]
sum_substitution_cases = [
(a * b + Sum(c * k, (k, 0, n)), {
'a': b,
'b': 2,
'k': 1,
'n': 2
}, b * 2 + c * (1 + 2)),
]
indexed_substitution_cases = [(a_[i] * b, {
'b': 3
}, a_[i] * 3),
(a_[i] * b, {
'a': sympy.Array([1, 2, 3])
}, sympy.Array([1, 2, 3])[i] * b),
(sympy.Array([1, 2, 3])[i] * b, {
'i': 1
}, 2 * b)]
vector_valued_cases = [
(a * b, {
'a': sympy.Array([1, 2, 3])
}, sympy.Array([1, 2, 3]) * b),
(a * b, {
'a': sympy.Array([1, 2, 3]),
'b': sympy.Array([4, 5, 6])
}, sympy.Array([4, 10, 18])),
]
inp1, inp2, inp3, inp4, f = sp.symbols('inp1 inp2 inp3 inp4 f')
#_Weights_definitions_________________________________________________________
neuronsByLayer = [4, 4, 4, 2, 1]
W = []
for k in range(len(neuronsByLayer) - 1):
auxLis = []
for i in range(neuronsByLayer[k]):
for j in range(neuronsByLayer[k + 1]):
exec('w' + str(k + 1) + str(i + 1) + str(j + 1) +
' = sp.symbols("w' + str(k + 1) + str(i + 1) + str(j + 1) +
'")')
exec('auxLis.append(w' + str(k + 1) + str(i + 1) + str(j + 1) +
')')
W.append(
sp.Array(auxLis).reshape(neuronsByLayer[k], neuronsByLayer[k + 1]))
#______________________________________________________________________________
#_function_definition_and_input__________
f = lambda x: 1 / (1 + sp.exp(-x))
input = sp.Array([inp1, inp2, inp3, inp4])
#_________________________________________
#_Outputs_____________________________________________________________________
o_l1 = [f(input[0]), f(input[1]), f(input[2]), f(input[3])]
o_l2 = tenCon(tenProd(o_l1, W[0]).applyfunc(f), (0, 1))
o_l3 = tenCon(tenProd(o_l2, W[1]).applyfunc(f), (0, 1))
o_l4 = tenCon(tenProd(o_l3, W[2]).applyfunc(f), (0, 1))
o_l5 = tenCon(tenProd(o_l4, W[3]).applyfunc(f), (0, 1))
#______________________________________________________________________________
def Z(self, value):
self.validation.Zsetter(self, value)
if value is None:
self._Z = None
else:
self._Z = sp.Array(value)
W = W.reshape(outputNeurons, inputNeurons)
return (W)
#_weights_between_layers_1_and_2____
W_l1tol2 = weights(1, 4, 4)
#_weights_between_layers_2_and_3____
W_l2tol3 = weights(2, 4, 4)
#_weights_between_layers_3_and_4____
W_l3tol4 = weights(3, 4, 2)
#_weights_between_layers_4_and_5____
W_l4tol5 = weights(4, 2, 1)
#______________________________________________________________________________
#_function_definition_and_input______________
f = lambda x: 1 / (1 + sp.exp(-x))
input = sp.Array([inp1, inp2, inp3, inp4])
#_____________________________________________
#_Outputs_____________________________________________________________________
o_l1 = sp.Array([[f(input[0]), f(input[1]), f(input[2]), f(input[3])]])
o_l2 = sp.Array([
sp.tensorcontraction(
sp.tensorproduct(o_l1, W_l1tol2)[0, :, :, :].applyfunc(f), (0, 1))
])
o_l3 = sp.Array([
sp.tensorcontraction(
sp.tensorproduct(o_l2, W_l2tol3)[0, :, :, :].applyfunc(f), (0, 1))
])
o_l4 = sp.Array([
sp.tensorcontraction(
sp.tensorproduct(o_l3, W_l3tol4)[0, :, :, :].applyfunc(f), (0, 1))
