def taxa_anuidade(E,P,a):
Esym,t,asym,Psym = sy.symbols('Esym t asym Psym')
fsym = Esym*t/( 12*(1 - 1/(1 + t/12)**(12*asym) ) ) - Psym
dfsym = sy.diff(fsym,t)
f = fsym.subs({'Esym':E,'asym':a,'Psym':P})
df = dfsym.subs({'Esym':E,'asym':a,'Psym':P})
ft = eval(lambdastr(t,f))
dft = eval(lambdastr(t,df))
# Aplicação do método de Newton
# parâmetros
t0 = 1.0 # estimativa inicial
tole=1e-3 # tolerância de erro
nmax=50 # número máximo de iterações
# método de Newton
t = newton(ft,t0,dft,tol=tole,maxiter=nmax)
msg = "A taxa do empréstimo é de {0:.2f}% a.a.".format(t*100)
return (t,msg)
def test_lambdify_Derivative_arg_issue_16468():
f = Function('f')(x)
fx = f.diff()
assert lambdify((f, fx), f + fx)(10, 5) == 15
assert eval(lambdastr((f, fx), f/fx))(10, 5) == 2
raises(SyntaxError, lambda:
eval(lambdastr((f, fx), f/fx, dummify=False)))
assert eval(lambdastr((f, fx), f/fx, dummify=True))(10, 5) == 2
assert eval(lambdastr((fx, f), f/fx, dummify=True))(10, 5) == S.Half
assert lambdify(fx, 1 + fx)(41) == 42
assert eval(lambdastr(fx, 1 + fx, dummify=True))(41) == 42
def test_lambdify_Derivative_arg_issue_16468():
f = Function('f')(x)
fx = f.diff()
assert lambdify((f, fx), f + fx)(10, 5) == 15
assert eval(lambdastr((f, fx), f / fx))(10, 5) == 2
raises(SyntaxError, lambda: eval(lambdastr(
(f, fx), f / fx, dummify=False)))
assert eval(lambdastr((f, fx), f / fx, dummify=True))(10, 5) == 2
assert eval(lambdastr((fx, f), f / fx, dummify=True))(S(10), 5) == S.Half
assert lambdify(fx, 1 + fx)(41) == 42
assert eval(lambdastr(fx, 1 + fx, dummify=True))(41) == 42
def do_filter(node):
def strip_index(toks):
indexing = 0
func_str = ''
for i, t in enumerate(toks):
if t == '[':
indexing = indexing + 1
if not indexing:
func_str = func_str + t
if t == ']':
indexing = indexing - 1
return func_str
def actual_func(cond_func, func, **kwargs):
cond_filter = cond_func(**kwargs)
filtered_args = {}
for k, v in kwargs.items():
filtered_args[k] = [
val for val, cond in zip(v, cond_filter) if cond
]
return func(**filtered_args)
assert node.val.scripted
start = node.val.toks.index(Names.listCond)
end = node.val.toks.index(')', start)
cond_toks = node.val.toks[start + 1:end]
cond_str = strip_index(cond_toks)
func_toks = node.val.toks[:start] + node.val.toks[end:]
func_str = strip_index(func_toks)
syms = {}
for name in node.val.names:
syms.update(SympyHelper.initSyms(name))
cond_exprs = SympyHelper.initExprs([cond_str], syms)
func_exprs = SympyHelper.initExprs([func_str], syms)
cond_sol = simplify(cond_exprs)[0]
func_sol = list(
solve(func_exprs,
exclude=list(
SympyHelper.initSyms(node.ordered_given).values()),
check=False,
manual=True,
rational=False)[0].values())[0]
cond_func = lambdify(tuple(node.ordered_given),
cond_sol,
modules=[{
'umin': umin,
'ufloor': ufloor
}, mcerp.umath, 'numpy', 'sympy'])
func = lambdify(tuple(node.ordered_given),
func_sol,
modules=[{
'umin': umin,
'ufloor': ufloor
}, mcerp.umath, 'numpy', 'sympy'])
node.func = functools.partial(actual_func,
cond_func=cond_func,
func=func)
node.func_str = lambdastr(tuple(node.ordered_given), func_sol)
def sym_to_num(func):
return njit()(eval(lambdastr(*func.args), {
**globals(),
**{
"sqrt": np.sqrt
}
}))
def numbafy(self,
expression,
coordinates=None,
parameters=None,
name='trial_func'):
if coordinates:
code_coord = []
for i, x in enumerate(coordinates):
code_coord.append(f'{x} = coordinates[{i}]')
code_coord = '\n '.join(code_coord)
if parameters:
code_param = []
for i, x in enumerate(parameters):
code_param.append(f'{x} = parameters[{i}]')
code_param = '\n '.join(code_param)
temp = lambdastr((), expression)
temp = temp[len('lambda : '):]
temp = temp.replace('math.exp', 'np.exp')
code_expression = f'{temp}'
template = f"""
@njit
def {name}(coordinates, parameters):
{code_coord}
{code_param}
return {code_expression}"""
print('Numba function template generated! Its name is ', name)
return template
def lambdify_mpf(dims, exprs):
# Perform the initial lambdification
ls = [lambdastr(dims, ex.evalf(mp.dps)) for ex in exprs]
csf = {}
# Locate all numerical constants in these lambdified expressions
for l in ls:
for m in re.findall('([0-9]*\.[0-9]+(?:[eE][-+]?[0-9]+)?)', l):
if m not in csf:
csf[m] = mp.mpf(m)
# Sort the keys by their length to prevent erroneous substitutions
cs = sorted(csf, key=len, reverse=True)
# Name these constants
csn = {s: '__c%d' % i for i, s in enumerate(cs)}
cnf = {n: csf[s] for s, n in csn.iteritems()}
# Substitute
lex = []
for l in ls:
for s in cs:
l = l.replace(s, csn[s])
lex.append(eval(l, cnf))
return lex
def make_return_string(expr):
"""Process a sympy expression into an evaluatable Python return statement.
