def construct_backdoor_estimand(self, estimand_type, treatment_name,
outcome_name, common_causes):
# TODO: outputs string for now, but ideally should do symbolic
# expressions Mon 19 Feb 2018 04:54:17 PM DST
# TODO Better support for multivariate treatments
expr = None
if estimand_type == "nonparametric-ate":
outcome_name = outcome_name[0]
num_expr_str = outcome_name
if len(common_causes)>0:
num_expr_str += "|" + ",".join(common_causes)
expr = "d(" + num_expr_str + ")/d" + ",".join(treatment_name)
sym_mu = sp.Symbol("mu")
sym_sigma = sp.Symbol("sigma", positive=True)
sym_outcome = spstats.Normal(num_expr_str, sym_mu, sym_sigma)
# sym_common_causes = [sp.stats.Normal(common_cause, sym_mu, sym_sigma) for common_cause in common_causes]
sym_treatment_symbols = [sp.Symbol(t) for t in treatment_name]
sym_treatment = sp.Array(sym_treatment_symbols)
sym_conditional_outcome = spstats.Expectation(sym_outcome)
sym_effect = sp.Derivative(sym_conditional_outcome, sym_treatment)
sym_assumptions = {
'Unconfoundedness': (
u"If U\N{RIGHTWARDS ARROW}{{{0}}} and U\N{RIGHTWARDS ARROW}{1}"
" then P({1}|{0},{2},U) = P({1}|{0},{2})"
).format(",".join(treatment_name), outcome_name, ",".join(common_causes))
}
else:
raise ValueError("Estimand type not supported. Supported estimand types are 'non-parametric-ate'.")
estimand = {
'estimand': sym_effect,
'assumptions': sym_assumptions
}
return estimand
def Hessiana(f, y):
fr = FunctionStandard(f)
g1 = sp.Derivative(fr, 'x1')
g1 = g1.doit()
H1 = sp.Derivative(g1, 'x1')
H2 = sp.Derivative(g1, 'x2')
H1 = H1.doit()
H2 = H2.doit()
H1 = H1.subs('x1', y[0])
H1 = H1.subs('x2', y[1])
H2 = H2.subs('x1', y[0])
H2 = H2.subs('x2', y[1])
g2 = sp.Derivative(fr, 'x2')
g2 = g2.doit()
H3 = sp.Derivative(g2, 'x1')
H4 = sp.Derivative(g2, 'x2')
H3 = H3.doit()
H4 = H4.doit()
H3 = H3.subs('x1', y[0])
H3 = H3.subs('x2', y[1])
H4 = H4.subs('x1', y[0])
H4 = H4.subs('x2', y[1])
hess = sp.Matrix([[H1, H2], [H3, H4]])
return hess
# numerical integral of function
lf = lambdify_t(f)
tt = np.arange(dt,t+dt,dt)
return lf(tt).sum() * dt
_gexpr = gamma_expr(5.4, 5.2) - 0.35 * gamma_expr(10.8, 7.35)
_gexpr = _gexpr / _getint(_gexpr)
_glover = lambdify_t(_gexpr)
glover = implemented_function('glover', _glover)
glovert = lambdify_t(glover(T))
# Derivative of Glover HRF
_dgexpr = _gexpr.diff(T)
dpos = sympy.Derivative((T >= 0), T)
_dgexpr = _dgexpr.subs(dpos, 0)
_dgexpr = _dgexpr / _getint(sympy_abs(_dgexpr))
_dglover = lambdify_t(_dgexpr)
dglover = implemented_function('dglover', _dglover)
dglovert = lambdify_t(dglover(T))
del(_glover); del(_gexpr); del(dpos); del(_dgexpr); del(_dglover)
# AFNI's HRF
_aexpr = ((T >= 0) * T)**8.6 * sympy.exp(-T/0.547)
_aexpr = _aexpr / _getint(_aexpr)
_afni = lambdify_t(_aexpr)
afni = implemented_function('afni', _afni)
afnit = lambdify_t(afni(T))
def _latex(self, *args):
varname = sympy.Symbol(self.varname)
t = sympy.Symbol('t')
sympy_expr = sympy.Eq(sympy.Derivative(varname, t),
str_to_sympy(self.expr))
return sympy.latex(sympy_expr)
t = np.arange(0, 10, 0.01)
y0 = [0, np.sqrt(2)]
#Solution scipy
def func(y, t):
return -2 * y
y_sci = odeint(func, y0, t)[:, 1]
#Solution sympy
x = sp.symbols('x')
y = sp.Function('y')
eq = sp.Eq(sp.Derivative(y(x), x), -2 * y(x))
f = sp.dsolve(eq, ics={y(0): sp.sqrt(2)})
y_sym = np.zeros(t.size)
for i in range(t.size):
y_sym[i] = (f.rhs).subs({x: t[i]}).n()
fig, (graph, dif) = plt.subplots(1, 2, figsize=(15, 15))
fig.subplots_adjust(hspace=0.2)
graph.plot(t, y_sci, label='sympy')
graph.plot(t, y_sym, label='scipy')
graph.set(title='Решения дифференциального уравнения')