import matplotlib.pyplot as plt
import numpy as np
import sympy
from sympy.parsing import sympy_parser
import lmfit
###############################################################################
# Instead of creating the SymPy symbols explicitly and building an expression
# with them, we will use the SymPy parser.
gauss_peak1 = sympy_parser.parse_expr('A1*exp(-(x-xc1)**2/(2*sigma1**2))')
gauss_peak2 = sympy_parser.parse_expr('A2*exp(-(x-xc2)**2/(2*sigma2**2))')
exp_back = sympy_parser.parse_expr('B*exp(-x/xw)')
model_list = sympy.Array((gauss_peak1, gauss_peak2, exp_back))
model = sum(model_list)
print(model)
###############################################################################
# We are using SymPy's lambdify function to make a function from the model
# expressions. We then use these functions to generate some fake data.
model_list_func = sympy.lambdify(list(model_list.free_symbols), model_list)
model_func = sympy.lambdify(list(model.free_symbols), model)
###############################################################################
# Generate synthetic data with noise and plot the data.
np.random.seed(1)
x = np.linspace(0, 10, 40)
param_values = dict(x=x,
def test_array_broadcast(self):
expected = sympy.Array([1, 2, a, b])
self.assertEqual(expected, Broadcast(list(expected), (4,)))
self.assertEqual(expected, Broadcast(tuple(expected), (4,)))
self.assertEqual(expected, Broadcast(expected, (4,)))
import numpy as np
import sympy as sp
import math
import matplotlib.pyplot as plt
xp = sp.Array([-1.5, -1, -0.5, 0, 0.5, 1, 1.5])
def func_np(x):
y = 8 * np.sin(1 / np.tan(3)) + (np.sin(5 * x))**2 / (5 * np.cos(10 * x))
return y
def func(x):
y = 8 * sp.sin(1 / sp.tan(3)) + (sp.sin(5 * x))**2 / (5 * sp.cos(10 * x))
return y
def func_six():
x = sp.Symbol('x')
r = sp.diff(func(x), x, xp.shape[0])
return sp.lambdify(x, r, 'numpy')
def newton_up():
n = xp.shape[0]
x = sp.Symbol('x')
res = 0
div = func(x)
for i in range(0, n, 1):
temp = div.evalf(subs={x: xp[i]})
from sympy import tensorcontraction as tenCon
from sympy import tensorproduct as tenProd
inp1, inp2, inp3, inp4, f = sp.symbols('inp1 inp2 inp3 inp4 f')
#_Weights_definitions_________________________________________________________
neuronsByLayer = [5,10,10,5,1]
W = []
for k in range(len(neuronsByLayer) - 1):
auxLis = []
for i in range(neuronsByLayer[k]):
for j in range(neuronsByLayer[k + 1]):
exec('w' + str(k+1) + str(i+1) + str(j+1) +
' = sp.symbols("w' + str(k+1) + str(i+1) + str(j+1) + '")')
exec('auxLis.append(w' + str(k+1) + str(i+1) + str(j+1) + ')')
W.append(sp.Array(auxLis).reshape(neuronsByLayer[k],
neuronsByLayer[k + 1]))
#______________________________________________________________________________
#_input_________________________________________________________
input = []
for i in range(neuronsByLayer[0]):
exec('inp' + str(i+1) + ' = sp.symbols("inp' + str(i+1) + '")')
exec('input.append(inp' + str(i+1) + ')')
#________________________________________________________________
#_function_definition_and_input__________
f = lambda x: 1 / (1 + sp.exp(-x))
#_________________________________________
def f(self, x, u, p, *, s):
"""ODE function."""