Parameters:
-----------
expr : sympy.Expr instance
Returns:
--------
output : string
A return string that can be used to assemble a Kwant value function.
func_symbols : set of sympy.Symbol instances
All space dependent functions that appear in the expression.
const_symbols : set of sympy.Symbol instances
All constants that appear in the expression.
"""
func_symbols = {sympy.Symbol(i.func.__name__) for i in
expr.atoms(AppliedUndef)}
free_symbols = {i for i in expr.free_symbols if i.name not in ['x', 'y', 'z']}
const_symbols = free_symbols - func_symbols
expr = expr.subs(sympy.I, sympy.Symbol('1.j')) # quick hack
output = lambdastr((), expr, printer=NumericPrinter)[len('lambda : '):]
output = output.replace('MutableDenseMatrix', 'np.array')
output = output.replace('ImmutableMatrix', 'np.array')
return 'return {}'.format(output), func_symbols, const_symbols
def do_generation(cur, do_solve=False):
assert isinstance(cur.val, Relation)
syms = {}
for name in cur.val.names:
syms.update(
SympyHelper.initSyms(
name, self.v2v[getBaseName(name)].vector
and cur.val.deferred))
exprs = SympyHelper.initExprs([cur.val.str], syms)
# Sympy bug workaround:
# Must set rational to False, otherwise piecewise function cannot be solved correctly.
if do_solve:
solutions = solve(exprs,
exclude=cur.ordered_given,
check=False,
manual=True,
rational=False)
solution = solutions[0].values()[0]
else:
solutions = simplify(exprs)
solution = solutions[0]
if len(solutions) != 1:
# We cannot handle multiplte solutions yet.
raise NotImplementedError
cur.func = lambdify(tuple(cur.ordered_given),
solution,
modules=[{
'umin': umin,
'ufloor': ufloor
}, mcerp.umath, 'numpy', 'sympy'])
cur.func_str = lambdastr(tuple(cur.ordered_given), solution)
def logL_alt(self, a, b, c):
subst = [(self.a, a), (self.b, b), (self.c, c)]
for x, kv in enumerate(self.kVec):
subst.append((self.k[x], kv))
subst.append((self.t[x], self.tVec[x]))
ll = lambdastr((self.a, self.b, self.c, self.k, self.t), self.logL_eqn.doit())
print(ll(a, b, c))
def make_return_string(expr):
"""Process a sympy expression into an evaluatable Python return statement.
Parameters:
-----------
expr : sympy.Expr instance
Returns:
--------
output : string
A return string that can be used to assemble a Kwant value function.
func_symbols : set of sympy.Symbol instances
All space dependent functions that appear in the expression.
const_symbols : set of sympy.Symbol instances
All constants that appear in the expression.
"""
func_symbols = {
sympy.Symbol(i.func.__name__)
for i in expr.atoms(AppliedUndef)
}
free_symbols = {
i
for i in expr.free_symbols if i.name not in ['x', 'y', 'z']
}
const_symbols = free_symbols - func_symbols
expr = expr.subs(sympy.I, sympy.Symbol('1.j')) # quick hack
output = lambdastr((), expr, printer=NumericPrinter)[len('lambda : '):]
output = output.replace('MutableDenseMatrix', 'np.array')
output = output.replace('ImmutableMatrix', 'np.array')
return 'return {}'.format(output), func_symbols, const_symbols
def bracket_product_scipy_tplquad(wave_function,
wave_function_prime,
operator=None):
"""
Compute the scalar product <bra|operator|ket> with scipy and the integrate algorithm tplquad
:param wave_function: bra
:param wave_function_prime: ket
:param operator: operator
:return: bracket product (float)
"""
from sympy.utilities.lambdify import lambdastr
r, theta, phi = sp.Symbol("r", real=True), sp.Symbol(
"theta", real=True), sp.Symbol("phi", real=True)
jacobian = (r**2) * sp.sin(theta)
integral_core = sp.conjugate(wave_function) * (
operator if operator is not None else 1) * wave_function_prime
# integral_core_expension = sp.expand(sp.FU["TR8"](jacobian * integral_core), func=True).simplify()
integral_core_expension = sp.expand(jacobian * integral_core,
func=True).simplify()
# Process the integral with scipy
integral_core_lambdify = sp.lambdify((theta, r, phi),
integral_core_expension,
modules="numpy")
print(lambdastr((theta, r, phi), integral_core_expension))
bracket_product = QuantumFactory.complex_quadrature(
sc.integrate.tplquad, integral_core_lambdify, 0, 2 * np.pi,
lambda b_phi: 0, lambda b_phi: np.inf, lambda b_theta, b_r: 0,
lambda b_theta, b_r: np.pi)[0]
# bracket_product = sc.integrate.tplquad(integral_core_lambdify,
# 0, 2 * np.pi, lambda b_phi: 0, lambda b_phi: np.inf,
# lambda b_theta, b_r: 0, lambda b_theta, b_r: np.pi)[0]
return bracket_product
def _expr2code(self, arg_list, expr):
"""Convert the given symbolic expression into code."""