graph.legend()
graph.grid()
dif.plot(t, y_sym - y_sci)
print(os.getcwd())
except:
pass
#%%
from IPython.display import display, Markdown, Math
import sympy as sp
sp.init_printing()
#%%
s_t = sp.Symbol("t")
s_i = sp.Symbol("i", integer=True)
s_z = sp.IndexedBase("z")
s_N = sp.Symbol("N", integer=True)
f_r = sp.IndexedBase(sp.Function("r")(s_t))
eq_27_1 = sp.Eq(sp.Derivative(f_r[s_i], s_t),
(sp.Rational(1, 2) * (s_z[s_i - 1] + s_z[s_i]) * (1 - s_t)**2 +
s_z[s_i] * 2 * (1 - s_t) * s_t + sp.Rational(1, 2) *
(s_z[s_i] + s_z[s_i + 1]) * s_t**2)**2,
evaluate=False)
display(eq_27_1)
#%%
def disabled():
s_j = sp.Symbol("j", integer=True)
r2 = sp.IndexedBase(sp.Function("r")(s_j / 4))
eq_27_1_bernstein = sp.Sum(
sp.binomial(4, s_j) * s_t**s_j * (1 - s_t)**(4 - s_j) * r2[s_i],
(s_j, 0, 4))
display(eq_27_1_bernstein)
der_g = g.diff(x, y)
der_g_suc = g.diff(x, 3, y, 4)
## Utilisation
expr = x**4 + x**3 + x**2 + x + 1
# expr.diff(x) ↦ 4x**3 + 3*x**2 + 2*x + 1
### Matrix
# Jacobian
# To specify a Jacobian for optimize.fsolve to use we need to define a function that evaluates the Jacobian for a given input vector
x, y = sy.symbols("x, y")
f_mat = sy.Matrix([y - x**3 - 2 * x**2 + 1, y + x**2 - 1])
#f_mat.jacobian(sympy.Matrix([x, y]))
## Autre
d = sy.Derivative(sy.exp(sy.cos(x)), x)
ddd = d.doit()
#### Faire une figure avec un zoom
#figure = True
#if figure == True :
# import matplotlib as mpl
# import matplotlib.pyplot as plt
# from mpl_toolkits.mplot3d.axes3d import Axes3D
# fig = plt.figure(figsize=(8, 4))
# def f(x):
# return 1/(1 + x**2) + 0.1/(1 + ((3 - x)/0.1)**2)
# def plot_and_format_axes(ax, x, f, fontsize):
# ax.plot(x, f(x), linewidth=2)
def test_Derivative():
simp = lambda expr: aesara_simplify(fgraph_of(expr))
assert theq(
simp(aesara_code_(sy.Derivative(sy.sin(x), x, evaluate=False))),
simp(aesara.grad(aet.sin(xt), xt)))
def diff(self, order):
self.k = 0
for var in self.vars:
self.k = self.k + sympy.Derivative(var, (var, self.order))
return self.k
from sympy import lambdify
from sympy.abc import a,b,c,d
import sys
t,r,theta,phi,tau = gravipy.symbols('t r \\theta \phi \\tau')
c = gravipy.Coordinates('\chi', [t, r, theta, phi])
m = 5
Metric2D = gravipy.diag(-(1-2*m/r), 1/(1-2*m/r), r**2, r**2)
g = gravipy.MetricTensor('g', c, Metric2D)
w = gravipy.Geodesic('w', g, tau)
d2t = sympy.solve(w(1),sympy.Derivative(t(tau),tau,tau))[0].subs({sympy.Derivative(t(tau),tau):a,sympy.Derivative(r(tau),tau):b,sympy.Derivative(phi(tau),tau):c,sympy.Derivative(theta(tau),tau):d})
d2r = sympy.solve(w(2),sympy.Derivative(r(tau),tau,tau))[0].subs({sympy.Derivative(t(tau),tau):a,sympy.Derivative(r(tau),tau):b,sympy.Derivative(phi(tau),tau):c,sympy.Derivative(theta(tau),tau):d})
d2theta = sympy.solve(w(3),sympy.Derivative(theta(tau),tau,tau))[0].subs({sympy.Derivative(t(tau),tau):a,sympy.Derivative(r(tau),tau):b,sympy.Derivative(phi(tau),tau):c,sympy.Derivative(theta(tau),tau):d})
d2phi = sympy.solve(w(4),sympy.Derivative(phi(tau),tau,tau))[0].subs({sympy.Derivative(t(tau),tau):a,sympy.Derivative(r(tau),tau):b,sympy.Derivative(phi(tau),tau):c,sympy.Derivative(theta(tau),tau):d})
d2tf = lambdify((a,b,c,d,t(tau),r(tau),theta(tau),phi(tau)),d2t,modules="numpy")
d2rf = lambdify((a,b,c,d,t(tau),r(tau),theta(tau),phi(tau)),d2r,modules="numpy")
d2thetaf = lambdify((a,b,c,d,t(tau),r(tau),theta(tau),phi(tau)),d2theta,modules="numpy")
d2phif = lambdify((a,b,c,d,t(tau),r(tau),theta(tau),phi(tau)),d2phi,modules="numpy")
def geo(x, tau):
'''
x[0] = t
x[1] = r
x[2] = phi
def curvesketch(jarvis, s):
"""
Prints useful information about a graph of a function.