u0 = 44.5695
wd = s.zw*s.w + (u0 + s.zq)*s.q + s.zeta*s.eta
qd = s.mw*s.w + s.mq*s.q + s.meta*s.eta
return sympy.Array([wd, qd])
def get_wind_ned_sym2(self, _x, _y, _z, _t):
wd = self.strength*sym.exp(-((_x-self.center[0])**2 + (_y-self.center[1])**2)/self.radius**2)
ret = sym.Array([self.cst[0], self.cst[1], self.cst[2]+wd])
#ret = sym.Array(self.cst)+[0, 0, wd] # nope
return ret
def get_wind_ned_sym2(self, _x, _y, _z, _t):
ret = sym.Array([0, 0, 0])
for atm in self.atms:
ret += atm.get_wind_ned_sym2(_x, _y, _z, _t)
return ret
def get_wind_ned_sym2(self, _n, _e, _d, _t):
wd = self.strength*sym.exp(-((_n-self.center[0])**2 + self.foo*(_e-self.center[1])**2)/self.radius**2)
ret = sym.Array([self.cst[0], self.cst[1], self.cst[2]+wd])
return ret
def get_wind_ned_sym2(self, _n, _e, _d, _t):
return sym.Array([0, self.s*sym.cos(_n/25.), 0])
def SARIMACorrelation(trendparams: tuple = (0, 0, 0),
seasonalparams: tuple = (0, 0, 0, 1),
trendAR=None,
trendMA=None,
seasonAR=None,
seasonMA=None):
p, d, q = trendparams
P, D, Q, m = seasonalparams
print(f'SARIMA({p},{d},{q})({P},{D},{Q},{m})')
assert type(p) is int, 'Input parameter "p" is not an integer type.'
assert type(d) is int, 'Input parameter "d" is not an integer type.'
assert type(q) is int, 'Input parameter "q" is not an integer type.'
assert type(P) is int, 'Input parameter "P" is not an integer type.'
assert type(D) is int, 'Input parameter "D" is not an integer type.'
assert type(Q) is int, 'Input parameter "Q" is not an integer type.'
assert type(m) is int, 'Input parameter "m" is not an integer type.'
if trendAR:
assert len(
trendAR
) == p, f'The len(trendAR) must be {p}. Reset the parameters.'
else:
trendAR = [0] * p
if trendMA:
assert len(
trendMA
) == q, f'The len(trendMA) must be {q}. Reset the parameters.'
else:
trendMA = [0] * q
if seasonAR:
assert len(
seasonAR
) == P, f'The len(seasonAR) must be {P}. Reset the parameters.'
else:
seasonAR = [0] * P
if seasonMA:
assert len(
seasonMA
) == Q, f'The len(seasonMA) must be {Q}. Reset the parameters.'
else:
seasonMA = [0] * Q
Y_order = p + P * m + d + D * m
e_order = q + Q * m
# define Y, e
Y, e = sympy.symbols('Y_t, e_t')
I, J = sympy.symbols('i, j')
Y_ = {}
e_ = {}
Y_['t'] = Y
Y__ = [[Y_['t']]]
e_['t'] = e
e__ = [[e_['t']]]
for i in range(1, Y_order + 1):
Y_[f't-{i}'] = sympy.symbols(f'Y_t-{i}')
Y__.append([
Y_[f't-{i}'] * (I**i)
]) # Y__ = [ [Y_['t']], [Y_['t-1']], ..., [Y_['t-(p+P*m+q+Q*m)']] ]
for i in range(1, e_order + 1):
e_[f't-{i}'] = sympy.symbols(f'e_t-{i}')
e__.append([
e_[f't-{i}'] * (J**i)
]) # e__ = [ [e_['t']], [e_['t-1']], ..., [e_['t-(q+Q*m)']] ]
# define L
L = sympy.symbols('L')
S_Lag = L**m
T_Lag = L
S_Lag_Diff = (1 - L**m)**D
T_Lag_Diff = (1 - L)**d
# define coefficients : phis(T), Phis(S), thetas(T), Thetas(S)
T_phi = {}
T_phis = []
L_byT_phi = []
S_phi = {}
S_phis = []
L_byS_phi = []
T_theta = {}
T_thetas = []
L_byT_theta = []
S_theta = {}
S_thetas = []
L_byS_theta = []
for p_ in range(0, p + 1):
T_phi[p_] = sympy.symbols(f'phi_{p_}')