code = lambdastr(arg_list, expr)
function_code = code.split(':')[1].strip()
#for arg in arg_list:
# function_code = function_code.replace(arg.name, 'self.'+arg.name)
return function_code
def _expr2code(self, arg_list, expr):
"""Convert the given symbolic expression into code."""
code = lambdastr(arg_list, expr)
function_code = code.split(':')[1].strip()
#for arg in arg_list:
# function_code = function_code.replace(arg.name, 'self.'+arg.name)
return function_code
def parse_math(expr):
if expr != '':
# вот здесь самая главная часть программы
res = lambdastr(x, expr)
res = eval(res)
# ----------
return res
else:
return False
def buildstr(exp):
exp=exp.subs(hzN,zN/zeta)
exp=exp.subs(hxN,xN/xi)
exp=exp.subs(zN,zN2zh)
exp=exp.subs(xN,xN2xb)
exp=exp.subs(Mki**2,-Mki**2)
args=(M,Mh,xb,zh,eta,Q,T.t,xi,zeta,dk.t,ki.t,Mki,Mkf)
out=lambdastr(args, exp)
out=out.replace('math.','')
return out
def make_jit_fn(args, fn_expr):
fn_str = lambdastr(args, fn_expr).replace('MutableDenseMatrix', '') \
.replace('(([[', '[') \
.replace(']]))', ']')
f = eval(fn_str)
try:
jit_fn = numba.jit(nopython=True)(f)
jit_fn(*np.ones(len(args), dtype=float))
except:
jit_fn = f
return jit_fn
def string_lambdastr_as_function(expr,
x,
a=None,
fun_name=None,
use_numpy=False,
include_imports=False):
if fun_name is None:
fun_name = "calc_expression"
s_lambda = lambdastr(x, expr)
x_name = get_symbol_name(x)
header, content = s_lambda.split(":")
x_tuple_str = string_inputs_as_tuple_descontruction(x)
if a is not None:
a_name = get_symbol_name(a)
a_tuple_str = string_inputs_as_tuple_descontruction(a)
header = "def {}({}, {}):\n".format(fun_name, x_name, a_name)
else:
header = "def {}({}):\n".format(fun_name, x_name)
a_tuple_str = ""
content = content.strip()
content = content[1:-1]
# Im not sure when math is applyed to the converted string or not by sympy
# FIXME: for now i will force removing math to them applying it again
content = content.replace("math.", "")
content = content.replace("sqrt(", "math.sqrt(")
content = content.replace("log(", "math.log(")
content = content.replace("cos(", "math.cos(")
content = content.replace(
"erf(", "math.erf("
) # this is an error, lambdastr should be keeping math or numpy, check later
if use_numpy:
content = content.replace("math", "numpy")
full = "{}\t{}\n\t{}\n\tr = {}\n\treturn r\n".format(
header, x_tuple_str, a_tuple_str, content)
if include_imports:
is_numpy = "import numpy\n" if "numpy" in full else ""
is_math = "import math\n" if "math" in full else ""
full = is_numpy + is_math + "\n" + full
return full
def display_orbital(n, l, m, n_o_c, box_size, resolution, isostep):
"""Diplays a 3D view of electron orbitals"""
P_tex = "" #initialize the LaTex string of the probabilities
#Validate the quantum numbers
assert (n >=
1), "n must be greater or equal to 1" #validate the value of n
assert (0 <= l <=
n - 1), "l must be between 0 and n-1" #validate the value of l
assert (-l <= m <=
l), "p must be between -l and l" #validate the value of p
#Determine the probability equation symbolically and convert
#it to a string
prob = lambdastr((radius, angle_phi, angle_theta),
P(radius, n, l, m, angle_phi, angle_theta))
# print(prob)
#record the probability equation as a LaTex string
P_eq = simplify(P(r, n, l, m, phi, theta))
P_tex += "$$P =" + latex(P_eq) + "$$ \n\n "
if '(nan)' in prob: #Check for errors in the equation
print("There is a problem with the probability function.")
raise ValueError
#Convert the finctions in the probability equation from the sympy
#library to the numpy library to allow for the use of matrix
#calculations
prob = prob.replace('math.sin', 'np.sin') #convert to numpy
prob = prob.replace('math.cos', 'np.cos') #convert to numpy
prob = prob.replace('math.Abs', 'np.abs') #convert to numpy
prob = prob.replace('math.pi', 'np.pi') #convert to numpy
prob = prob.replace('math.exp', 'np.exp') #convert to numpy
# print("Sybolic Prob function: ")
# print(prob)
#convert the converted string to a callable functio
Prob = eval(prob)
#go and let the marching boxes do their thing and create the isosurface mesh
main(Prob, r_fun, phi_fun, theta_fun, n_o_c, box_size, resolution, isostep)
return
def __init__(self, expr, all_vars, variable_list):
'''
Sets up the initial class, accepts a sympy expression and a list of variables.