* Limit
* Intersection x/y axis
* Derivative and Integral
* Minima / Maxima / Turning point
-- Example:
curve sketch y=x**2+10x-5
curve sketch y=sqrt((x+1)(x-1))
curve sketch y=1/3x**3-2x**2+3x
"""
if len(s) == 0:
jarvis.say(
"Missing parameter: function (e.g. call 'curve sketch y=x**2+10x-5')")
return
def section(jarvis, headline):
jarvis.say("\n{:#^50}".format(" {} ".format(headline)), Fore.MAGENTA)
term = remove_equals(jarvis, s)
term = format_expression(term)
term = solve_y(term)
def get_y(x_val, func=term):
x = sympy.Symbol('x')
return func.evalf(subs={x: x_val})
section(jarvis, s)
section(jarvis, "Graph")
jarvis.eval('plot {}'.format(s))
section(jarvis, "Limit")
jarvis.eval('limit {}'.format(term))
section(jarvis, "Intersection x-axis")
jarvis.eval('solve {}'.format(term))
section(jarvis, "Intersection y-axis")
jarvis.say(str(get_y(0).round(9)), Fore.BLUE)
section(jarvis, "Factor")
jarvis.eval('factor {}'.format(term))
section(jarvis, "Derivative")
x = sympy.Symbol('x')
derivative_1 = sympy.Derivative(term, x).doit()
derivative_2 = sympy.Derivative(derivative_1, x).doit()
derivative_3 = sympy.Derivative(derivative_2, x).doit()
jarvis.say("1. Derivative: {}".format(derivative_1), Fore.BLUE)
jarvis.say("2. Derivative: {}".format(derivative_2), Fore.BLUE)
jarvis.say("3. Derivative: {}".format(derivative_3), Fore.BLUE)
section(jarvis, "Integral")
jarvis.say("F(x) = {} + t".format(sympy.Integral(term, x).doit()), Fore.BLUE)
section(jarvis, "Maxima / Minima")
try:
critical_points = sympy.solve(derivative_1)
except NotImplementedError:
jarvis.say("Sorry, cannot solve...", Fore.RED)
critical_points = []
for x in critical_points:
y = str(get_y(x).round(9))
try:
isminmax = float(get_y(x, func=derivative_2))
except ValueError:
isminmax = None
except TypeError:
# probably complex number
isminmax = None
if isminmax is None:
minmax = "unknown"
elif isminmax > 0:
minmax = "Minima"
elif isminmax < 0:
minmax = "Maxima"
else:
minmax = "/"
jarvis.say("({}/{}) : {}".format(x, y, minmax), Fore.BLUE)
section(jarvis, "Turning Point")
try:
critical_points = sympy.solve(derivative_2)
except NotImplementedError:
jarvis.say("Sorry, cannot solve...", Fore.RED)
critical_points = []
for x in critical_points:
y = get_y(x)
try:
is_turning_point = float(get_y(x, func=derivative_3))
except ValueError:
is_turning_point = -1
if is_turning_point != 0:
jarvis.say("({}/{})".format(x, y.round(9)), Fore.BLUE)
section(jarvis, "")
def __call__(self, equations, variables=None):
if equations.is_stochastic:
raise UnsupportedEquationsException('Cannot solve stochastic '
'equations with this state '
'updater')
if variables is None:
variables = {}
diff_eqs = equations.get_substituted_expressions(variables)
t = Symbol('t', real=True, positive=True)
dt = Symbol('dt', real=True, positive=True)
t0 = Symbol('t0', real=True, positive=True)
f0 = Symbol('f0', real=True)
# TODO: Shortcut for simple linear equations? Is all this effort really
# worth it?
code = []
for name, expression in diff_eqs:
rhs = str_to_sympy(expression.code, variables)
# We have to be careful and use the real=True assumption as well,
# otherwise sympy doesn't consider the symbol a match to the content
# of the equation
var = Symbol(name, real=True)
f = sp.Function(name)
rhs = rhs.subs(var, f(t))
derivative = sp.Derivative(f(t), t)
diff_eq = sp.Eq(derivative, rhs)
# TODO: simplify=True sometimes fails with 0.7.4, see:
# https://github.com/sympy/sympy/issues/2666
try:
general_solution = sp.dsolve(diff_eq, f(t), simplify=True)
except RuntimeError:
general_solution = sp.dsolve(diff_eq, f(t), simplify=False)
# Check whether this is an explicit solution
if not getattr(general_solution, 'lhs', None) == f(t):
raise UnsupportedEquationsException(
'Cannot explicitly solve: ' + str(diff_eq))
# Solve for C1 (assuming "var" as the initial value and "t0" as time)
if general_solution.has(Symbol('C1')):
if general_solution.has(Symbol('C2')):
raise UnsupportedEquationsException(
'Too many constants in solution: %s' %
str(general_solution))
constant_solution = sp.solve(general_solution, Symbol('C1'))
if len(constant_solution) != 1:
raise UnsupportedEquationsException(
("Couldn't solve for the constant "
"C1 in : %s ") % str(general_solution))
constant = constant_solution[0].subs(t, t0).subs(f(t0), var)
solution = general_solution.rhs.subs('C1', constant)
else:
solution = general_solution.rhs.subs(t, t0).subs(f(t0), var)
# Evaluate the expression for one timestep
solution = solution.subs(t, t + dt).subs(t0, t)
# only try symplifying it -- it sometimes raises an error
try:
solution = solution.simplify()
except ValueError:
pass
code.append(name + ' = ' + sympy_to_str(solution))
return '\n'.join(code)
def test_Derivative():
simp = lambda expr: theano_simplify(fgraph_of(expr))