T_phis.append(
-T_phi[p_]) # T_phis = [T_phi[0], T_phi[1], ..., T_phi[p]]
L_byT_phi.append([T_Lag**p_
]) # L_byT_phi = [[L**0], [L**1], ..., [L**p]]
for P_ in range(0, P + 1):
S_phi[P_] = sympy.symbols(f'Phi_{P_}')
S_phis.append(
-S_phi[P_]) # S_phis = [S_phi[0], S_phi[1], ..., S_phi[P]]
L_byS_phi.append([
S_Lag**P_
]) # L_byS_phi = [[(L**m)**0], [(L**m)**1], ..., [(L**m)**P]]
for q_ in range(0, q + 1):
T_theta[q_] = sympy.symbols(f'theta_{q_}')
T_thetas.append(
T_theta[q_]
) # T_thetas = [T_theta[0], T_theta[1], ..., T_theta[q]]
L_byT_theta.append([T_Lag**q_
]) # L_byT_theta = [[L**0], [L**1], ..., [L**q]]
for Q_ in range(0, Q + 1):
S_theta[Q_] = sympy.symbols(f'Theta_{Q_}')
S_thetas.append(
S_theta[Q_]
) # S_thetas = [T_theta[0], T_theta[1], ..., T_theta[Q]]
L_byS_theta.append([
S_Lag**Q_
]) # L_byS_theta = [[(L**m)**0], [(L**m)**1], ..., [(L**m)**Q]]
T_phi_Lag = sympy.Matrix([T_phis]) * sympy.Matrix(L_byT_phi)
S_phi_Lag = sympy.Matrix([S_phis]) * sympy.Matrix(L_byS_phi)
T_theta_Lag = sympy.Matrix([T_thetas]) * sympy.Matrix(L_byT_theta)
S_theta_Lag = sympy.Matrix([S_thetas]) * sympy.Matrix(L_byS_theta)
Y_operator = (T_phi_Lag * S_phi_Lag * T_Lag_Diff * S_Lag_Diff).subs(
T_phi[0], -1).subs(S_phi[0], -1)[0]
e_operator = (T_theta_Lag * S_theta_Lag).subs(T_theta[0],
1).subs(S_theta[0], 1)[0]
Y_operation = sympy.collect(Y_operator.expand(), L)
e_operation = sympy.collect(e_operator.expand(), L)
Y_coeff = sympy.Poly(Y_operation, L).all_coeffs()[::-1]
e_coeff = sympy.Poly(e_operation, L).all_coeffs()[::-1]
Y_term = sympy.Matrix([Y_coeff]) * sympy.Matrix(Y__) # left-side
e_term = sympy.Matrix([e_coeff]) * sympy.Matrix(e__) # right-side
Time_Series = {}
Time_Series['Y_t(i,j)'] = sympy.Poly(Y - Y_term[0] + e_term[0], (I, J))
Time_Series['Y_t'] = Time_Series['Y_t(i,j)'].subs(I, 1).subs(J, 1)
for i in range(1, int(p + P * m + d + D * m) + 1):
Time_Series['Y_t'] = sympy.collect(Time_Series['Y_t'],
Y_[f't-{i}']).simplify()
for i in range(1, int(q + Q * m) + 1):
Time_Series['Y_t'] = sympy.collect(Time_Series['Y_t'],
e_[f't-{i}']).simplify()
print('* Time Series Equation(Analytic Form)')
sympy.pprint(Time_Series['Y_t'])
print()
Time_Series['Analytic_Coeff_of_Y'] = Time_Series['Y_t(i,j)'].subs(
J, 0).all_coeffs()[::-1]
Time_Series['Analytic_Coeff_of_e'] = Time_Series['Y_t(i,j)'].subs(
I, 0).all_coeffs()[::-1]
Time_Series['Numeric_Coeff_of_Y'] = Time_Series['Y_t(i,j)'].subs(
J, 0) - e_['t']
Time_Series['Numeric_Coeff_of_e'] = Time_Series['Y_t(i,j)'].subs(I, 0)
for i, (phi, Np) in enumerate(zip(list(T_phi.values())[1:], trendAR)):
Time_Series['Numeric_Coeff_of_Y'] = Time_Series[
'Numeric_Coeff_of_Y'].subs(phi, Np)
for i, (Phi, NP) in enumerate(zip(list(S_phi.values())[1:], seasonAR)):
Time_Series['Numeric_Coeff_of_Y'] = Time_Series[
'Numeric_Coeff_of_Y'].subs(Phi, NP)
for i, (theta, Nt) in enumerate(zip(list(T_theta.values())[1:], trendMA)):
Time_Series['Numeric_Coeff_of_e'] = Time_Series[
'Numeric_Coeff_of_e'].subs(theta, Nt)