'''
try:
self.expr = expr # self.expr stores the function to be evaluated, this stores a sympy function
# this is limited right now to a rank 1 tensor that can be embeded into a R^n vector-space
self.dim = len(
self.expr
) # the dimension of the function, IE f is a member of R^2 or a member of R^9001
self.variable_list = variable_list # this stores the list of variables to be used in the function
self.in_args = [
i for i in range(len(self.variable_list))
] # this enumerates the absolute order of the variable in the evaluation list
# IE: f such that R^5 -> R^5 where a,b,x,z,p belong to R; therefore f(a,b,x,z,p) maps to R^5 and "a" has the absolute position in
# the evaluation list of f as 0, b as 1, x as 2, z as 3, and p as 4
self.results = [
{} for x in self.in_args
] # assign an empty set to each function output, this is kept as an empty set so
# that tensors of rank 2 or more can be evaluated at a later version of this module
self.vars = {
variable: 0
for variable in all_vars
} # set the variable as a key into a dictionary with an initial value of 0
self.var_dict = {
i: self.variable_list[i]
for i in range(self.dim)
} # link each variable to a enumeratable key value instead of the name
# IE: {a: 1, b: 2, c: 3} allows f to be evaluated by positional arguments f(1,2,3) and calls the values from self.vars
self.lambda_expr = [
lambdastr(self.variable_list[i], self.expr[key])
for i, key in enumerate(self.var_dict)
] # c style lambda function for
# quicker evaluation of mathematical functions over pure python evaluation. Makes use of cython or direct evaluation of static bytecode
return
except Exception as e:
print(
'Error in Symbolic Function module : __init__.\n{}'.format(e))
PrintException()
def vegafy(self, expression, coordinates=None, name='vega_func'):
if coordinates:
code_coord = []
for i, x in enumerate(coordinates):
code_coord.append(f'{x} = coordinates[:,{i}]')
code_coord = '\n '.join(code_coord)
temp = lambdastr((), expression)
temp = temp[len('lambda : '):]
temp = temp.replace('math.exp', 'np.exp')
code_expression = f'{temp}'
template = f"""
@vegas.batchintegrand
def {name}(coordinates):
{code_coord}
return {code_expression}"""
print('Vega function template generated! Its name is ', name)
return template
def numbafy(expression,
parameters=None,
constants=None,
use_cse=False,
new_function_name='numbafy_func'):
cse = sym.cse(expression)
code_parameters = ''
code_constants = ''
code_cse = ''
if parameters:
code_parameters = ', '.join(f'{p}' for p in parameters)
if constants:
code_constants = []
for k, v in constants.items():
code_constants.append(f'{k} = {v}')
code_constants = '\n '.join(code_constants)
if use_cse:
expressions = sym.cse(expression)
code_cse = []
for e in expressions[0]:
k, v = e
code_cse.append(f'{k} = {v}')
code_cse = '\n '.join(code_cse)
code_expression = f'{expressions[1][0]}'
else:
temp = lambdastr((), expression)
temp = temp[len('lambda : '):]
code_expression = f'{temp}'
template = f"""@jit
def {new_function_name}({code_parameters}):
{code_constants}
{code_cse}
return {code_expression}"""
return template
def display_orbital(n, l, m_, no_of_contours=16, Opaque=0.5):
"""Diplays a 3D view of electron orbitals"""
# The plot density settings (don't mess with unless you are sure)
rng = 12 * n * 1.5 # This determines the size of the box
_steps = 55j # (it needs to be bigger with n).
_x, _y, _z = np.ogrid[-rng:rng:_steps, -rng:rng:_steps, -rng:rng:_steps]
# Plot tweaks
color = (0, 1.0, 1.0) # relative RGB color (0-1.0 vs 0-255)
mlab.figure(bgcolor=color) # set the background color of the plot
P_tex = "" # initialize the LaTex string of the probabilities
# Validate the quantum numbers
# validate the value of n
assert (n >= 1), "n must be greater or equal to 1"
# validate the value of l
assert (0 <= l <= n - 1), "l must be between 0 and n-1"
# validate the value of p
assert (-l <= max(m_) <= l), "p must be between -l and l"
# validate the value of p
assert (-l <= min(m_) <= l), "p must be between -l and l"
for m in m_:
# Determine the probability equation symbolically and convert
# it to a string
prob = lambdastr((radius, angle_phi, angle_theta),
P(radius, n, l, m, angle_phi, angle_theta))
# record the probability equation as a LaTex string
P_eq = simplify(P(r, n, l, m, phi, theta))
P_tex += "$$P =" + latex(P_eq) + "$$ \n\n "
# print("prob before substitution = \n\n".format(prob)) #for debugging
if '(nan)' in prob: # Check for errors in the equation
print("There is a problem with the probability function.")
raise ValueError
# Convert the finctions in the probability equation from the sympy
# library to the numpy library to allow for the use of matrix
# calculations
# prob = prob.replace('sin', 'np.sin') # convert to numpy
# prob = prob.replace('cos', 'np.cos') # convert to numpy
# prob = prob.replace('Abs', 'np.abs') # convert to numpy
# prob = prob.replace('pi', 'np.pi') # convert to numpy
# prob = prob.replace('exp', 'np.exp') # convert to numpy
# print("prob after substitution = \n\n".format(prob)) #for debugging
# convert the converted string to a callable function
Prob = eval(prob)
# generate a set of data to plot the contours of.
w = Prob(r_fun(_x, _y, _z), phi_fun(_x, _y, _z), theta_fun(_x, _y, _z))
# add the generated data to the plot
mlab.contour3d(w,
contours=no_of_contours,
opacity=Opaque,
transparent=True)
mlab.colorbar()
mlab.outline()
mlab.show() # this pops up a interactive window that allows you to
# rotate the view
# Information used for the 2D slices below
limits = []
lengths = []
for cor in (_x, _y, _z):
limit = (np.min(cor), np.max(cor))
limits.append(limit)
# print(np.size(cor))
lengths.append(np.size(cor))
# print(limit)
return (limits, lengths, _x, _y, _z, P_tex)
