def get_poly(order, dim, is_simplex=False):
"""
Construct a polynomial of given `order` in space dimension `dim`,
and integrate it symbolically over a rectangular or simplex domain
for coordinates in [0, 1].
"""
try:
xs = sm.symarray('x', dim)
except:
xs = sm.symarray(dim, 'x')
opd = max(1, int((order + 1) / dim))
poly = 1.0
oo = 0
for x in xs:
if (oo + opd) > order:
opd = order - oo
poly *= (x**opd + 1)
oo += opd
limits = [[xs[ii], 0, 1] for ii in range(dim)]
if is_simplex:
for ii in range(1, dim):
for ip in range(0, ii):
limits[ii][2] -= xs[ip]
integral = sm.integrate(poly, *reversed(limits))
return xs, poly, limits, integral
def main(y0='1,0', mu=1.0, tend=10., nt=50, savefig='None', plot=False,
savetxt='None', integrator='scipy', dpi=100, kwargs='',
verbose=False):
assert nt > 1
y = sp.symarray('y', 2)
p = sp.Symbol('p', real=True)
f = [y[1], -y[0] + p*y[1]*(1 - y[0]**2)]
odesys = SymbolicSys(zip(y, f), params=[p], names=True)
tout = np.linspace(0, tend, nt)
y0 = list(map(float, y0.split(',')))
kwargs = dict(eval(kwargs) if kwargs else {})
xout, yout, info = odesys.integrate(
tout, y0, [mu], integrator=integrator, **kwargs)
if verbose:
print(info)
if savetxt != 'None':
np.savetxt(stack_1d_on_left(xout, yout), savetxt)
if plot:
import matplotlib.pyplot as plt
odesys.plot_result()
plt.legend()
if savefig != 'None':
plt.savefig(savefig, dpi=dpi)
else:
plt.show()
def main(init_conc='1e-7,1e-7,1e-7,1,55.5',
lnKa=-21.28, lnKw=-36.25,
savetxt='None', verbose=False,
rref=False, charge=False, solver='scipy'):
# H+, OH- NH4+, NH3, H2O
iHp, iOHm, iNH4p, iNH3, iH2O = init_conc = map(float, init_conc.split(','))
lHp, lOHm, lNH4p, lNH3, lH2O = x = sp.symarray('x', 5)
Hp, OHm, NH4p, NH3, H2O = map(sp.exp, x)
# lHp + lOHm - lH2O - lnKw,
# lHp + lNH3 - lNH4p - lnKa,
coeffs = [[1, 1, 0, 0, -1], [1, 0, -1, 1, 0]]
vals = [lnKw, lnKa]
lp = linear_exprs(coeffs, x, vals, rref=rref)
f = lp + [
Hp + OHm + 4*NH4p + 3*NH3 + 2*H2O - (
iHp + iOHm + 4*iNH4p + 3*iNH3 + 2*iH2O), # H
NH4p + NH3 - (iNH4p + iNH3), # N
OHm + H2O - (iOHm + iH2O)
]
if charge:
f += [Hp - OHm + NH4p - (iHp - iOHm + iNH4p)]
neqsys = SymbolicSys(x, f)
x, sol = neqsys.solve([0]*5, solver=solver)
if verbose:
print(np.exp(x), sol)
else:
print(np.exp(x))
assert sol.success
def initialize(self, args):
self.E = args[0]
self.A = args[1]
self.I = args[2]
self.r = args[3]
self.rho = args[4]
self.l = args[5]
self.g = args[6]
q = sym.Matrix(sym.symarray('q',6))
E, A, I, r, rho, l, g = sym.symbols('E A I r rho l g')
theta = sym.Matrix(['theta_1','theta_2'])
omega = sym.Matrix(['omega_1','omega_2'])
# Load symbolic needed matricies and vectors
M_sym = pickle.load( open( "gebf-mass-matrix.dump", "rb" ) )
beta_sym = pickle.load( open( "gebf-force-vector.dump", "rb" ) )
Gamma1_sym = pickle.load( open( "gebf-1c-matrix.dump", "rb" ) )
Gamma2_sym = pickle.load( open( "gebf-2c-matrix.dump", "rb" ) )