assert theq(simp(theano_code_(sy.Derivative(sy.sin(x), x, evaluate=False))),
simp(theano.grad(tt.sin(xt), xt)))
# 测试配置未知数符号
# from sympy import *
# var('x, y, z')
# print()
# 测试将sympy 求极限函数转换为latex语句
# from sympy import *
# x = symbols('x')
# expr = S('limit(x**2 - 6*x + 10, x, 2)')
# print(latex(expr, mode='inline'))
# 测试计算嵌套语句
# from sympy import *
# import sympy
# x = sympy.var('x')
# expr = sympy.Derivative((sympy.log((1-x)/(1+x), sympy.E)), x, 1).doit()
# print(sympy.simplify(expr))
# 测试字符串公式
import sympy
# from sympy import log, E
x = sympy.var('x')
expr = 'sympy.log((1-x)/(1+x), E)'
result = sympy.Derivative(sympy.sympify(expr), x, 1).doit()
print(sympy.simplify(result))
def __init__(self, scheme):
# TODO: add source terms
t, x, y, z = sp.symbols("t x y z", type='real')
consm = list(scheme.consm.keys())
nconsm = len(scheme.consm)
self.consm = sp.Matrix(consm)
self.dim = scheme.dim
space = [x, y, z]
LA = scheme.symb_la
if not LA:
LA = scheme.la
func = []
for i in range(nconsm):
func.append(sp.Function('f{}'.format(i))(t, x, y, z)) #pylint: disable=not-callable
func = sp.Matrix(func)
sublist = [(i, j) for i, j in zip(consm, func)]
sublist_inv = [(j, i) for i, j in zip(consm, func)]
eq_func = sp.Matrix(scheme.EQ[nconsm:]).subs(sublist)
s = sp.Matrix(scheme.s[nconsm:])
all_vel = scheme.stencil.get_all_velocities()
Lambda = []
for i in range(all_vel.shape[1]):
vd = LA * sp.diag(*all_vel[:, i])
Lambda.append(scheme.M * vd * scheme.invM)
phi1 = sp.zeros(s.shape[0], 1) #pylint: disable=unsubscriptable-object
sigma = sp.diag(*s).inv() - sp.eye(len(s)) / 2
gamma_1 = sp.zeros(nconsm, 1)
self.coeff_order1 = []
for dim, lambda_ in enumerate(Lambda):
A, B = sp.Matrix([lambda_[:nconsm, :nconsm]
]), sp.Matrix([lambda_[:nconsm, nconsm:]])
C, D = sp.Matrix([lambda_[nconsm:, :nconsm]
]), sp.Matrix([lambda_[nconsm:, nconsm:]])
self.coeff_order1.append(A * func + B * eq_func)
alltogether(self.coeff_order1[-1], nsimplify=True)
for i in range(nconsm):
gamma_1[i] += sp.Derivative(self.coeff_order1[-1][i],
space[dim])
dummy = -C * func - D * eq_func
alltogether(dummy, nsimplify=True)
for i in range(dummy.shape[0]):
phi1[i] += sp.Derivative(dummy[i], space[dim])
self.coeff_order2 = [[sp.zeros(nconsm) for _ in range(scheme.dim)]
for _ in range(scheme.dim)]
for dim, lambda_ in enumerate(Lambda):
A, B = sp.Matrix([lambda_[:nconsm, :nconsm]
]), sp.Matrix([lambda_[:nconsm, nconsm:]])
meq = sp.Matrix(scheme.EQ[nconsm:])
jac = meq.jacobian(consm)
jac = jac.subs(sublist)
delta1 = jac * gamma_1
phi1_ = phi1 + delta1
sphi1 = B * sigma * phi1_
sphi1 = sphi1.doit()
alltogether(sphi1, nsimplify=True)
for i in range(scheme.dim):
for jc in range(nconsm):
for ic in range(nconsm):
self.coeff_order2[dim][i][
ic, jc] += sphi1[ic].expand().coeff(
sp.Derivative(func[jc], space[i]))
for ic, c in enumerate(self.coeff_order1):
self.coeff_order1[ic] = c.subs(sublist_inv).doit()
for ic, c in enumerate(self.coeff_order2):
for jc, cc in enumerate(c):
self.coeff_order2[ic][jc] = cc.subs(sublist_inv).doit()
def derivative(self):
derivative = sympy.Derivative(self.function, self.x, self.n)
return sympy.Eq(derivative, derivative.doit())
def eval_derivativerule(integrand, symbol):
# isinstance(integrand, Derivative) should be True
if len(integrand.args) == 2:
return integrand.args[0]
else:
return sympy.Derivative(integrand.args[0], *integrand.args[1:-1])
import sympy as sym
from sympy.matrices import Matrix
x, y = sym.symbols('x y')
expr1 = input("enter 1st function:")
sym.sympify(expr1)
expr2 = input("enter 2nd function:")
sym.sympify(expr2)
expr3 = input("enter 3rd function:")
sym.sympify(expr3)
print("1st Expression : {} ".format(expr1))
print("2nd Expression : {} ".format(expr2))
print("3rd Expression : {} ".format(expr3))
xexpr1_diff1 = sym.Derivative(expr1, x)
xexpr2_diff1 = sym.Derivative(expr2, x)
xexpr3_diff1 = sym.Derivative(expr3, x)
xexpr1_diff2 = sym.Derivative(xexpr1_diff1, x)
xexpr2_diff2 = sym.Derivative(xexpr2_diff1, x)
xexpr3_diff2 = sym.Derivative(xexpr3_diff1, x)
yexpr1_diff1 = sym.Derivative(expr1, y)
yexpr2_diff1 = sym.Derivative(expr2, y)
yexpr3_diff1 = sym.Derivative(expr3, y)
yexpr1_diff2 = sym.Derivative(yexpr1_diff1, y)
yexpr2_diff2 = sym.Derivative(yexpr2_diff1, y)
yexpr3_diff2 = sym.Derivative(yexpr3_diff1, y)
m = sym.Matrix([[expr1, expr2, expr3],
def __call__(self, equations, variables=None, method_options=None):
logger.warn(
"The 'independent' state updater is deprecated and might be "
"removed in future versions of Brian.",
'deprecated_independent',
once=True)
extract_method_options(method_options, {})
if equations.is_stochastic:
raise UnsupportedEquationsException("Cannot solve stochastic "
"equations with this state "
"updater")
if variables is None:
variables = {}
diff_eqs = equations.get_substituted_expressions(variables)
t = Symbol('t', real=True, positive=True)
