1, 0]])
B = bezier(a)
I receive this IndexError:
---------------------------------------------------------------------------
IndexError Traceback (most recent call last)
<ipython-input-16-fce87c9f1c04> in <module>()
1 a = np.array([[0, 0], [0.5, 1], [1, 0]])
----> 2 B = bezier(a)
<ipython-input-13-3bb3bb02cc87> in bezier(a)
11 for j in range(0, n + 1):
12 # YOUR CODE HERE
---> 13 terms[j,:] = ((1 - t[i]) ** 3 * a[0,:] + 3 * t[i] * (1-t[i]) ** 2 * a[1,:] + 3 * t[i] ** 2 * (1-t[i]) * a[2,:] + t[i] ** 3 * a[3,:])
14 #sum terms to find Bezier curve
15 B[i, :] = sum(terms, 0)
IndexError: index 3 is out of bounds for axis 0 with size 3
So I figure it is trying to access something outside the container but I can't see where it is I need to change the code.

You array a = np.array([[0, 0], [0.5, 1], [1, 0]] does not have element with index 3. Add another point to the array. You need four points for a Bezier curve, anyway.

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