import sympy
from sympy.utilities.lambdify import lambdastr
# A line through the 3 dimensional vector a in direction s has points
# (ax+t*sx, ay+t*sy, az+t*sz). To find the distance from a point at
# the origin, where the derivative of the distance function (as a
# function of t) has its inflection.
a = sympy.DeferredVector('a')
s = sympy.DeferredVector('x')
t = sympy.Symbol('t')
dist2 = (a[0] + s[0] * t)**2 + (a[1] + s[1] * t)**2 + (a[2] + s[2] * t)**2
ddist2_dt = sympy.diff(dist2, t)
func = sympy.solvers.solve(ddist2_dt, t)
#print func
print lambdastr((a, s), func[0])
import sympy
from sympy.utilities.lambdify import lambdastr
# A line through the 3 dimensional vector a in direction s has points
# (ax+t*sx, ay+t*sy, az+t*sz). To find the distance from a point at
# the origin, where the derivative of the distance function (as a
# function of t) has its inflection.
a = sympy.DeferredVector('a')
s = sympy.DeferredVector('x')
t=sympy.Symbol('t')
dist2 = (a[0]+s[0]*t)**2 + (a[1]+s[1]*t)**2 + (a[2]+s[2]*t)**2
ddist2_dt = sympy.diff(dist2,t)
func = sympy.solvers.solve(ddist2_dt, t)
#print func
print lambdastr((a,s),func[0])
e_equil = -((-(9.81 * rod_mass) / rod_spring) + e_offset_scalar)
if j == 0:
p = ((r21_x * x + a2_x * r1_x) * z - r21_z * x**2 +
(r1_z * r2_x - r1_x * r2_z + a2_x * r21_z - a2_z * r2_x) * x +
r1_x * (a2_x * r2_z - a2_z * r2_x))
equation_all = p.subs({
H: workspace_width,
a: plate_width,
b: plate_height,
phi: theta,
e: e_equil
}).evalf()
equation_all = equation_all.simplify()
lambda_str1 = lambdastr((theta), equation_all)
lambda_str2 = lambda_str1.replace('sin', 'np.sin')
lambda_str3 = lambda_str2.replace('cos', 'np.cos')
lambda_str4 = lambda_str3.replace('x', 'x_temp')
lambda_str = lambda_str4.replace('z', 'z_temp')
func1 = eval(lambda_str)
# In[4]:
x_values_heat = np.linspace(x_begin, x_end, 100)
z_values_heat = np.linspace(z_begin, z_end, 100)
theta_values_heat = []
x_values_heatp = []
z_values_heatp = []
for j in range(100):
def __init__(self, model, *setup, **operation):
# Collect parameter values
environment = dict()
for s in setup + (operation,):
if isinstance(s, dict):
environment.update(s)
elif isinstance(s, System):
environment.update(s.environment)
else:
environment.update(s.__dict__)
E = self.environment = environment
M = self.model = (sympify(pythonify(model), self._skip.copy())
if isinstance(model, str) else
(model[0] if len(model) == 1 else Matrix(model))
if isinstance(model, list) else model)
S = symbols = (dict([(a.name, a) for a in M.atoms(Symbol)])
if isinstance(M, Basic) else dict())
# Find applicable substitutions and remaning undefined parameters.
A = self.applied = dictmap(set(S).intersection(E), E.get)
I = self.implicit = dict()
R = self.remaining = dictmap(set(S).difference(A), S.get)
# S = A + R now.
# Use I for objects in A that sympy can't handle.
if A:
# Evaluate all sub-expressions. May expand A + I + R.
for n, v in A.copy().items():
if isinstance(v, Basic) and not isinstance(v, Symbol):
try:
assert len(v.atoms(Symbol)) > 1
except:
continue
v = v(E) if isinstance(v, System) else System(v, E)
A.update(v.applied)
A[n] = v.result
I.update(v.implicit)
R.update(v.remaining)
# M /. E == M /. A now. A may contain sympy-incompatible objects.
# Categorize A.
Q = dict([(n, v) for n, v in A.items()
if isinstance(v, Quantity) and not n[0] == '_'])
I.update([(n, v) for n, v in A.items()
if isinstance(v, Iterable) and not n in Q])
OK = dictmap(set(A).difference(Q).difference(I), A.get)
# M /. E == M /. (OK + I + Q) now. Only OK is sympy-compatible.
# Eliminate Q by expanding OK and I.
for n, q in Q.items():
q = SymQuantity(q, n)
I.update(q.implicit)
OK[n] = q.result
# M /. E == M /. (OK + I) == (M /. OK) /. I now.
# Evalute M /. OK symbolically.
try:
result = M.subs(OK).evalf()
except Exception as err:
raise Exception('%s /. %s: %s' % (M, params(OK), err))
# M /. E == result /. I now.
# If possible, evaluate result /. I numerically.
if not R and isinstance(result, Basic):
try:
result = eval(lambdastr(I, result))(*I.values())
except Exception as err:
raise Exception('%s /. %s: %s' % (result, params(I), err))
self.implicit = dict()
self.result = result
else:
self.result = M
self.implicit = dict()
def __repr__(self) -> str:
'''Returns a string of the vector field in angle-bracket notation with lambda parameters.'''
x, y = sym.symbols('x y')
return '{} = <{}, {}>'.format(self.name, lambdastr((x, y),
self.__usym),
lambdastr((x, y), self.__vsym))
def display_orbital(n, l, m_, no_of_contours=16, Opaque=0.5):
"""Diplays a 3D view of electron orbitals"""
#The plot density settings (don't mess with unless you are sure)
rng = 12 * n * 1.5 #This determines the size of the box
_steps = 100j # (it needs to be bigger with n).