for i, (Theta, NT) in enumerate(zip(list(S_theta.values())[1:], seasonMA)):
Time_Series['Numeric_Coeff_of_e'] = Time_Series[
'Numeric_Coeff_of_e'].subs(Theta, NT)
print('* Time Series Equation(Numeric Form)')
sympy.pprint((Time_Series['Numeric_Coeff_of_Y'] +
Time_Series['Numeric_Coeff_of_e']).subs(I, 1).subs(J, 1))
Time_Series['Numeric_Coeff_of_Y'] = sympy.Poly(
Time_Series['Numeric_Coeff_of_Y'], I).all_coeffs()[::-1]
Time_Series['Numeric_Coeff_of_e'] = sympy.Poly(
Time_Series['Numeric_Coeff_of_e'], J).all_coeffs()[::-1]
final_coeffs = [[], []]
print('\n* Y params')
print(f'- TAR({trendparams[0]}) phi : {trendAR}')
print(f'- TMA({trendparams[2]}) theta : {trendMA}')
for i, (A_coeff_Y, N_coeff_Y) in enumerate(
zip(Time_Series['Analytic_Coeff_of_Y'],
Time_Series['Numeric_Coeff_of_Y'])):
if i == 0:
pass
elif i != 0:
A_coeff_Y = A_coeff_Y.subs(Y_[f"t-{i}"], 1)
N_coeff_Y = N_coeff_Y.subs(Y_[f"t-{i}"], 1)
print(f't-{i} : {A_coeff_Y} > {round(N_coeff_Y, 5)}')
final_coeffs[0].append(N_coeff_Y)
print('\n* e params')
print(f'- SAR({seasonalparams[0]}) Phi : {seasonAR}')
print(f'- SMA({seasonalparams[2]}) Theta : {seasonMA}')
for i, (A_coeff_e, N_coeff_e) in enumerate(
zip(Time_Series['Analytic_Coeff_of_e'],
Time_Series['Numeric_Coeff_of_e'])):
if i == 0:
A_coeff_e = A_coeff_e.subs(e_[f"t"], 1)
N_coeff_e = N_coeff_e.subs(e_[f"t"], 1)
print(f't-{i} : {A_coeff_e} > {1}')
elif i != 0:
A_coeff_e = A_coeff_e.subs(e_[f"t-{i}"], 1)
N_coeff_e = N_coeff_e.subs(e_[f"t-{i}"], 1)
print(f't-{i} : {A_coeff_e} > {round(N_coeff_e, 5)}')
final_coeffs[1].append(N_coeff_e)
print('\n* Quasi-Convergence Factor')
try:
print(
'Y :',
sympy.tensorcontraction(
sympy.Array(final_coeffs[0]).applyfunc(lambda x: x**2), (0, )))
print(
'e :',
sympy.tensorcontraction(
sympy.Array(final_coeffs[1]).applyfunc(lambda x: x**2), (0, )))
except:
pass
_, axes = plt.subplots(5, 1, figsize=(12, 15))
ar_params = np.array(final_coeffs[0])
ma_params = np.array(final_coeffs[1])
ar, ma = np.r_[1, -ar_params], np.r_[1, ma_params]
y = smt.ArmaProcess(ar, ma).generate_sample(300, burnin=50)
axes[0].plot(y, 'o-')
axes[0].grid(True)
axes[1].stem(smt.ArmaProcess(ar, ma).acf(lags=40))
axes[1].set_xlim(-1, 41)
axes[1].set_ylim(-1.1, 1.1)
axes[1].set_title(
"Theoretical autocorrelation function of an SARIMA process")
axes[1].grid(True)
axes[2].stem(smt.ArmaProcess(ar, ma).pacf(lags=40))
axes[2].set_xlim(-1, 41)
axes[2].set_ylim(-1.1, 1.1)
axes[2].set_title(
"Theoretical partial autocorrelation function of an SARIMA process")
axes[2].grid(True)
smt.graphics.plot_acf(y, lags=40, ax=axes[3])
axes[3].set_xlim(-1, 41)
axes[3].set_ylim(-1.1, 1.1)
axes[3].set_title(
"Experimental autocorrelation function of an SARIMA process")
axes[3].grid(True)
smt.graphics.plot_pacf(y, lags=40, ax=axes[4])
axes[4].set_xlim(-1, 41)
axes[4].set_ylim(-1.1, 1.1)
axes[4].set_title(
"Experimental partial autocorrelation function of an SARIMA process")
axes[4].grid(True)
plt.tight_layout()
plt.show()
def to_equation(self):
"""Returns the value returned by the node when evaluated."""