# Create numeric function of needed matrix and vector quantities
# this is MUCH MUCH faster than .subs()
# .subs() is unusably slow !!
self.M = lambdify((E, A, I, r, rho, l, g, q, theta, omega), M_sym, "numpy")
self.beta = lambdify((E, A, I, r, rho, l, g, q, theta, omega), beta_sym, "numpy")
self.Gamma1 = lambdify((E, A, I, r, rho, l, g, q, theta, omega), Gamma1_sym, "numpy")
self.Gamma2 = lambdify((E, A, I, r, rho, l, g, q, theta, omega), Gamma2_sym, "numpy")
def construct_matrix_and_entries(prefix_name,shape):
A_entries = matrix(sympy.symarray(prefix_name,shape))
A = sympy.Matrix.zeros(shape[0],shape[1])
for r in range(A.rows):
for c in range(A.cols):
A[r,c] = A_entries[r,c]
return A, A_entries
def __init__(self, modulus, inertia, radius, density, length, state):
"""
Initializes all of the inertial propertias of the element and assigns
values to physical and material properties
"""
# symbolic system parameters
E, I, r, rho, l, = sym.symbols('E I r rho l')
# Load symbolic mass matrix and substitute material properties
M = pickle.load( open( "gebf-mass-matrix.dump", "rb" ) ).subs([ \
(E, modulus), (I, inertia), (r, radius), (rho, density), (l, length)])
# load the body and applied force vector and substitute material properties
self.beta = pickle.load( open( "gebf-beta.dump", "rb" ) ).subs([ \
(E, modulus), (I, inertia), (r, radius), (rho, density), (l, length)])
# Substitute State Variables
# re-make symbols for proper substitution and create the paird list
q = sym.Matrix(sym.symarray('q',2*8))
q_sub = [(q, qi) for q, qi in zip(q, state)]
# Substitute state variables
beta = self.beta.subs(q_sub)
M = M.subs(q_sub)
# Form the binary-DCA algebraic quantities
# Partition mass matrix
M11 = np.array(M[0:4,0:4])
M12 = np.array(M[0:4,4:8])
M21 = np.array(M[4:8,0:4])
M22 = np.array(M[4:8,4:8])
# For now use these definitions to cast Fic (constraint forces between GEBF elements)
# into generalized constraint forces
self.gamma11 = np.eye(4)
self.gamma12 = np.zeros((4,4))
self.gamma22 = np.eye(4)
self.gamma21 = np.zeros((4,4))
# partition beta into lambda13 and lambda23
self.gamma13 = np.array(beta[0:4])
self.gamma23 = np.array(beta[4:8])
# Commonly inverted quantities
iM11 = inv(self.M11)
iM22 = inv(self.M22)
Gamma1 = inv(self.M11 - self.M12*iM22*self.M21)
Gamma2 = inv(self.M22 - self.M21*iM11*self.M12)
# Compute all terms of the two handle equations
self.z11 = Gamma1.dot(self.gamma11 - self.M12.dot(iM22.dot(self.gamma21)))
self.z12 = Gamma1.dot(self.gamma12 - self.M12.dot(iM22.dot(self.gamma22)))
self.z13 = Gamma1.dot(self.gamma13 - self.M12.dot(iM22.dot(self.gamma23)))
self.z21 = Gamma2.dot(self.gamma21 - self.M21.dot(iM11.dot(self.gamma11)))
self.z22 = Gamma2.dot(self.gamma22 - self.M21.dot(iM11.dot(self.gamma12)))
self.z23 = Gamma2.dot(self.gamma23 - self.M21.dot(iM11.dot(self.gamma13)))
def intProps(self,args):
u = args[0]
# symbolic system parameters
E, A, I, r, rho, l, = sym.symbols('E A, I r rho l')
# Load symbolic mass matrix and substitute material properties
self.M = pickle.load( open( "gebf-mass-matrix.dump", "rb" ) ).subs([ \
(E, self.E), (A, self.A), (I, self.I), (r, self.r), (rho, self.rho), (l, self.l)])
# load the body and applied force vector and substitute material properties
self.beta = pickle.load( open( "gebf-beta.dump", "rb" ) ).subs([ \
(E, self.E), (A, self.A), (I, self.I), (r, self.r), (rho, self.rho), (l, self.l)])
# Substitute State Variables
# re-make symbols for proper substitution and create the paird list
q = sym.Matrix(sym.symarray('q',2*3))
qe = [self.theta1] + u[:2] + [self.theta2] + u[2:]
q_sub = [(q, qi) for q, qi in zip(q, qe)]
# Substitute state variables
self.beta = self.beta.subs(q_sub).evalf()
self.M = self.M.subs(q_sub).evalf()
# Form the binary-DCA algebraic quantities
# Partition mass matrix
self.M11 = np.array(self.M[0:3,0:3])
self.M12 = np.array(self.M[0:3,3:6])
self.M21 = np.array(self.M[3:6,0:3])
self.M22 = np.array(self.M[3:6,3:6])
# For now use these definitions to cast Fic (constraint forces between GEBF elements)
# into generalized constraint forces
self.gamma11 = np.eye(3)
self.gamma12 = np.zeros((3,3))
self.gamma22 = np.eye(3)
self.gamma21 = np.zeros((3,3))
# partition beta into lambda13 and lambda23
self.gamma13 = np.array(self.beta[0:3])
self.gamma23 = np.array(self.beta[3:6])
# Commonly inverted quantities
self.iM11 = inv(self.M11)
self.iM22 = inv(self.M22)
self.Gamma1 = inv(self.M11 - self.M12*self.iM22*self.M21)
self.Gamma2 = inv(self.M22 - self.M21*self.iM11*self.M12)
# Compute all terms of the two handle equations
self.z11 = self.Gamma1.dot(self.gamma11 - self.M12.dot(self.iM22.dot(self.gamma21)))
self.z12 = self.Gamma1.dot(self.gamma12 - self.M12.dot(self.iM22.dot(self.gamma22)))
self.z13 = self.Gamma1.dot(self.gamma13 - self.M12.dot(self.iM22.dot(self.gamma23)))
self.z21 = self.Gamma2.dot(self.gamma21 - self.M21.dot(self.iM11.dot(self.gamma11)))
self.z22 = self.Gamma2.dot(self.gamma22 - self.M21.dot(self.iM11.dot(self.gamma12)))
self.z23 = self.Gamma2.dot(self.gamma23 - self.M21.dot(self.iM11.dot(self.gamma13)))
def test_SymbolicSys_bateman(band):
tend, k, y0 = 2, [4, 3], (5, 4, 2)
y = sp.symarray('y', len(k)+1)
dydt = decay_dydt_factory(k)
f = dydt(0, y)
odesys = SymbolicSys(zip(y, f), band=band)
xout, yout, info = odesys.integrate(tend, y0, integrator='scipy')
ref = np.array(bateman_full(y0, k+[0], xout-xout[0], exp=np.exp)).T
assert np.allclose(yout, ref)
def test_symarray():
"""Test creation of numpy arrays of sympy symbols."""
import numpy as np
import numpy.testing as npt
syms = symbols('_0 _1 _2')
s1 = symarray(3)
s2 = symarray(3)
npt.assert_array_equal (s1, np.array(syms, dtype=object))
assert s1[0] is s2[0]
a = symarray(3, 'a')
b = symarray(3, 'b')
assert not(a[0] is b[0])
asyms = symbols('a_0 a_1 a_2')
npt.assert_array_equal (a, np.array(asyms, dtype=object))
# Multidimensional checks
a2d = symarray((2,3), 'a')
assert a2d.shape == (2,3)
a00, a12 = symbols('a_0_0, a_1_2')
assert a2d[0,0] is a00
assert a2d[1,2] is a12
a3d = symarray((2,3,2), 'a')
assert a3d.shape == (2,3,2)
a000, a120, a121 = symbols('a_0_0_0, a_1_2_0 a_1_2_1')
assert a3d[0,0,0] is a000
assert a3d[1,2,0] is a120
assert a3d[1,2,1] is a121
def test_linear_exprs():
a, b, c = x = sp.symarray('x', 3)
coeffs = [[1, 3, -2],
[3, 5, 6],
[2, 4, 3]]
vals = [5, 7, 8]
exprs = linear_exprs(coeffs, x, vals)
known = [1*a + 3*b - 2*c - 5,
3*a + 5*b + 6*c - 7,
2*a + 4*b + 3*c - 8]
assert all([(rt - kn).simplify() == 0 for rt, kn in zip(exprs, known)])
rexprs = linear_exprs(coeffs, x, vals, rref=True)
rknown = [a + 15, b - 8, c - 2]
assert all([(rt - kn).simplify() == 0 for rt, kn in zip(rexprs, rknown)])
def __init__(self,name,func,argnames=None):
"""
Args:
name (str): the symbolic module name of the function.