dt = Symbol('dt', real=True, positive=True)
t0 = Symbol('t0', real=True, positive=True)
code = []
for name, expression in diff_eqs:
rhs = str_to_sympy(expression.code, variables)
# We have to be careful and use the real=True assumption as well,
# otherwise sympy doesn't consider the symbol a match to the content
# of the equation
var = Symbol(name, real=True)
f = sp.Function(name)
rhs = rhs.subs(var, f(t))
derivative = sp.Derivative(f(t), t)
diff_eq = sp.Eq(derivative, rhs)
# TODO: simplify=True sometimes fails with 0.7.4, see:
# https://github.com/sympy/sympy/issues/2666
try:
general_solution = sp.dsolve(diff_eq, f(t), simplify=True)
except RuntimeError:
general_solution = sp.dsolve(diff_eq, f(t), simplify=False)
# Check whether this is an explicit solution
if not getattr(general_solution, 'lhs', None) == f(t):
raise UnsupportedEquationsException(
f"Cannot explicitly solve: {str(diff_eq)}")
# Solve for C1 (assuming "var" as the initial value and "t0" as time)
if general_solution.has(Symbol('C1')):
if general_solution.has(Symbol('C2')):
raise UnsupportedEquationsException(
f'Too many constants in solution: {str(general_solution)}'
)
constant_solution = sp.solve(general_solution, Symbol('C1'))
if len(constant_solution) != 1:
raise UnsupportedEquationsException(
("Couldn't solve for the constant "
"C1 in : %s ") % str(general_solution))
constant = constant_solution[0].subs(t, t0).subs(f(t0), var)
solution = general_solution.rhs.subs('C1', constant)
else:
solution = general_solution.rhs.subs(t, t0).subs(f(t0), var)
# Evaluate the expression for one timestep
solution = solution.subs(t, t + dt).subs(t0, t)
# only try symplifying it -- it sometimes raises an error
try:
solution = solution.simplify()
except ValueError:
pass
code.append(f"{name} = {sympy_to_str(solution)}")
return '\n'.join(code)
"""
Created on Mon Nov 11 12:27:52 2019
@author: UO264982
"""
import matplotlib.pyplot as plt
import sympy as S
S.init_printing(use_unicode=True)
x,y,z = S.symbols('x y z',real=True)
funcion = (x+y)**z
a = S.diff(funcion,z,3)
b = funcion.diff(x,1,y,2)
c = S.Derivative(funcion,z,3)
c.doit()
import numpy as N
import scipy.misc as SM
#Número de puntos ha de ser impar
x = S.symbols('x',real=True)
funsym = S.exp(-x**2)
funnum = S.lambdify(x,funsym,"numpy")
z= 2
solsym = funsym.diff(x,1).subs(x,z).evalf()
solnum = SM.derivative(funnum,z, dx=1e-6)
pesos = SM.central_diff_weights(3)
print(solnum,pesos)
def print_Function(self, rule):
with self.new_step():
self.append("Trivial:")
self.append(self.format_math_display(
sympy.Eq(sympy.Derivative(rule.context, rule.symbol),
diff(rule))))
derivDictMi[aDerivInt]['evalNP'] = sy.lambdify((x,y,z),derivDictMi[aDerivInt]['eval'],modules=[{'ImmutableDenseMatrix':np.matrix},'numpy'])
Mf = sy.Function('M',commutative=False)(x,y,z)
Mfi = sy.Function('Mi',commutative=False)(x,y,z)
dxMfe = sy.Function('dxM',commutative=False)
dyMfe = sy.Function('dyM',commutative=False)
dzMfe = sy.Function('dzM',commutative=False)
dydxMfe = sy.Function('dydxM',commutative=False)
dzdxMfe = sy.Function('dzdxM',commutative=False)
dzdydxMfe = sy.Function('dzdydxM',commutative=False)
subsDict1 = dict([(sy.Derivative(Mfi,ax),-(Mfi*sy.diff(Mf,ax)*Mfi)) for ax in [x,y,z]])
subsList2 = ((sy.Derivative(Mf,x,y,z),"dzdydxM"),(sy.Derivative(Mf,x,z,y),"dzdydxM"),(sy.Derivative(Mf,y,x,z),"dzdydxM"),(sy.Derivative(Mf,y,z,x),"dzdydxM"),(sy.Derivative(Mf,z,x,y),"dzdydxM"),(sy.Derivative(Mf,z,y,x),"dzdydxM"),
(sy.Derivative(Mf,x,y),"dydxM"),(sy.Derivative(Mf,y,x),"dydxM"),
(sy.Derivative(Mf,x,z),"dzdxM"),(sy.Derivative(Mf,z,x),"dzdxM"),
(sy.Derivative(Mf,y,z),"dzdxM"),(sy.Derivative(Mf,z,y),"dzdyM"),
(sy.Derivative(Mf,x),"dxM"),(sy.Derivative(Mf,y),"dyM"),(sy.Derivative(Mf,z),"dzM"),
(Mfi,"Mi"),(Mf,"M"))
subsList2 = tuple((str(a),b) for (a,b) in subsList2)
dxMfi = sy.diff(Mfi,x)
dxMfi_s = dxMfi.subs(subsDict1)
print(dxMfi_s)
print(sy.simplify(dxMfi_s))
def calculate_line():
# We have a line going from s_q[s_i-1] to s_q[s_i]
# So let's fix the endpoints
eq_line1 = sp.Eq(f_r[s_i].subs(s_t, 0), s_q[s_i - 1])
eq_line2 = sp.Eq(f_r[s_i].subs(s_t, 1), s_q[s_i])
display(eq_line1)
display(eq_line2)
# Then the first derivative
eq_line3 = sp.Eq(
sp.Derivative(f_r[s_i], s_t).subs(s_t, 0), s_q[s_i] - s_q[s_i - 1])
eq_line4 = sp.Eq(
sp.Derivative(f_r[s_i], s_t).subs(s_t, 1), s_q[s_i] - s_q[s_i - 1])
display(eq_line3)
display(eq_line4)
# Finally the second derivatives should be zero at the endpoints
eq_line5 = sp.Eq(sp.Derivative(f_r[s_i], s_t, 2).subs(s_t, 0), 0)
eq_line6 = sp.Eq(sp.Derivative(f_r[s_i], s_t, 2).subs(s_t, 1), 0)
display(eq_line5)
display(eq_line6)
expanded_curve = eq_r_bezier.rhs.subs(s_K, 5).doit()
expanded_curve = expanded_curve.subs(eq_p5.lhs, eq_p5.rhs)
expanded_curve = expanded_curve.subs(eq_p4.lhs, eq_p4.rhs)