entries_list = self.entries.tolist()
return sympy.Array(entries_list)
def g(self, x, u, p, *, s):
"""Measurement log likelihood."""
az = s.zw*s.w + s.zq*s.q + s.zeta*s.eta
return sympy.Array([s.w, s.q, az])
(a*b, {'a': c}, b*c),
(a*b, {'a': b, 'b': a}, a*b),
(a*b, {'a': 1, 'b': 2}, 2),
]
elem_func_substitution_cases = [
(a*b + sin(c), {'a': b, 'c': sympy.pi/2}, b**2 + 1),
]
sum_substitution_cases = [
(a*b + Sum(c * k, (k, 0, n)), {'a': b, 'b': 2, 'k': 1, 'n': 2}, b*2 + c*(1 + 2)),
]
indexed_substitution_cases = [
(a_[i]*b, {'b': 3}, a_[i]*3),
(a_[i]*b, {'a': sympy.Array([1, 2, 3])}, sympy.Array([1, 2, 3])[i]*b),
(sympy.Array([1, 2, 3])[i]*b, {'i': 1}, 2*b)
]
vector_valued_cases = [
(a*b, {'a': sympy.Array([1, 2, 3])}, sympy.Array([1, 2, 3])*b),
(a*b, {'a': sympy.Array([1, 2, 3]), 'b': sympy.Array([4, 5, 6])}, sympy.Array([4, 10, 18])),
]
full_featured_cases = [
(Sum(a_[i], (i, 0, Len(a) - 1)), {'a': sympy.Array([1, 2, 3])}, 6),
]
##################################################### SYMPIFY ##########################################################
simple_sympify = [
]
sum_substitution_cases = [
(a * b + Sum(c * k, (k, 0, n)), {
'a': b,
'b': 2,
'k': 1,
'n': 2
}, b * 2 + c * (1 + 2)),
]
indexed_substitution_cases = [(a_[i] * b, {
'b': 3
}, a_[i] * 3),
(a_[i] * b, {
'a': sympy.Array([1, 2, 3])
}, sympy.Array([1, 2, 3])[i] * b),
(sympy.Array([1, 2, 3])[i] * b, {
'i': 1
}, 2 * b)]
vector_valued_cases = [
(a * b, {
'a': sympy.Array([1, 2, 3])
}, sympy.Array([1, 2, 3]) * b),
(a * b, {
'a': sympy.Array([1, 2, 3]),
'b': sympy.Array([4, 5, 6])
}, sympy.Array([4, 10, 18])),
(a + b, {
'a': sympy.Array([1, 2, 3])
import sympy as sp
neuronsByLayer = [4, 4, 4, 2, 1]
W = []
for k in range(len(neuronsByLayer) - 1):
auxLis = []
for i in range(neuronsByLayer[k]):
for j in range(neuronsByLayer[k + 1]):
exec('w' + str(k + 1) + str(i + 1) + str(j + 1) +
' = sp.symbols("w' + str(k + 1) + str(i + 1) + str(j + 1) +
'")')
exec('auxLis.append(w' + str(k + 1) + str(i + 1) + str(j + 1) +
')')
W.append(
sp.Array(auxLis).reshape(neuronsByLayer[k], neuronsByLayer[k + 1]))
R[0, 1, 0, 0, 1] = rri
R[0, 1, 1, 0, 1] = rti
R[0, 1, 0, 1, 1] = rsi
R[0, 1, 1, 1, 1] = rpi
R[1, 1, 0, 0, 1] = rrj
R[1, 1, 0, 1, 1] = rtj
R[1, 1, 1, 0, 1] = rsj
R[1, 1, 1, 1, 1] = rpj
R[0, 0, :, :, :] = mi
R[0, :, :, :, 0] = mi
R[1, 0, :, :, :] = mj
R[1, :, :, :, 0] = mj
R = sp.Array(R)
# %% Benefit and Cost substitiutions
N = sp.symbols("N")
Nd = sp.symbols("N_D")
b, c = sp.symbols("b c")
f = sp.symbols("f")
bc_reward_subs = {