func (sympy.Function): the Sympy function
argnames (list of strs, optional): provided if you don't
want to use 'x','y', 'z' for the argument names.
"""
assert isinstance(func,sympy.FunctionClass)
if hasattr(func,'nargs'):
if len(func.nargs) > 1:
print("SympyFunctionAdaptor: can't yet handle multi-argument functions")
nargs = None
for i in func.nargs:
nargs = i
else:
nargs = 1
if argnames is None:
if nargs == 1:
argnames = ['x']
elif nargs <= 3:
argnames = [['x','y','z'][i] for i in len(nargs)]
else:
argnames = ['arg'+str(i+1) for i in len(nargs)]
Function.__init__(self,name,func,argnames)
self.deriv = [None]*len(argnames)
self.jacobian = [None]*len(argnames)
self.sympy_jacobian = [None]*len(argnames)
self.sympy_jacobian_funcs = [None]*len(argnames)
xs = sympy.symarray('x',nargs)
for i,arg in enumerate(argnames):
self.sympy_jacobian[i] = func(*xs).diff(xs[i])
for i in range(len(argnames)):
def cache_jacobian(*args):
if self.sympy_jacobian_funcs[i] is None:
#print "Creating jacobian function",name + "_jac_" + argnames[i]
self.sympy_jacobian_funcs[i] = SympyFunction(name + "_jac_" + argnames[i], self.sympy_jacobian[i],argnames)
return OperatorExpression(self.sympy_jacobian_funcs[i],args)
self.jacobian[i] = cache_jacobian
def generate_jac_lambda(self):
"""
translates the symbolic Jacobian to a function using SymPy’s `lambdify <http://docs.sympy.org/latest/modules/utilities/lambdify.html>`_ tool. If the symbolic Jacobian has not been generated, it is generated by calling `generate_jac_sym`.
"""
if self.helpers:
warn("Lambdification handles helpers by pluggin them in. This may be very ineficient")
self._generate_jac_sym()
jac_matrix = sympy.Matrix([ [entry for entry in line] for line in self.jac_sym ])
t,y = provide_basic_symbols()
Y = sympy.symarray("Y", self.n)
substitutions = self.helpers[::-1] + [(y(i),Y[i]) for i in range(self.n)]
jac_subsed = jac_matrix.subs(substitutions)
JAC = sympy.lambdify([t]+[Yentry for Yentry in Y], jac_subsed)
self.jac = lambda t,ypsilon: array(JAC(t,*ypsilon))
def calculate_moment_expansion_trick(args,
expr,
order,
delete_first_moment=False):
# sad hack
expansion = expr
index_start = 1
if delete_first_moment:
index_start = 2
moment_indices = range(index_start, order + 1)
spectral_functions = []
for k in moment_indices:
deriv_list = moment_derivatives(args, [expr / args[0]], k)
spectral_functions += clean_derivatives(deriv_list)
moments = list(sym.symarray('w', len(spectral_functions)))
if spectral_functions:
expansion += sym.Matrix(moments).dot(sym.Matrix(spectral_functions))
return moments, expansion
def trou_formula(self, value_list):
can_list = [] # 参数
temp = []
max = len(value_list)
_print(type(value_list))
for num in range(1, max + 1):
can_list.append([num**x for x in range(1, max + 1)])
a = sympy.Matrix(can_list)
b = sympy.Matrix(value_list)
x = sympy.symarray('x', (max, 1))
dict = sympy.solve(a * x - b)
# return dict
for items in dict.items():
temp.append(items[1])
n = sympy.Symbol('n')
value = 0
temp_len = len(temp)
for i in range(0, temp_len):
letter = sympy.simplify(temp[i] * n**(temp_len - i - 1))
value += letter
return value
def differentials(f,x,y):
"""
Returns a basis of the holomorphic differentials defined on the
Riemann surface `X: f(x,y) = 0`.
Input:
- f: a Sympy object describing a complex plane algebraic curve.
- x,y: the independent and dependent variables, respectively.
"""
d = f.as_poly().total_degree()
n = sympy.degree(f,y)
# coeffiecients and general adjoint polynomial
c_arr = sympy.symarray('c',(d-3,d-3)).tolist()
c = dict( (cij,(c_arr.index(ci),ci.index(cij)))
for ci in c_arr for cij in ci )
P = sum( c_arr[i][j]*x**i*y**j
for i in range(d-3) for j in range(d-3) if i+j <= d-3)
S = singularities(f,x,y)
differentials = set([])
for (alpha, beta, gamma), (m,delta,r) in S:
g,u,v,u0,v0 = _transform(f,x,y,(alpha,beta,gamma))
# if delta > m*(m-1)/2:
if True:
# Use integral basis method.
b = integral_basis(g,u,v)
for bi in b:
monoms = _adjoint_monomials(f,x,y,bi,P,c)
differentials.add(monom for monom in monoms)
else:
# Use Puiseux series method
b = integral_basis(g,u,v)
return [differential/sympy.diff(f,y) for differential in differentials]
def give_coords(self, distances):
distances = np.array(distances)
n = len(distances)
X = sympy.symarray('x', (n, n - 1))
for row in range(n):
X[row, row:] = [0] * (n - 1 - row)
for point2 in range(1, n):
expressions = []
for point1 in range(point2):
expression = np.sum((X[point1] - X[point2])**2)
expression -= distances[point1, point2]**2
expressions.append(expression)
X[point2, :point2] = sympy.solve(expressions,
list(X[point2, :point2]))[1]
return X
def preprocess(self, p, _x, config):
if not p.eq:
return p, _x
free_vars = symarray('x', p.n, commutative=False)
free_vars = Matrix(free_vars)
rhs = self.solve_equitions(to_matrix(p.eq.A), to_matrix(p.eq.b),
free_vars)
p.eq = None
self.vars_mapping = rhs
new_free_vars = list(rhs.free_symbols)
new_x = self.new_x(free_vars, new_free_vars, _x)
self.free_vars = new_free_vars
new_free_vars = Matrix(new_free_vars)
A, b = self.linear_equitions_var_subs(to_matrix(p.le.A),
to_matrix(p.le.b), rhs,
new_free_vars)
p.le.A = p.le.A.new(A)
p.le.b = p.le.b.new(b)
if isinstance(p.obj, LinearObjective):
b, c = self.linear_expr_var_subs(to_matrix(p.obj.b),
float(p.obj.c), rhs,
new_free_vars)
p.obj.b = p.obj.b.new(b)
p.obj.c = c
if isinstance(p.obj, QuadraticObjective):
A, b, c = self.quadratic_expr_var_subs(to_matrix(p.obj.A),
to_matrix(p.obj.b),
float(p.obj.c), rhs,
new_free_vars)
p.obj.A = p.obj.A.new(A)
p.obj.b = p.obj.b.new(b)
p.obj.c = c
return p, new_x
def generate_f_lambda(self, simplify=True):
"""
translates the symbolic derivative to a function using SymPy’s `lambdify <http://docs.sympy.org/latest/modules/utilities/lambdify.html>`_ tool.
Parameters
----------
simplify : boolean
Whether the derivative should be `simplified <http://docs.sympy.org/dev/modules/simplify/simplify.html>`_ (with `ratio=1.0`) before translating to C code. The main reason why you could want to disable this is if your derivative is already optimised and so large that simplifying takes a considerable amount of time.
"""
if self.helpers:
warn("Lambdification does not handle helpers in an efficient manner.")
t,y = provide_basic_symbols()
Y = sympy.symarray("Y", self.n)
substitutions = self.helpers[::-1] + [(y(i),Y[i]) for i in range(self.n)]
f_sym_wc = (entry.subs(substitutions) for entry in f_sym)
if simplify:
f_sym_wc = (entry.simplify(ratio=1.0) for entry in f_sym)
F = sympy.lambdify([t]+[Yentry for Yentry in Y], list(f_sym_wc))
self.f = lambda t,ypsilon: array(F(t,*ypsilon)).flatten()
def model_format(mode, num_cluster, bound_limit=1e-9):
data = np.sort(data, axis = 0)
data_length = len(data)
raw_probability = np.array([(i - 0.3) / data_length for i in range(1, data_length+1)])
plot_p = np.log(-np.log(1 - raw_probability))
if mode == 'GM': # Gaussian Mixture
t = sy.symbols('t')
u = sy.symarray('u', num_cluster)
var = sy.symarray('var', num_cluster)
PDF = [1/(sqrt(2*pi)*j) * sy.exp(((t-i)/(sqrt(2)*j))**2) for i,j in zip(u, var)]
variable_list = [t, u, var]
bound = ((bound_limit, 1/bound_limit), (None, None), (None, None))
elif mode == 'WM': # Weibull Mixture
t = sy.symbols('t')
k = sy.symarray('k', num_cluster)
tor = sy.symarray('tor', num_cluster)
PDF = sy.diff(1 - sy.exp(- (t/tor)**k), t)
variable_list = [t, k, tor]
bound = ((bound_limit, 1/bound_limit), (None, None), (None, None))
elif mode == 'CM':
t = sy.symbols('t')
a = sy.symarray('a', num_cluster)
b = sy.symarray('b', num_cluster)
tor = sy.symarray('tor', num_cluster)
PDF = sy.diff(1 - (1 + 1/a*(t/tor)**(b))**(-a), t)
variable_list = [t, a, b, tor]
bound = ((bound_limit, 1/bound_limit), (None, None), (None, None), (bound_limit, 1/bound_limit))
else:
print('Mode Error')
PDF = None
variable_list = None
bound = None
return PDF, variable_list, bound
def generate_f_lambda(self, simplify=True):
"""
translates the symbolic derivative to a function using SymPy’s `lambdify <http://docs.sympy.org/latest/modules/utilities/lambdify.html>`_ tool.
Parameters
----------
simplify : boolean
Whether the derivative should be `simplified <http://docs.sympy.org/dev/modules/simplify/simplify.html>`_ (with `ratio=1.0`) before translating to C code. The main reason why you could want to disable this is if your derivative is already optimised and so large that simplifying takes a considerable amount of time.
"""
if self.helpers:
warn("Lambdification does not handle helpers in an efficient manner.")
t,y = provide_basic_symbols()
Y = sympy.symarray("Y", self.n)
substitutions = self.helpers[::-1] + [(y(i),Y[i]) for i in range(self.n)]
f_sym_wc = (entry.subs(substitutions) for entry in self.f_sym())
if simplify:
f_sym_wc = (entry.simplify(ratio=1.0) for entry in f_sym_wc)
F = sympy.lambdify([t]+[Yentry for Yentry in Y], list(f_sym_wc))
self.f = lambda t,ypsilon: array(F(t,*ypsilon)).flatten()
def generate_jac_lambda(self):
"""
translates the symbolic Jacobian to a function using SymPy’s `lambdify <http://docs.sympy.org/latest/modules/utilities/lambdify.html>`_ tool. If the symbolic Jacobian has not been generated, it is generated by calling `generate_jac_sym`.