expanded_curve = expanded_curve.subs(eq_p3.lhs, eq_p3.rhs)
expanded_curve = expanded_curve.subs(eq_p2.lhs, eq_p2.rhs)
expanded_curve = expanded_curve.subs(eq_p1.lhs, eq_p1.rhs)
expanded_curve = expanded_curve.subs(eq_p0.lhs, eq_p0.rhs)
expanded_curve = expanded_curve.simplify()
eq_expanded_r = sp.Eq(f_r[s_i], expanded_curve)
eq_expanded_r_d1 = sp.Eq(sp.Derivative(eq_expanded_r.lhs, s_t),
eq_expanded_r.rhs.diff(s_t)).simplify()
eq_expanded_r_d2 = sp.Eq(sp.Derivative(eq_expanded_r_d1.lhs, s_t),
eq_expanded_r_d1.rhs.diff(s_t)).simplify()
display(eq_expanded_r)
display(eq_expanded_r_d1)
display(eq_expanded_r_d2)
eq_line1 = sp.Eq(eq_expanded_r.rhs.subs(s_t, 0), eq_line1.rhs)
eq_line2 = sp.Eq(eq_expanded_r.rhs.subs(s_t, 1), eq_line2.rhs)
eq_line3 = sp.Eq(eq_expanded_r_d1.rhs.subs(s_t, 0), eq_line3.rhs)
eq_line4 = sp.Eq(eq_expanded_r_d1.rhs.subs(s_t, 1), eq_line4.rhs)
eq_line5 = sp.Eq(eq_expanded_r_d2.rhs.subs(s_t, 0), eq_line5.rhs)
eq_line6 = sp.Eq(eq_expanded_r_d2.rhs.subs(s_t, 1), eq_line6.rhs)
display("eq 1", eq_line1) # Always true
display("eq 2", eq_line2)
display("eq 3", eq_line3)
display("eq 4", eq_line4)
display("eq 5", eq_line5)
display("eq 6", eq_line6)
#If we arrange and simplify a bit
eq_line7 = sp.Eq(s_w[s_i, 0]**2, sp.solve(eq_line5, s_w[s_i, 0]**2)[0])
display("eq 7", eq_line7)
eq_line8 = sp.Eq(s_w[s_i, 2]**2, sp.solve(eq_line6, s_w[s_i, 2]**2)[0])
display("eq 8", eq_line8)
#It's easy to see that
eq_line9 = sp.Eq(sp.Eq(s_w[s_i, 0], s_w[s_i, 1]),
s_w[s_i, 2],
evaluate=False)
display("eq 9", eq_line9)
#So let's use substitute
eq_p0_line = eq_p0.subs(s_w[s_i, 1], s_w[s_i,
0]).subs(s_w[s_i, 2], s_w[s_i, 0])
eq_p1_line = eq_p1.subs(s_w[s_i, 1], s_w[s_i,
0]).subs(s_w[s_i, 2], s_w[s_i, 0])
eq_p2_line = eq_p2.subs(s_w[s_i, 1], s_w[s_i,
0]).subs(s_w[s_i, 2], s_w[s_i, 0])
eq_p3_line = eq_p3.subs(s_w[s_i, 1], s_w[s_i,
0]).subs(s_w[s_i, 2], s_w[s_i, 0])
eq_p4_line = eq_p4.subs(s_w[s_i, 1], s_w[s_i,
0]).subs(s_w[s_i, 2], s_w[s_i, 0])
eq_p5_line = eq_p5.subs(s_w[s_i, 1], s_w[s_i,
0]).subs(s_w[s_i, 2], s_w[s_i, 0])
display(eq_p0_line)
display(eq_p1_line)
display(eq_p2_line)
display(eq_p3_line)
display(eq_p4_line)
display(eq_p5_line)
# And the final form
eq_p0_line = eq_p0_line.subs(eq_line3.lhs, eq_line3.rhs)
eq_p1_line = eq_p1_line.subs(eq_line3.lhs, eq_line3.rhs)
eq_p2_line = eq_p2_line.subs(eq_line3.lhs, eq_line3.rhs)
eq_p3_line = eq_p3_line.subs(eq_line3.lhs, eq_line3.rhs)
eq_p4_line = eq_p4_line.subs(eq_line3.lhs, eq_line3.rhs)
eq_p5_line = eq_p5_line.subs(eq_line3.lhs, eq_line3.rhs)
display(eq_p0_line)
display(eq_p1_line)
display(eq_p2_line)
display(eq_p3_line)
display(eq_p4_line)
display(eq_p5_line)
return eq_p0_line, eq_p1_line, eq_p3_line, eq_p4_line, eq_p5_line
u_exact = sp.lambdify((t, x), u_exact, 'numpy')
# All data
t_domain = np.linspace(0, T_end, n_samples)
x_domain = np.linspace(-0.5, 0.5, n_samples)
T, X = np.meshgrid(t_domain, x_domain, indexing='ij')
u_data = u_exact(T, X)
# To build the model we sample the data (in interior indices) to get
# the derivative right. For the rhs we keep it simple
u_grid = GridFunction([t_domain, x_domain], u_data)
x_grid = Coordinate([t_domain, x_domain], 1)
lhs_stream = compile_stream((sp.Derivative(u, t), ), {u: u_grid})
foos = polynomials([u], 3)
derivatives = Dx(u, 1, [3])[1:] # Don't include 0;th
# Combine these guys
columns = [(1./sp.cos(x))*sp.Derivative(u, x)] + \
list(foos) + \
list(derivatives) + \
[reduce(operator.mul, cs) for cs in itertools.product(foos, derivatives)]
rhs_stream = compile_stream(columns, {u: u_grid})
# A subset of this data is now used for training; where we can eval deriv
# safely
train_indices = map(
tuple, np.random.choice(np.arange(5, n_samples - 5), (n_train, 2)))
def inverse_laplace_product(expr, s, t, **assumptions):
# Handle expressions with a function of s, e.g., V(s) * Y(s), V(s)
# / s etc.
if assumptions.get('causal', False):
# Assume that all functions are causal in the expression.
t1 = sym.S.Zero
t2 = t
else:
t1 = -sym.oo
t2 = sym.oo
const, expr = factor_const(expr, s)
factors = expr.as_ordered_factors()
if len(factors) < 2:
raise ValueError('Expression does not have multiple factors: %s' %
expr)
if (len(factors) > 2 and not
# Help s * 1 / (s + R * C) * I(s)
isinstance(factors[1], AppliedUndef) and
isinstance(factors[2], AppliedUndef)):
factors = [factors[0], factors[2], factors[1]] + factors[3:]
if isinstance(factors[1], AppliedUndef):
# Try to expose more simple cases, e.g. (R + s * L) * V(s)
terms = factors[0].as_ordered_terms()
if len(terms) >= 2:
result = sym.S.Zero
for term in terms:
result += inverse_laplace_product(factors[1] * term, s, t)
return result * const
cresult, uresult = inverse_laplace_term1(factors[0], s, t)
result = cresult + uresult
intnum = 0
for m in range(len(factors) - 1):
if m == 0 and isinstance(factors[1], AppliedUndef):