steps = _steps.imag
_x, _y, _z = np.ogrid[-rng:rng:_steps, -rng:rng:_steps, -rng:rng:_steps]
P_tex = "" #initialize the LaTex string of the probabilities
#Validate the quantum numbers
assert (n >=
1), "n must be greater or equal to 1" #validate the value of n
assert (0 <= l <=
n - 1), "l must be between 0 and n-1" #validate the value of l
assert (-l <= max(m_) <=
l), "p must be between -l and l" #validate the value of p
assert (-l <= min(m_) <=
l), "p must be between -l and l" #validate the value of p
for m in m_:
#Determine the probability equation symbolically and convert
#it to a string
prob = lambdastr((radius, angle_phi, angle_theta),
P(radius, n, l, m, angle_phi, angle_theta))
#record the probability equation as a LaTex string
P_eq = simplify(P(r, n, l, m, phi, theta))
P_tex += "$$P =" + latex(P_eq) + "$$ \n\n "
if '(nan)' in prob: #Check for errors in the equation
print("There is a problem with the probability function.")
raise ValueError
#Convert the finctions in the probability equation from the sympy
#library to the numpy library to allow for the use of matrix
#calculations
prob = prob.replace('sin', 'np.sin') #convert to numpy
prob = prob.replace('cos', 'np.cos') #convert to numpy
prob = prob.replace('Abs', 'np.abs') #convert to numpy
prob = prob.replace('pi', 'np.pi') #convert to numpy
prob = prob.replace('exp', 'np.exp') #convert to numpy
#convert the converted string to a callable function
Prob = eval(prob)
#generate a set of data to plot the contours of.
w = Prob(r_fun(_x, _y, _z), phi_fun(_x, _y, _z), theta_fun(_x, _y, _z))
#Remove nan's in grid w
w[np.isnan(w)] = 0
#Determine minimum and maximum value (probability density) in the grid w
minw = np.nanmin(w)
maxw = np.nanmax(w)
# print(minw, maxw)
#Determine value slices
colorSlice = (maxw - minw) / (no_of_contours)
#Create contour lookup array
pr = np.linspace(minw, maxw, no_of_contours)
# print(pr)
buildGridAndCreate(int(steps), pr, w)
#########
#select object as active and change toggle to editmode
bpy.context.scene.objects.active = bpy.data.objects['Orbital']
bpy.ops.object.mode_set(mode='EDIT')
# Get the active mesh
ob = bpy.context.edit_object
me = ob.data
# Get a BMesh representation
bm = bmesh.from_edit_mesh(me)
# Modify the BMesh, can do anything here...
for v in bm.verts:
v.co.x += 1.0
# Show the updates in the viewport
# and recalculate n-gon tessellation.
bmesh.update_edit_mesh(me, True)
bpy.ops.object.mode_set(mode='OBJECT')
########
#Information used for the 2D slices below
limits = []
lengths = []
for cor in (_x, _y, _z):
limit = (np.min(cor), np.max(cor))
limits.append(limit)
#print(np.size(cor))
lengths.append(np.size(cor))
#print(limit)
return (limits, lengths, _x, _y, _z, P_tex)
f1 = (_E1.T * _E1 - ONE)**2 + (_E2.T * _E2 - ONE)**2 + (_E1.T * _E2)**2
f2 = (R.T * R - ONE)**2 + (_E1.T * R - E1.T * R)**2 + (_E2.T * R - E2.T * R)**2
f3 = (P_i.T * _E1 - Matrix([x2_s]))**2 + (P_i.T * _E2 - Matrix([y2_s]))**2
#f1 = sq(_E1.T * _E1, ONE) + sq(_E2.T * _E2, ONE) + sq(_E1.T * _E2, ZERO)
print(f1)
substitution = [(_E1, F1), (_E2, F2), (_R, R)]
f = f1 + f2 + f3
func = Matrix(f.subs(substitution))
#f = f.subs(_E2, F2)
#f.subs(_R, R)
lam_f = lambdify(var, func, 'numpy')
print(lambdastr(var,func))
def lam(x2, y2, p, e_1, e_2):
return lambda a1,b1,c1,a2,b2,c2,t,s: \
lam_f(x2, y2, p, e_1, e_2, a1, b1, c1, a2, b2, c2, t, s)
arr_init = np.array([1, 0, 0, 0, 1, 0, 1, 1])
print("lambda: ready")
######## Graph Drawing ########
root = Tk()
w = Canvas(root, width=_wid, height=_hei, bg='White')
w.pack()
circles = []
lines = []
r = 10
_E1.dot(_E1) - 1,
_E2.dot(_E2) - 1,
_E1.dot(_E2),
R.dot(R) - 1,
_E1.dot(R) - sp.Matrix(E1).dot(R),
_E2.dot(R) - sp.Matrix(E2).dot(R),
sp.Matrix(P_i).dot(_E1) - x2_s,
sp.Matrix(P_i).dot(_E2) - y2_s
])
[e * e for e in [e1, e2, e3]]
func = sp.simplify(sp.Matrix.norm(f))
lam_f = lambdify(var, func, 'numpy')
print(lambdastr(var, func))
print(lambdastr(var, _E1))
print(lambdastr(var,_E1.dot(_E1)))
def lam(x2, y2, p, e_1, e_2):
return lambda a1,b1,c1,a2,b2,c2,t,s: \
lam_f(x2, y2, p, e_1, e_2, a1, b1, c1, a2, b2, c2, t, s)
arr = np.array([1, 1, 1, 1, 1, 1, 1, 1])
print("lambda: ready")
#X_sample = 3 * np.random.random_sample((100, 1)) - 1.5
#Y_sample = 3 * np.random.random_sample((100, 1)) - 1.5
print("ready")
"""
def make_njit_fn(args, fn_expr):
fn_str = lambdastr(args, fn_expr).replace('MutableDenseMatrix', '')\
.replace('(([[', '[') \
.replace(']]))', ']')
jit_fn = numba.njit(parallel=True)(eval(fn_str))
return jit_fn
def gen_ode_model(self):
"""
Generate a lambda (self.f_model) from the Sympy expressions in self.equations that takes the values for all
species and parameters and returns dX/dt.