"""
if self.helpers:
warn(
"Lambdification handles helpers by pluggin them in. This may be very ineficient"
)
self._generate_jac_sym()
jac_matrix = sympy.Matrix([[entry for entry in line]
for line in self.jac_sym])
t, y = provide_basic_symbols()
Y = sympy.symarray("Y", self.n)
substitutions = self.helpers[::-1] + [(y(i), Y[i])
for i in range(self.n)]
jac_subsed = jac_matrix.subs(substitutions)
JAC = sympy.lambdify([t] + [Yentry for Yentry in Y], jac_subsed)
self.jac = lambda t, ypsilon: array(JAC(t, *ypsilon))
import sympy as sp
import numpy as np
import matplotlib.pyplot as plt
from sympy.interactive import printing
from sympy.parsing.sympy_parser import parse_expr
from sympy.printing.latex import latex
import typing
x = sp.symarray('x', 5)
#C, P, R, T == x[1], x[2], x[3], x[4] respectively
xpr = parse_expr
some_var = 606
#equation parameters
k_c, mu_c, C_0, lambda_c, K_c = sp.symbols("k_c, mu_c, C_0, lambda_c, K_c")
k_p, mu_p, K_p, P_0, lambda_p = sp.symbols("k_p, mu_p, K_p, P_0, lambda_p")
k_r, lambda_r, gamma_p, gamma_c = sp.symbols("k_r, lambda_r, gamma_p, gamma_c")
k_t, K_t, lambda_t = sp.symbols("k_t, K_t, lambda_t")
params = {'C_0' : 10**6, 'P_0' : 10**5,'k_c' : 7.5e-2, 'K_c' : 0.1, 'lambda_c' : 10**(-7), \
'k_p' : 0.2, 'lambda_p' : 0.15, 'k_r' : 0.2, 'lambda_r' : 0.22, 'k_t' : 3300, 'lambda_t' : 0.3}
params['mu_c'] = 20 * params['k_c'] / params['P_0']
params['mu_p'] = 20 * params['k_p']
params['K_p'] = params['C_0'] / 100
params['gamma_p'] = 0.02 * params['lambda_r'] / (
params['P_0'] * (1 - params['lambda_p'] / params['k_p']))
params['gamma_c'] = params['gamma_p']
params['K_t'] = params['K_c']
# f1 = (k_c + mu_c*x[2]) * x[1]**(3/4) * (1 - (x[1]/C_0)**(1/4) ) - \
# Kinematic values of previos nodes (generic)
# e.g., omega_node = omega + qdot
theta = sym.Matrix(['theta_1','theta_2'])
omega = sym.Matrix(['omega_1','omega_2'])
alpha = sym.Matrix(['alpha_1','alpha_2'])
# coordinates of the point in the 2D cross-section
# of nodes one and two
s1 = sym.Matrix(['r_2','r_3'])
s2 = sym.Matrix(['r_2','r_3'])
s = sym.Matrix.vstack(s1,s2)
# generalized coordinates
# one rotation and two displacements per-node (two nodes per element)
# in this version generalzied speeds are qdots
q = sym.Matrix(sym.symarray('q',6))
qdot = sym.Matrix(sym.symarray('qdot',len(q)))
qddot = sym.Matrix(sym.symarray('qddot',len(q)))
# Deformations of Nodes (u's are not generalized speeds)
u = sym.Matrix([q[1:3,0], q[4:6,0]])
udot = sym.Matrix([qdot[1:3,0], qdot[4:8,0]])
uddot = sym.Matrix([qddot[1:3,0], qddot[4:6,0]])
# ### Needed Matrix Quantities
"""
Some cheating here:
q0,q3 and q0dot,q3dot are really theta_1j, theta_2j and omega_1j, omega_2j
"""
# angular position and velocity for 2D using relative coordinates
# the sum of the respective quantites of bodies 1 - bodyj-1
def __init__(
self,
params: Dict[str, Any],
states: Dict[str, Any],
rhs_sympy: Callable[[sym.Symbol, np.ndarray, np.ndarray], Dict[str,
Any]],
derivative_params: List[Path],
coords: Optional[Dict[str, pd.Index]] = None,
simplify: Optional[Callable[[sym.Expr], sym.Expr]] = None,
):
self.params_subset = dtypesubset.DTypeSubset(
params,
derivative_params,
fixed_dtype=basic.data_dtype,
coords=coords)
self.coords = self.params_subset.coords
self.params_dtype = self.params_subset.dtype
self.state_subset = dtypesubset.DTypeSubset(
states, [], fixed_dtype=basic.data_dtype, coords=self.coords)
self.state_dtype = self.state_subset.dtype
self._rhs_sympy_func = rhs_sympy
if simplify is None:
simplify = lambda x: x
self._simplify = np.vectorize(simplify)
def check_dtype(dtype: np.dtype, path: Optional[str] = None) -> None:
if dtype.fields is None:
if dtype.base != basic.data_dtype:
raise ValueError(
'Derivative param %s has incorrect dtype %s. Should be %s'
% (path, dtype.base, basic.data_dtype))
return
for name, (dt, _) in dtype.fields.items():
if path is None:
path_ = name
else:
path_ = '.'.join([path, name])
check_dtype(dt, path_)
check_dtype(self.params_subset.subset_dtype)
self._sym_time = sym.Symbol('time', real=True)
n_fixed = self.params_subset.n_items - self.n_params
def make_vars(var_shapes: Dict[Path, Shape],
**kwargs: Any) -> Dict[Path, sym.Symbol]:
vars = {}
for path, shape in var_shapes.items():
name = '_'.join(path)
var = sym.symarray(name, shape, **kwargs)
vars[path] = var
return vars
self._sym_states = make_vars(self.state_subset.flat_shapes,
positive=True)
self._sym_params = make_vars(self.params_subset.flat_shapes, real=True)
self._varmap: Dict[str, Tuple[Any, ...]] = {}
for path, vars in self._sym_states.items():
for idxs in product(*[range(i) for i in vars.shape]):
var = vars[idxs]
if idxs == ():
self._varmap[var.name] = ('state', *path)
else:
self._varmap[var.name] = ('state', *path, idxs)