rri: (N - Nd) * b - c,
rti: (N - Nd - 1) * b,
rsi: b - c,
rpi: 0,
rrj: (N - Nd) * b - c,
rsj: (N - Nd - 1) * b - c,
rtj: b,
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.widgets import Slider, Button
import matplotlib.animation as animation
import sympy as sp
from sympy.abc import u, v
from sympy.utilities.lambdify import lambdify
N = 300
X = (1 + v * sp.cos(u / 2)) * sp.cos(u)
Y = (1 + v * sp.cos(u / 2)) * sp.sin(u)
Z = v * sp.sin(u / 2)
sigma_sym = sp.Array([X, Y, Z])
sigma_u = lambdify((u, v), sigma_sym.diff(u), "numpy")
sigma_v = lambdify((u, v), sigma_sym.diff(v), "numpy")
f = 0.5
sigma = lambda u, v: (
(1 + w * v / 2 * np.cos(u / f)) * np.cos(u),
(1 + w * v / 2 * np.cos(u / f)) * np.sin(u), w * v / 2 * np.sin(u / f))
def generate_data(nbr_iterations):
dims = (3, 1)
start_positions = np.array(np.sigma(0, 0.5))
def dX(self):
if self.X is None:
return None
return sp.Array([x.dt for x in self.X])
def simplify_sympy_array(arr):
flattened_list = _flatten_list(arr.tolist())
simplified_flattened_list = [sympy.simplify(e) for e in flattened_list]
return sympy.Array(simplified_flattened_list, arr.shape)
def getPhaseStateVector(dim):
state = []
for i in xrange(dim):
state.append(sp.Symbol('x%s' % str(i + 1).zfill(2)))
return sp.Array(state)
def test_to_equation(self):
self.assertEqual(
pybamm.Array([1, 2]).to_equation(), sympy.Array([[1.0], [2.0]]))
def __init__(
self,
expression: ExpressionType,
signature: Optional[Sequence[Union[str, List[str]]]] = None,
*,
user_funcs: Optional[Dict[str, Callable]] = None,
consts: Optional[Dict[str, NumberOrArray]] = None,
explicit_symbols: Sequence[str] = None,
):
"""
Warning:
{WARNING_EXEC}
Args:
expression (str or float):
The expression, which is either a number or a string that sympy
can parse
signature (list of str):
The signature defines which variables are expected in the
expression. This is typically a list of strings identifying
the variable names. Individual names can be specified as list,
in which case any of these names can be used. The first item in
such a list is the definite name and if another name of the list
is used, the associated variable is renamed to the definite
name. If signature is `None`, all variables in `expressions`
are allowed.
user_funcs (dict, optional):
A dictionary with user defined functions that can be used in the
expression.
consts (dict, optional):
A dictionary with user defined constants that can be used in the
expression. The values of these constants should either be numbers or
:class:`~numpy.ndarray`.