Returns
-------
None
"""
# Prune any term from an equation that is not either a constant or the LHS of another equation.
pruned_eqs = []
lhs = [eq.lhs.args[0] for eq in self.equations]
for eq in self.equations:
for arg in self.ravel_expression(eq.rhs):
if arg not in lhs and not isinstance(arg, Constant):
eq = eq.subs(arg, 0)
pruned_eqs.append(eq)
self.equations = pruned_eqs
# Process
rhss = [i.rhs for i in self.equations]
all_args = []
for eq in rhss:
args = self.ravel_expression(eq)
for a in args:
all_args.append(a)
# remove duplicates
unique_sels = []
unique_consts = []
seen_sel = []
seen_const = []
for val in all_args:
if isinstance(val, Selector):
if val.selector not in seen_sel:
seen_sel.append(val.selector)
unique_sels.append(val)
elif isinstance(val, Constant):
if val.name not in seen_const:
seen_const.append(val.name)
unique_consts.append(val)
else:
print('unable to handle %s' % val)
self.species = unique_sels
self.params = unique_consts
# We need the rhss in the same order as the same order as the unique_sels to pass to lambdify.
ordered_rhss = []
# In the process, we will rearrange self.equations to be in the same order.
ordered_equations = []
for arg in unique_sels:
for eq in self.equations:
if eq.lhs.args[0].selector == arg.selector:
ordered_rhss.append(eq.rhs)
ordered_equations.append(eq)
self.equations = ordered_equations
unique_args = unique_sels + unique_consts
self.unique_args = unique_args
self.ordered_rhss = ordered_rhss
self.f_model = sy.lambdify(unique_args, ordered_rhss)
self.f_model = njit(self.f_model)
print(
"celltx ODELayer: Successfully compiled self.f_model to C via njit."
)
self.lambda_string = lambdastr(unique_args, ordered_rhss, dummify=True)
# Also generate starting conditions self.x0 (zeros for each species)
self.x0 = np.zeros(len(self.species))
def compute(self, maximize=False, constraints=None):
""" Solves the system and computes the responses.
"""
# Expand expressions on first time.
if not self.exprs:
self.exprs = self.parser.expand(self.exprs_str, self.idx_bounds, self.syms)
u_math = umath if not self.analytical else a_umath
# Generate an ordering for inputs.
if not self.ordered_given:
for (k, v) in self.given.iteritems():
self.ordered_given.append(k)
q_ordered_given = []
# Ordered given list fed to optimization, might be different from ordered_given.
opt_ordered_given = []
opt_q_ordered_given = []
self.opts = []
for (k, v) in self.given.iteritems():
if isinstance(v, str) and v == 'opt':
self.opts.append(k)
else:
opt_ordered_given.append(k)
opt_q_ordered_given.append(v)
# Do minimization if needed.
if self.opts:
opt_given = []
for k in self.opts:
opt_given.append(k)
opt_ordered_given = opt_given + opt_ordered_given
opt_val = self.optimize(opt_ordered_given, opt_q_ordered_given, maximize)
# Assemble q_ordered_given according to ordered_given.
for k in self.ordered_given:
if isinstance(self.given[k], str) and self.given[k] == 'opt':
q_ordered_given.append(opt_val[k])
else:
q_ordered_given.append(self.given[k])
# Solve for intermediate solution set, use cached version if possible.
if not self.sol_inter_set:
self.sol_inter_set = solve(self.exprs, exclude=self.ordered_given,
check=False, manual=True)[0]
""" Uncomment following code to print intermediate solution set.
logging.debug('Sheet -- Partial Solutions:')
for k, s in self.sol_inter_set.iteritems():
logging.debug('Inter -- {}: {}'.format(k, s))
"""
# Generate intermediate funcs, use cached version if possible.
if not self.inter_funcs:
for var in self.intermediates.keys():
if var in self.sol_inter_set:
logging.debug('{} Lambdification -- {}'.format(var,
self.sol_inter_set[var]))
self.inter_funcs[var] = lambdify(tuple(self.ordered_given),
self.sol_inter_set[var],
modules=[self.sym2func, u_math])
else:
print "WARN: ignoring %r, no solution found!" % var
else:
# Make sure that all intermediates have a solution.
for k in self.intermediates.keys():
assert(k in self.inter_funcs)
# Before generating quantities for intermediates,
# if we are performing analytical analysis,
# we need to convert any MC given to analytical form when needed.
if self.analytical:
q_ordered_given = self.conv2analytical(q_ordered_given)
for q in q_ordered_given:
assert(not isinstance(q, UncertainFunction))
# Organize intermediates based on an ordering.
ordered_intermediates = []
q_ordered_intermediates = []
for (k, v) in self.intermediates.iteritems():
if k in self.inter_funcs:
ordered_intermediates.append(k)
mean_intermediate = float(self.inter_funcs[k](*tuple(q_ordered_given)))
assert(callable(v))
q_ordered_intermediates.append(v(mean_intermediate))
logging.debug('Sheet -- Inter {}: {}'.format(
ordered_intermediates[-1], q_ordered_intermediates[-1]))
# Solve for final solution set, use cached version if possible.
if not self.sol_final_set:
self.sol_final_set = solve(self.exprs,
exclude=self.ordered_given + ordered_intermediates, check=False, manual=True)[0]
""" Uncomment the following to print final solution set.