for path, vars in self._sym_params.items():
for idxs in product(*[range(i) for i in vars.shape]):
var = vars[idxs]
if idxs == ():
self._varmap[var.name] = ('params', *path)
else:
self._varmap[var.name] = ('params', *path, idxs)
deriv_params = {
k: v
for k, v in self._sym_params.items()
if k in self.params_subset.subset_paths
}
if deriv_params:
raveled_deriv = np.concatenate(
[var.ravel() for var in deriv_params.values()])
else:
raveled_deriv = np.zeros((0, ))
fixed_params = {
k: v
for k, v in self._sym_params.items()
if k not in self.params_subset.subset_paths
}
if fixed_params:
raveled_fixed = np.concatenate(
[var.ravel() for var in fixed_params.values()])
else:
raveled_fixed = np.zeros((0, ))
def item_map(item: np.ndarray) -> np.ndarray:
if hasattr(item, 'shape') and item.shape == ():
return item.item()
return item
self._sym_deriv_paramsvec = raveled_deriv
self._sym_params = self.params_subset.as_dataclass('Params',
raveled_deriv,
raveled_fixed,
item_map=item_map)
self._sym_statevec = np.concatenate(
[var.ravel() for var in self._sym_states.values()])
self._sym_states = self.state_subset.as_dataclass('State', [],
self._sym_statevec,
item_map=item_map)
dydt = self._make_dydt()
self._sym_dydt = np.array(dydt).ravel()
self._sym_sens = sym.symarray('sens', (self.n_params, self.n_states))
self._sym_lamda = sym.symarray('lamda', self.n_states)
for idxs in product(*[range(i) for i in self._sym_lamda.shape]):
var = self._sym_lamda[idxs]
self._varmap[var.name] = ('lamda', idxs)
for idxs in product(*[range(i) for i in self._sym_sens.shape]):
var = self._sym_sens[idxs]
self._varmap[var.name] = ('sens', idxs)
self._sym_dydt_jac = np.array(dydt.jacobian(self._sym_statevec))
#self._sym_dydp = np.array([val.diff(self._sym_deriv_paramsvec) for val in self._sym_dydt])
self._sym_dydp = np.array(dydt.jacobian(self._sym_deriv_paramsvec))
#jacprotsens = (self._sym_dydt_jac * self._sym_sens.T.as_explicit()).as_explicit()
#self._sym_rhs_sens = (jacprotsens + self._sym_dydp).as_explicit().T
self._sym_dlamdadt = -self._sym_lamda @ self._sym_dydt_jac
self._sym_quad_rhs = self._sym_lamda @ self._sym_dydp
self.user_data_dtype = np.dtype([
('params', self.params_subset.dtype),
('tmp_nstates_nstates', np.float64, (self.n_states,
self.n_states)),
('tmp_nparams_nstates', np.float64, (self.n_params,
self.n_states)),
('tmp2_nparams_nstates', np.float64, (self.n_params,
self.n_states)),
('error_states', self.state_dtype),
('error_rhs', np.float64, (self.n_states, )),
('error_jac', np.float64, (self.n_states, self.n_states)),
])
sp.linsolve(ftz_from_F_c_expr.values(),
l_robot["ftz"].values()).args[0])
# A = l_robot["thrusters_mapping"]
# ftz_from_F_c_result = (A.T*A).inv()*A.T*F_c # Least squares
return {
"lambda": sp.lambdify((F_c, ), ftz_from_F_c_result, 'numpy'),
"s_lambda": sp.lambdify((F_c, ), ftz_from_F_c_result, 'sympy'),
"result": ftz_from_F_c_result,
"expr": ftz_from_F_c_expr,
"J_wrt_F_c": ftz_from_F_c_result.jacobian(F_c)
}
w_c = sp.Matrix(sp.symarray('w_c', 3))
v_c = sp.Matrix(sp.symarray('v_c', 3))
V_c = sp.Matrix([w_c, v_c])
t_sc = sp.Matrix(sp.symarray('t_sc', 3))
q_s_cog = sp.Matrix(sp.symbols('q_sc_w q_sc_x q_sc_y q_sc_z')) # world to COG
q_sc = sophus.Quaternion(q_s_cog[0], q_s_cog[1:4, :])
T_sc = sophus.Se3(sophus.So3(q_sc), t_sc)
t_st = sp.Matrix(sp.symarray('t_st', 3))
q_s_target = sp.Matrix(
sp.symbols('q_st_w q_st_x q_st_y q_st_z')) # world to target
q_st = sophus.Quaternion(q_s_target[0], q_s_target[1:4, :])
T_st = sophus.Se3(sophus.So3(q_st), t_st)
"""
# Consider the TSP problem with cost matrix:
"""
$\begin{bmatrix}
0 & 1 & 2 & 9 & 2 & 1 \\
1 & 0 & 1 & 2 & 9 & 2 \\
2 & 1 & 0 & 1 & 2 & 9 \\
9 & 2 & 1 & 0 & 1 & 2 \\
2 & 9 & 2 & 1 & 0 & 1 \\
1 & 2 & 9 & 2 & 1 & 0
\end{bmatrix}$
"""
# The following equations are just a few of the many which arise
C = sympy.symarray("C", (7, 7), nonnegative=True)
eqns = [
sympy.Eq(C[1][2] + C[2][3] + C[3][4] + C[4][5] + C[5][6] + C[6][1], 6),
sympy.Eq(C[1][2] + C[2][3] + C[3][4] + C[4][6] + C[6][5] + C[5][1], 8),
sympy.Eq(C[1][2] + C[2][3] + C[3][5] + C[5][4] + C[4][6] + C[6][1], 8),
sympy.Eq(C[1][2] + C[2][4] + C[4][3] + C[3][5] + C[5][6] + C[6][1], 8),
sympy.Eq(C[1][3] + C[3][2] + C[2][4] + C[4][5] + C[5][6] + C[6][1], 8),
sympy.Eq(C[1][3] + C[3][4] + C[4][5] + C[5][6] + C[6][2] + C[2][1], 8),
sympy.Eq(C[1][6] + C[6][2] + C[2][3] + C[3][4] + C[4][5] + C[5][1], 8)
]
# If we use sympy to solve, we find multiple solutions, one of which
# is the original C.
print("Solutions to equations from the original problem:")
print(sympy.solve(eqns))
def n_bar_pendulum(N=1, param_values=dict()):
'''
Returns the mass matrix :math:`M` and right hand site :math:`B` of motion equations
.. math::
M * (d^2/dt^2) x = B
for the :math:`N`\ -bar pendulum.