explicit_symbols (list of str):
List of symbols that need to be interpreted as general sympy symbols
"""
from sympy.tensor.array.ndim_array import ImmutableNDimArray
# parse the expression
if isinstance(expression, TensorExpression):
# copy constructor
sympy_expr = copy.copy(expression._sympy_expr)
if user_funcs is None:
user_funcs = expression.user_funcs
else:
user_funcs.update(expression.user_funcs)
elif isinstance(expression, (np.ndarray, list, tuple)):
# expression is a constant array
sympy_expr = sympy.Array(sympy.sympify(expression))
elif isinstance(expression, ImmutableNDimArray):
# expression is an array of sympy expressions
sympy_expr = expression
else:
# parse expression as a string
sympy_expr_raw = parse_expr_guarded(
str(expression),
symbols=[signature, consts, explicit_symbols],
functions=user_funcs,
)
sympy_expr = sympy.Array(sympy_expr_raw)
super().__init__(
expression=sympy_expr,
signature=signature,
user_funcs=user_funcs,
consts=consts,
)
return sym.simplify(dv)
##================test metric derivatives (must = 0) ==============
#print('-----metric derivatives------')
#for i in range(dim):
# for j in range(i+1):
# for k in range(dim):
# vd = tensor_up_derivative(gup, i, j, k)
# if(not vd == 0):
# print(i, j, k, vd)
##==================================================================
print("--------------vec down-------------------")
#covariant vector
vec = sym.Array([sym.sin(u[0]), sym.sin(u[1])])
for i in range(dim):
print(vec[i])
print('-----vector derivatives------')
div = 0 #to save divergence
for i in range(dim):
for j in range(dim):
vd = vector_down_derivative(vec, i, j)
if (vd != 0):
print(i, j, vd)
div += vd * gup[i, j]
print("divergence = " + str(div))
#print("divergence at (0, 0):"+str(div.subs(u[0],0).subs(u[1],0).evalf()))
print("--------------vec up-------------------")
#contravariant vector
def upscale_kernel(__method: str, __kernel: np.ndarray, __coarse_scale: int, __fine_scale: int, __zero_padding=True,
P=None, R=None) -> np.ndarray:
if P is None and R is None:
__P = sp.Matrix(
get_prolongation(__method, __coarse_scale, __fine_scale, __zero_padding=False, __kernel_size=len(__kernel)))
__R = sp.Matrix(
get_restriction(__method, __fine_scale, __coarse_scale, __zero_padding=False, __kernel_size=len(__kernel)))
else:
__P, __R = P, R
# create a kernel with unknown variables
__num_of_unknowns = len(__kernel)
__unknowns = symbols('x0:%d' % __num_of_unknowns)
__K_h = sp.Matrix(utils.get_toeplitz(np.array(sp.Array(__unknowns)), __fine_scale, zero_padding=True))
__K_H = sp.Matrix(np.transpose(utils.get_toeplitz(__kernel, __coarse_scale, zero_padding=__zero_padding)))
__B = __R * __K_h * __P
if __method == 'linear':
__K_H.col_del(0)
__K_H.row_del(-1)
__B.col_del(0)
__B.row_del(-1)
sp.pprint(round_expr(__K_H, 3))
sp.pprint(round_expr(__B,3))
######################################
# __M = __K_H - __B
# __equations = []
#
# for __r in range(__M.shape[0]):
# for __c in range(__M.shape[1]):
# __equation = sp.Eq(__M[__r, __c])
#
# if __equation not in __equations and type(__equation) == sp.Eq:
# __equations.append(__equation)
###############################################
# define the matrix equation
__eq = sp.Eq(__K_H, __B)
__solution = sp.solve(__eq, __unknowns)
__new_kernel = []
try:
# if there is a solution
if __solution != []:
i = 0
for x in __unknowns:
__new_kernel.append(np.float32(__solution[x]))
i += 1
return np.array(__new_kernel)
else:
print("We cannot upscale this kernel")
__new_kernel = __solution
return __new_kernel
except KeyError:
print('Not unique solution')
return __solution
except TypeError:
print('Not unique solution')
return __solution
def Mvp(self, value):
self.validation.Mvpsetter(self, value)
if value is None:
self._Mvp = None
else:
self._Mvp = sp.Array(value)
y1, y2 = sy.symbols('y_1 y_2', real=True)
v11 = sy.symbols('v_11', real=True)
v22 = sy.symbols('v_22', real=True)
x12 = 0
v12 = 0
v21 = 0
x21 = 0
v11 = 1
v22 = 1
r11 = y1 - x11
r12 = y2 - x12
r21 = y1 - x21
r22 = y2 - x22
r1 = sy.Array([r11, r12])
r2 = sy.Array([r21, r22])
rr1 = sy.Matrix([r11, r12])
rr2 = sy.Matrix([r21, r22])
v1 = sy.Matrix([v11, v12])
v2 = sy.Matrix([v21, v22])
r1r1 = sy.tensorproduct(r1, r1).tomatrix()
r2r2 = sy.tensorproduct(r2, r2).tomatrix()
G1 = 0.01 * (3 / 4) * (r1r1 / (rr1.norm()**3) + sy.eye(2) / (rr1.norm()))
G2 = 0.01 * (3 / 4) * (r2r2 / (rr2.norm()**3) + sy.eye(2) / (rr2.norm()))
u1 = G1 * v1
u2 = G2 * v2