# Note, as_ordered_factors puts powers of s before the functions.
if factors[0] == s:
# Handle differentiation
# Convert s * V(s) to d v(t) / dt
result = laplace_func(factors[1], s, t, True)
result = sym.Derivative(result, t)
continue
elif factors[0].is_Pow and factors[0].args[
0] == s and factors[0].args[1] > 0:
# Handle higher order differentiation
# Convert s ** 2 * V(s) to d^2 v(t) / dt^2
result = laplace_func(factors[1], s, t, True)
result = sym.Derivative(result, t, factors[0].args[1])
continue
elif factors[0].is_Pow and factors[0].args[0] == s and factors[
0].args[1] == -1:
# Handle integration 1 / s * V(s)
tau = dummyvar(intnum, 'tau')
intnum += 1
result = laplace_func(factors[1], s, tau, True)
result = sym.Integral(result, (tau, t1, t))
continue
# Convert product to convolution
tau = dummyvar(intnum, 'tau')
intnum += 1
cresult, uresult = inverse_laplace_term1(factors[m + 1], s, t)
expr2 = cresult + uresult
result = sym.Integral(
result.subs(t, t - tau) * expr2.subs(t, tau), (tau, t1, t2))
return result * const
def __call__(self, equations, variables=None):
if variables is None:
variables = {}
if equations.is_stochastic:
raise ValueError(
'Cannot solve stochastic equations with this state updater')
diff_eqs = equations.substituted_expressions
t = Symbol('t', real=True, positive=True)
dt = Symbol('dt', real=True, positive=True)
t0 = Symbol('t0', real=True, positive=True)
f0 = Symbol('f0', real=True)
# TODO: Shortcut for simple linear equations? Is all this effort really
# worth it?
code = []
for name, expression in diff_eqs:
rhs = expression.sympy_expr
non_constant = _non_constant_symbols(rhs.atoms(), variables) - set(
[name])
if len(non_constant):
raise ValueError(('Equation for %s referred to non-constant '
'variables %s') % (name, str(non_constant)))
# We have to be careful and use the real=True assumption as well,
# otherwise sympy doesn't consider the symbol a match to the content
# of the equation
var = Symbol(name, real=True)
f = sp.Function(name)
rhs = rhs.subs(var, f(t))
derivative = sp.Derivative(f(t), t)
diff_eq = sp.Eq(derivative, rhs)
general_solution = sp.dsolve(diff_eq, f(t))
# Check whether this is an explicit solution
if not getattr(general_solution, 'lhs', None) == f(t):
raise ValueError('Cannot explicitly solve: ' + str(diff_eq))
# Solve for C1 (assuming "var" as the initial value and "t0" as time)
if Symbol('C1') in general_solution:
if Symbol('C2') in general_solution:
raise ValueError('Too many constants in solution: %s' %
str(general_solution))
constant_solution = sp.solve(general_solution, Symbol('C1'))
if len(constant_solution) != 1:
raise ValueError(("Couldn't solve for the constant "
"C1 in : %s ") % str(general_solution))
constant = constant_solution[0].subs(t, t0).subs(f(t0), var)
solution = general_solution.rhs.subs('C1', constant)
else:
solution = general_solution.rhs.subs(t, t0).subs(f(t0), var)
# Evaluate the expression for one timestep
solution = solution.subs(t, t + dt).subs(t0, t)
# only try symplifying it -- it sometimes raises an error
try:
solution = solution.simplify()
except ValueError:
pass
code.append(name + ' = ' + sympy_to_str(solution))
return '\n'.join(code)
def testDerivative(self):
self.compare(
mathics.Expression('D', mathics.Symbol('Global`x'),
mathics.Symbol('Global`y')),
sympy.Derivative(sympy.Symbol('_Mathics_User_Global`x'),
sympy.Symbol('_Mathics_User_Global`y')))
def main(dh_params=None):
if not dh_params:
# TODO connect this with the GUI
dh_params = input(
'Please input all the DH parameters separated by commas and ; to separate rows:\n'
)
rows = dh_params.split(';')
dh_params = np.zeros([len(rows), 4])
dh_params = [[0 for j in range(4)] for i in range(len(rows))]
#dh_params = sym.Matrix(dh_params)
for i, row in enumerate(rows):
cols = row.split(',')
cols = [remove_whitespace(item) for item in cols]
for j, col in enumerate(cols):
try:
col = float(col)
dh_params[i][j] = col
#dh_params[i,j] = col
except:
# recognizes that the cell isn't numerical
#print(col)
non_sympy = remove_sympy(col)
# need to trim whitespace, NOT remove
non_sympy = remove_punctuation(
remove_numericals(non_sympy)).strip()
non_sympy = non_sympy.split()
#lcl = locals() # TODO maybe need to define this as global to preventb
#lcl = globals()
# TODO python from confusing two 'locally' named same name variables?
for var in non_sympy:
#if not lcl.__contains__(var):
# lcl[var] = sym.Symbol(var)
define_sympy_vars(var)
# at this point, all non-sympy vars have been defined, can now directly evaluate!
dh_params[i][j] = eval(col)
#dh_params[i, j] = eval(col)
#print(dh_params)
for item in dh_params:
print(type(item))
T_matrices = [dh_transformer(joint) for joint in dh_params]
print(T_matrices[0])
T_forward = 1
for T in T_matrices:
T_forward *= T
print(T_forward)
T_inverse = create_transformation_matrix(
find_rotation_matrix(yaw='psi').T, ['x', 'y', 'z'])
omegas = []
vels = []
omega = sym.Matrix([0, 0, 0])
vel = sym.Matrix([0, 0, 0])
for i, joint in enumerate(dh_params):
alpha, a, d, theta = joint
if theta == 0:
pass
# this is a prismatic joint
#omegas.append()
#vels.append()
else:
# this is revolute
R = T_matrices[
0][:3, :
3].T # need to transpose, because we are going in the opposite direction!