Parameters
----------
N : int
Number of bars.
param_values : dict
Numeric values for the system parameters,
such as lengths, masses and gravitational acceleration.
Returns
-------
sympy.Matrix
The mass matrix `M`
sympy.Matrix
The right hand site `B`
list
List of symbols for state variables
list
List with symbol for input variable
'''
# first we have to create some symbols
F = sp.Symbol('F') # the force that acts on the car
g = sp.Symbol('g') # the gravitational acceleration
m = sp.symarray('m', N+1) # masses of the car (`m0`) and the bars
l = sp.symarray('l', N+1)#[1:] # length of the bars (`l0` is not needed nor used)
phi = sp.symarray('phi', N+1)#[1:] # deflaction angles of the bars (`phi0` is not needed nor used)
dphi = sp.symarray('dphi', N+1)#[1:] # 1st derivative of the deflaction angles (`dphi0` is not needed nor used)
if param_values.has_key('F'):
F = param_values['F']
elif param_values.has_key(F):
F = param_values[F]
if param_values.has_key('g'):
g = param_values['g']
elif param_values.has_key(g):
g = param_values[g]
else:
g = 9.81
for i, mi in enumerate(m):
if param_values.has_key(mi.name):
m[i] = param_values[mi.name]
elif param_values.has_key(mi):
m[i] = param_values[mi]
for i, li in enumerate(l):
if param_values.has_key(li.name):
l[i] = param_values[li.name]
elif param_values.has_key(li):
l[i] = param_values[li]
C = np.empty((N,N), dtype=object)
S = np.empty((N,N), dtype=object)
I = np.empty((N), dtype=object)
for i in xrange(1,N+1):
for j in xrange(1,N+1):
C[i-1,j-1] = cos(phi[i] - phi[j])
S[i-1,j-1] = sin(phi[i] - phi[j])
for i in xrange(1,N+1):
if param_values.has_key('I_%d'%i):
I[i-1] = param_values['I_%d'%i]
#elif param_values.has_key(Ii):
# I[i] = param_values[Ii]
else:
I[i-1] = 4.0/3.0 * m[i] * l[i]**2
#-------------#
# Mass matrix #
#-------------#
M = np.empty((N+1, N+1), dtype=object)
# 1st row
M[0,0] = m.sum()
for j in xrange(1,N):
M[0,j] = (m[j] + 2*m[j+1:].sum()) * l[j] * cos(phi[j])
M[0,N] = m[N] * l[N] * cos(phi[N])
# rest of upper triangular part, except last column
for i in xrange(1,N):
M[i,i] = I[i-1] + (m[i] + 4.0*m[i+1:].sum()) * l[i]**2
for j in xrange(i+1,N):
M[i,j] = 2.0*(m[j] + 2.0*m[j+1:].sum())*l[i]*l[j]*C[j-1,i-1]
# the last column
for i in xrange(1,N):
M[i,N] = 2.0*(m[N]*l[i]*l[N]*C[N-1,i-1])
M[N,N] = I[N-1] + m[N]*l[N]**2
# the rest (lower triangular part)
for i in xrange(N+1):
for j in xrange(i,N+1):
M[j,i] = 1 * M[i,j]
#-----------------#
# Right hand site #
#-----------------#
B = np.empty((N+1), dtype=object)
# first row
B[0] = F
for j in xrange(1,N):
B[0] += (m[j] + 2.0*m[j+1:].sum())*l[j]*sin(phi[j]) * dphi[j]**2
B[0] += (m[N]*l[N]*sin(phi[N])) * dphi[N]**2
# rest except for last row
for i in xrange(1,N):
B[i] = (m[i] + 2.0*m[i+1:].sum())*g*l[i]*sin(phi[i])
for j in xrange(1,N):
B[i] += (2.0*(m[j] + 2.0*m[j+1:].sum())*l[j]*l[i]*S[j-1,i-1]) * dphi[j]**2
B[i] += (2.0*m[N]*l[N]*l[N]*S[N-1,i-1]) * dphi[N]**2
# last row
B[N] = m[N]*g*l[N]*sin(phi[N])
for j in xrange(1,N+1):
B[N] += (2.0*m[N]*l[j]*l[N]*S[j-1,N-1]) * dphi[j]**2
# build lists of state and input variables
x, dx = sp.symbols('x, dx')
state_vars = [x, dx]
for i in xrange(1,N+1):
state_vars.append(phi[i])
state_vars.append(dphi[i])
input_vars = [F]
# return stuff
return sp.Matrix(M), sp.Matrix(B), state_vars, input_vars
import sympy as sym # we import the sympy package
import math # we import the math package (not used here, but has useful constants such math.pi which might be needed in other cases)
import sys # import the sys module used below
from pylab import *
from gravipy import *
############################################################################################################################################
# if using an integrated environment we recommend restarting the python console after running this script to make sure updates are found
location = "/home/jwr/Code/PyTransport/" # this should be the location of the PyTransport folder
sys.path.append(location) # we add this location to the python path
import PyTransSetup # the above commands allows python to find the PyTransSetup module and import it
############################################################################################################################################
nF=3 # number of fields needed to define the PseudoScalar potential
nP=3 # number of parameters needed to define the PseudoScalar potential
f=sym.symarray('f',nF) # an array representing the nF fields present for this model
p=sym.symarray('p',nP) # an array representing the nP parameters needed to define this model (that we might wish to change) if we don't
V= p[0] * f[0]**2 + p[1] * f[1]**2 + p[2] * f[2]**2 # this is the potential written in sympy notation
R=(0.9)/((sym.cosh(2.0*((1.0*f[0])-7.0)/0.12))**(2.0))
G=Matrix([[1,R,0],[R,1,0],[0,0,1]]) # selecting the field space metric in this instance.