logging.debug('Sheet -- Final Solutions:')
for k, s in self.sol_final_set.iteritems():
logging.debug('Final -- {}: {}'.format(k, s))
"""
# Generate target funcs, use cached version if possible.
if not self.target_funcs:
for var in self.response:
logging.debug('{} Lambdification -- {}'.format(var,
lambdastr(tuple(self.ordered_given + ordered_intermediates),
self.sol_final_set[var])))
self.target_funcs[var] = lambdify(tuple(self.ordered_given + ordered_intermediates),
self.sol_final_set[var], modules=[self.sym2func, u_math])
# Before generating quantities for final responses,
# we need to convert any MC intermediate quantities to analytical form when needed.
if self.analytical:
q_ordered_intermediates = self.conv2analytical(q_ordered_intermediates)
for q in q_ordered_intermediates:
assert(not isinstance(q, UncertainFunction))
# Compute response.
q_response = {}
for var in self.response:
logging.debug('Solving {}'.format(str(var)))
perf = self.target_funcs[var](*tuple(q_ordered_given + q_ordered_intermediates))
q_response[str(var)] = perf
return q_response
def _print_sympy(expr):
return lambdastr((), expr, printer=_NumericPrinter)[len('lambda : '):]
def compute(self, maximize=False, constraints=None):
"""
Solve the system and apply the quantification.
"""
# Expand expressions on first time.
if not self.exprs:
self.exprs = self.parser.expand(self.exprs_str, self.idx_bounds, self.syms)
u_math = umath if not self.analytical else a_umath
# Generate an ordering for inputs.
if not self.ordered_given:
for (k, v) in self.given.iteritems():
self.ordered_given.append(k)
q_ordered_given = []
# Ordered given list fed to optimization, might be different from ordered_given.
opt_ordered_given = []
opt_q_ordered_given = []
self.opts = []
for (k, v) in self.given.iteritems():
if isinstance(v, str) and v == 'opt':
self.opts.append(k)
else:
opt_ordered_given.append(k)
opt_q_ordered_given.append(v)
# Do minimization if needed.
if self.opts:
opt_given = []
for k in self.opts:
opt_given.append(k)
opt_ordered_given = opt_given + opt_ordered_given
opt_val = self.optimize(opt_ordered_given, opt_q_ordered_given, maximize)
# Assemble q_ordered_given according to ordered_given.
for k in self.ordered_given:
if isinstance(self.given[k], str) and self.given[k] == 'opt':
q_ordered_given.append(opt_val[k])
else:
q_ordered_given.append(self.given[k])
# Solve for final solution set, use cached version if possible.
if not self.sol_set:
sol_sets = solve(self.exprs,
exclude=self.ordered_given, check=False, manual=True)
assert len(sol_sets) == 1, 'Multiple solutios possible, consider rewrite model.'
self.sol_set = sol_sets[0]
logging.debug('Sheet -- Given: {}'.format(self.ordered_given))
logging.debug('Sheet -- Solutions:')
for k, s in self.sol_set.iteritems():
logging.debug('\t{}: {}'.format(k, s))
# Generate target funcs, use cached version if possible.
for var in self.response:
if var not in self.target_funcs:
self.target_funcs[var] = (lambdify(tuple(self.ordered_given),
self.sol_set[var], modules=[self.sym2func, u_math]))
logging.debug('Lamdification {} {} --\n\t{}'.format(var,
self.target_funcs[var],
lambdastr(tuple(self.ordered_given), self.sol_set[var])))
# Compute response.
q_response = {}
for var in self.response:
logging.debug('Solving {}'.format(str(var)))
logging.debug('Params:\n{}\n{}'.format(
self.ordered_given, q_ordered_given))
logging.debug('Calling {}'.format(self.target_funcs[var]))
perf = self.target_funcs[var](*tuple(q_ordered_given))
q_response[str(var)] = perf
return q_response
d_inner_by_dt = partial_L_by_partial_qdot.jacobian(Matrix(
[q])) * qdot + partial_L_by_partial_qdot.jacobian(Matrix([qdot])) * qddot
# Euler-Lagrange equation
lagrange_eq = partial_L_by_partial_q - d_inner_by_dt + Matrix([f, 0, 0])
# solve the lagrange equation for qddot and simplify
print("Calculations take a while...")
r = sympy.solvers.solve(simplify(lagrange_eq), Matrix([qddot]))
qddot_0 = simplify(r[qddot_0])
qddot_1 = simplify(r[qddot_1])
qddot_2 = simplify(r[qddot_2])
print('qddot_0 = {}\n'.format(qddot_0))
print('qddot_1 = {}\n'.format(qddot_1))
print('qddot_2 = {}\n'.format(qddot_2))
# generate python function
s = lambdastr(
(q_0, q_1, q_2, qdot_0, qdot_1, qdot_2, f, r_1, r_2, m_c, m_1, m_2, g),
[qdot_0, qdot_1, qdot_2, qddot_0, qddot_1, qddot_2])
f_gen = open("dpc_dynamics_generated.py", 'w')
f_gen.write(
"import math\ndef dpc_dynamics_generated(q_0, q_1, q_2, qdot_0, qdot_1, qdot_2, f, r_1, r_2, m_c, m_1, m_2, g):\n\tfun="
+ s +
"\n\treturn fun(q_0, q_1, q_2, qdot_0, qdot_1, qdot_2, f, r_1, r_2, m_c, m_1, m_2, g)"
)
f_gen.close()