def third():
A = Matrica([[0, -2], [1, -3]])
B = Matrica([[2, 0], [0, 3]])
x = Matrica([[1], [3]])
r = Matrica([[1], [1]])
T = 0.01
tmax = 10
f = [runge_kutta, euler, reverse_euler, trapeze, pece, pece2]
fig, ax = plt.subplots(6)
fig.suptitle("Third task")
fig.set_size_inches(10, 10)
i = 0
for fi in f:
print(fi.__name__)
x1, x2, t = fi(A, x, T, tmax, B=B, r=r, print_step=tmax / T)
ax[i].plot(t, x1)
ax[i].plot(t, x2)
ax[i].set_title(fi.__name__)
ax[i].set_xlabel("time")
ax[i].set_ylabel("X")
i += 1
plt.show()
def runge_kutta(A, x, T, tmax, B=None, r=None, print_step=0):
B = B if B != None else 0 #Matrica([[0 for _ in range(A.get_dimension())] for _ in range(A.get_dimension())])
t = 0
xk = x
x1 = []
x2 = []
t_array = []
i = 1
while t <= tmax:
t_array.append(t)
r1 = Matrica([[t], [t]]) if r == None else r
m1 = A * xk + B * r1
r2 = Matrica([[t + T / 2], [t + T / 2]]) if r == None else r
m2 = A * (xk + m1 * (T / 2)) + B * r2
r3 = Matrica([[t + T / 2], [t + T / 2]]) if r == None else r
m3 = A * (xk + m2 * (T / 2)) + B * r3
r4 = Matrica([[t + T], [t + T]]) if r == None else r
m4 = A * (xk + m3 * T) + B * r4
xk1 = xk + (m1 + m2 * 2 + m3 * 2 + m4) * T * (1.0 / 6)
xk = xk1
x1.append(xk1[0][0])
x2.append(xk1[1][0])
t += T
if i % print_step == 0:
print("xk+1:" + str(xk1[0][0]) + ", " + str(xk1[1][0]))
return x1, x2, t_array
def pece2(A, x, T, tmax, B=None, r=None, print_step=0):
U = indentity_matrix(A.get_dimension())
M = U + A * T
N = B * T if B != None else 0
P = ~(U - A * T)
Q = (~(U - A * T)) * B * T if B != None else 0
t = 0
xk = x
x1 = []
x2 = []
t_array = []
i = 1
while t <= tmax:
t_array.append(t)
#Euler
r = Matrica([[t], [t]]) if r == None else r
x_p = M * xk + N * r
#Reverse Euler
r = Matrica([[t + T], [t + T]]) if r == None else r
x_c = P * x_p + Q * r
#Reverse Euler
xk1 = P * x_c + Q * r
xk = xk1
x1.append(xk1[0][0])
x2.append(xk1[1][0])
t += T
if i % print_step == 0:
print("xk+1:" + str(xk1[0][0]) + ", " + str(xk1[1][0]))
return x1, x2, t_array
def pece(A, x, T, tmax, B=None, r=None, print_step=0):
U = indentity_matrix(A.get_dimension())
M = U + A * T
N = B * T if B != None else 0
R = (~(U - A * 0.5 * T)) * (U - A * 0.5 * T)
S = (~(U - A * 0.5 * T)) * 0.5 * T * B if B != None else 0
t = 0
xk = x
x1 = []
x2 = []
t_array = []
i = 1
while t <= tmax:
t_array.append(t)
#Euler
r = Matrica([[t], [t]]) if r == None else r
x_p = M * xk + N * r
#Trapeze
r = Matrica([[2 * t + T], [2 * t + T]]) if r == None else r
xk1 = R * x_p + S * r
xk = xk1
x1.append(xk1[0][0])
x2.append(xk1[1][0])
t += T
if i % print_step == 0:
print("xk+1:" + str(xk1[0][0]) + ", " + str(xk1[1][0]))
return x1, x2, t_array
def first():
A = Matrica([[0, 1], [-1, 0]])
x = Matrica([[1], [1]])
r = 0
T = 0.01
tmax = 10
f = [runge_kutta, euler, reverse_euler, trapeze, pece, pece2]
t = 0
xa1 = []
xa2 = []
while t <= tmax:
xa1.append(x[0][0] * cos(t) + x[1][0] * sin(t))
xa2.append(x[1][0] * cos(t) - x[0][0] * sin(t))
t += T
fig, ax = plt.subplots(7)
fig.suptitle("First task")
fig.set_size_inches(10, 10)
i = 0
for fi in f:
print(fi.__name__)
x1, x2, t = fi(A, x, T, tmax, r=r, print_step=tmax / T)
ax[i].plot(t, x1)
ax[i].plot(t, x2)
ax[i].set_title(fi.__name__)
ax[i].set_xlabel("time")
ax[i].set_ylabel("X")
i += 1
error = 0
for j in range(len(x1)):
error += abs(xa1[j] - x1[j]) + abs(xa2[j] - x2[j])
print(fi.__name__ + " Error: " + str(error))
ax[6].plot(t, xa1)
ax[6].plot(t, xa2)
ax[6].set_title("Analitical")
ax[6].set_xlabel("time")
ax[6].set_ylabel("X")
plt.show()
def second():
A = Matrica([[0, 1], [-200, -102]])
x = Matrica([[1], [-2]])
r = 0
T = 0.1
tmax = 1
f = [runge_kutta, euler, reverse_euler, trapeze, pece, pece2]
fig, ax = plt.subplots(6)
fig.suptitle("Second task")
fig.set_size_inches(10, 10)
i = 0
for fi in f:
print(fi.__name__)
x1, x2, t = fi(A, x, T, tmax, r=r, print_step=10000)
ax[i].plot(t, x1)
ax[i].plot(t, x2)
ax[i].set_title(fi.__name__)
ax[i].set_xlabel("time")
ax[i].set_ylabel("X")
i += 1
plt.show()
def euler(A, x, T, tmax, B=None, r=None, print_step=0):
U = indentity_matrix(A.get_dimension())
M = U + A * T
N = B * T if B != None else 0
t = 0
xk = x
x1 = []
x2 = []
t_array = []
i = 1
while t <= tmax:
t_array.append(t)
r = Matrica([[t], [t]]) if r == None else r
xk1 = M * xk + N * r
xk = xk1
x1.append(xk1[0][0])
x2.append(xk1[1][0])
t += T
if i % print_step == 0:
print("xk+1:" + str(xk1[0][0]) + ", " + str(xk1[1][0]))
return x1, x2, t_array
def indentity_matrix(n):
return Matrica([[1 if i == j else 0 for j in range(n)] for i in range(n)])