t = sym.Symbol('t')
# TODO does not work programmatically! Unless you can be sure of the coordinates and axes
# and where they belong!
# looks like you're blocked atm!
vel = R * (vel + omega.cross())
vels.append()
omega = R * omega + sym.Matrix(
[0, 0, sym.Derivative(theta, t)])
omegas.append(omega)
# TODO this part does not work at all, so skipped for now!
eqns = [
sym.simplify(exp1 - exp2)
for exp1, exp2 in zip(T_forward, T_inverse)
if sym.simplify(exp1 - exp2) != 0
]
vars = ['theta1', 'd2', 'theta3']
# TODO add object oriented approach where solution first seen, then user can specify next calculations
# TODO not very smart!
# TODO need to think of a smart method to do this!
# TODO need to save var that is being saved!
# THEN use simultaneous! SHOULD not be TOO hard!
#print(sym.solve(eqns, (theta1, d2, theta3)))
# rule based solver? maybe you need simultaneous equations??
# given joint variables, needs to solve the equations?
# maybe can isolate joint variables to find the simplest ones? starting
# TODO need to add an input for this too, currently has no input feature!
# TODO do this again, this is not correct! We want sympy for the definitions of the transformation matrices!
# TODO needs to recognize: symbols, numbers, and sympy sumbols (i.e. cos or pi)
""" Given dh parameters for a robot, calculates the forward kinematics """
def __init__(self, cct, node_voltages=True, branch_currents=False):
if not node_voltages and not branch_currents:
raise ValueError('No outputs')
inductors = []
capacitors = []
independent_sources = []
# Determine state variables (current through inductors and
# voltage across acapacitors) and replace inductors with
# current sources and capacitors with voltage sources.
sscct = cct._new()
cpt_map = {}
for key, elt in cct.elements.items():
ssnet = elt._ss_model()
sselt = sscct._add(ssnet)
name = elt.name
cpt_map[name] = sselt.name
if elt.is_inductor:
if sselt.name in cct.elements:
raise ValueError(
'Name conflict %s, either rename the component or improve the code!'
% sselt.name)
inductors.append(elt)
elif elt.is_capacitor:
if sselt.name in cct.elements:
raise ValueError(
'Name conflict %s, either rename the component or improve the code!'
% sselt.name)
capacitors.append(elt)
elif isinstance(elt, (I, V)):
independent_sources.append(elt)
self.cct = cct
self.sscct = sscct
# sscct can be analysed in the time domain since it has no
# reactive components. However, for large circuits
# this can take a long time due to inversion of the MNA matrix.
dotx_exprs = []
statevars = []
statenames = []
initialvalues = []
for elt in inductors + capacitors:
name = cpt_map[elt.name]
if isinstance(elt, L):
# Inductors v = L di/dt so need v across the L
expr = sscct[name].v / elt.cpt.L
var = -sscct[name].isc
x0 = elt.cpt.i0
else:
# Capacitors i = C dv/dt so need i through the C
# The current is negated since it is from a source V_Cx
expr = -sscct[name].i / elt.cpt.C
var = sscct[name].voc
x0 = elt.cpt.v0
dotx_exprs.append(expr)
statevars.append(var)
statenames.append(name)
initialvalues.append(x0)
statesyms = sympify(statenames)
# Determine independent sources.
sources = []
sourcevars = []
sourcenames = []
for elt in independent_sources:
name = cpt_map[elt.name]
if isinstance(elt, V):
expr = elt.cpt.voc
var = sscct[name].voc
else:
expr = elt.cpt.isc
var = sscct[name].isc
sources.append(expr)
sourcevars.append(var)
sourcenames.append(name)
sourcesyms = sympify(sourcenames)
# linear_eq_to_matrix expects only Symbols and not AppliedUndefs,
# so substitute.
subsdict = {}
for var, sym1 in zip(statevars, statesyms):
subsdict[var.expr] = sym1
for var, sym1 in zip(sourcevars, sourcesyms):
subsdict[var.expr] = sym1
for m, expr in enumerate(dotx_exprs):
dotx_exprs[m] = expr.subs(subsdict).expr.expand()
A, b = sym.linear_eq_to_matrix(dotx_exprs, *statesyms)
if sourcesyms != []:
B, b = sym.linear_eq_to_matrix(dotx_exprs, *sourcesyms)
else:
B = []
# Determine output variables.
yexprs = []
y = []
if node_voltages:
for node in cct.node_list:
if node == '0':
continue
yexprs.append(self.sscct[node].v.subs(subsdict).expand().expr)
# Note, this can introduce a name conflict
y.append(Vt('v_%s(t)' % node))
if branch_currents:
for name in cct.branch_list:
# Perhaps ignore L since the current through it is a
# state variable?
name2 = cpt_map[name]
yexprs.append(self.sscct[name2].i.subs(subsdict).expand().expr)
y.append(It('i_%s(t)' % name))
Cmat, b = sym.linear_eq_to_matrix(yexprs, *statesyms)
if sourcesyms != []:
D, b = sym.linear_eq_to_matrix(yexprs, *sourcesyms)
else:
D = []
# Rewrite vCanon1(t) as vC(t) etc if appropriate.
_hack_vars(statevars)
_hack_vars(sources)
# Note, Matrix strips the class from each element...
self.x = tMatrix(statevars)
self.x0 = Matrix(initialvalues)
self.dotx = tMatrix([sym.Derivative(x1, t) for x1 in self.x])
self.u = tMatrix(sources)
self.A = Matrix(A)
self.B = Matrix(B)
# Perhaps could use v_R1(t) etc. as the output voltages?
self.y = Matrix(y)
self.C = Matrix(Cmat)
self.D = Matrix(D)
def dotx(self):
"""Time derivative of state variable vector."""
return TimeDomainMatrix([sym.Derivative(x1, t) for x1 in self.x])