PyTransSetup.potential(V,nF,nP,False,G) # writes this potential and its derivatives into C++ file potential.h when run
PyTransSetup.compileName("Pseudo",True) # this compiles a python module using the C++ code, including the edited potential.h file, called PyTransPseudo
# and places it in the location folder, ready for use
############################################################################################################################################
variableit = input("elemento de matriz M: ")
vcm1.append(variableit)
for j in range(a * b):
variableit2 = input("elemento de matriz V: ")
vcm2.append(variableit2)
NewM = sy.Matrix(a, b, vcm1)
NewV = sy.Matrix(a, b, vcm2)
w2 = sy.symbols("w2")
m_v = w2 * NewM - NewV
autovalor = sy.solve(m_v.det(), w2)
glist = list(sy.symarray("a", a))
vec = [0 for i in range(a)]
md0 = sy.Matrix(vec)
M_x = []
for i in range(len(autovalor)):
z = autovalor[i]
M_0 = z * NewM - NewV
sol = list(sy.linsolve((M_0, md0), glist))
M_x.append(sol)
print("autovalor :", autovalor[i], ", autovector :", sol)
#print (NewM)
def add_link(self, dh):
new_joint = Joint(dh)
if isinstance(dh[0], Expr):
self.link_list.append(new_joint)
self.transform_list.append(new_joint)
self.sym_theta_list = symarray("theta", len(self.link_list))
def __init__(self, f, x, k, f_args=None):
self.f = f
self.x = x
self.k = k
if np.array(x).shape == ():
n, x = 1, [x]
else:
n = len(x)
# call to f
if f_args is None:
f_eval = f(*x)
else:
f_eval = f(*(list(x) + list(f_args)))
if np.array(f_eval).shape == ():
m = 1
else:
# call to f
m = len(f_eval)
self.m = m
self.n = n
# list of symbolic variables
var = sympy.symarray('x', (n, ))
# polynomial basis
pb = bases.poly_basis(var)
# vector basis
vb = bases.vf_basis(pb, m)
# k-jet of f centered at x
# call to f
self.jet = jet.jet(f, x, k, f_args, var, pb)
# fundamental operator of normal form theory, Lie bracket with f'
self.L1 = lie_operator.lie_operator(self.jet.fun_deg[1], var, 1, pb,
vb)
# work space of coefficients
n_terms = combinatorics.simplicial_list(n, k)
wrk = [[np.zeros(m * n_terms[i + j + 1]) for j in range(k - i)]
for i in range(k)]
# initialize first row of workspace as k-jet
for j in range(k):
wrk[0][j] = np.concatenate(self.jet.coeff[j + 1])
# generators
g = []
# Lie brackets with generators
L = []
# equivariant vector fields
eqv = []
# list of factorials
fac = combinatorics.factorial_list(k)
# algorithm based on Murdock
for deg in range(2, k + 1):
# update workspace and solve for generator
for j, l in enumerate(L):
wrk[1][deg - 2] += l[deg - 1 - j].dot(wrk[0][deg - 2 - j])
f_coeff = np.zeros(m * n_terms[deg])
for i in range(deg):
f_coeff += wrk[i][deg - 1 - i] / fac[i]
g_coeff = np.linalg.lstsq(self.L1[deg], f_coeff)[0]
# normal form coefficients
h_coeff = f_coeff - self.L1[deg].dot(g_coeff)
# represent normal form term in L1.T nullspace basis
u, s, v = np.linalg.svd(self.L1[deg])
rank = min(self.L1[deg].shape) - np.isclose(s, 0).sum()
perp_basis = u[:, rank:]
e_coeff = perp_basis.T.conj().dot(h_coeff)
e = [
sympy.Matrix(perp_basis[:, i].reshape(
m, perp_basis[:, i].shape[0] / m)) * pb[deg]
for i in range(perp_basis.shape[1])
]
# truncate roundoff error
for coeff in [e_coeff, f_coeff, g_coeff, h_coeff]:
coeff[np.isclose(coeff, 0)] = 0
# store generator
g.append(
sympy.Matrix(np.reshape(g_coeff,
(m, len(g_coeff) / m))) * pb[deg])
# update series coeff
self.jet.coeff[deg] = np.reshape(h_coeff, (m, len(h_coeff) / m))
# store equivariant vector fields
eqv.append((e_coeff, e))
# store Lie operator
L.append(lie_operator.lie_operator(g[-1], var, deg, pb, vb))
# update workspace
wrk[1][deg - 2] += L[-1][1].dot(wrk[0][0])
for i in range(2, k - deg + 2):
for j, l in enumerate(L):
wrk[i][deg - 2] += l[deg - 2 + i - j].dot(
wrk[i - 1][deg - 2 - j])
self.L = L
self.g = g
self.eqv = eqv
# update series symbolic and lambdified representation
self.jet.update_fun()
# make jet.fun accessible from this class
self.fun = self.jet.fun
def __init__(self, f, x, k, f_args=None, var=None, pb=None):
"""initialize the jet"""
self.f = f
self.x = x
self.k = k
if np.array(x).shape == ():
n, x = 1, [x]
else:
n = len(x)
# call to f
if f_args is None:
f_eval = f(*x)
else:
f_eval = f(*(list(x) + list(f_args)))
if np.array(f_eval).shape == ():
m = 1
else:
# call to f
m = len(f_eval)
self.m = m
self.n = n
if var is None:
var = sympy.symarray('x', (n, ))
if pb is None:
pb = bases.poly_basis(var)
self.var = var
self.pb = pb
# number of terms per degree of expanded form
n_terms = combinatorics.simplicial_list(n, k)
coeff = [np.empty([m, n_terms[deg]]) for deg in range(k + 1)]
basis = [sympy.ones(n_terms[deg], 1) for deg in range(k + 1)]
# call to f
if f_args is None:
f_eval = f(*var)
else:
f_eval = f(*(list(var) + list(f_args)))
coeff[0][:, 0] = list(sympy.Matrix([f_eval]).subs(zip(var, x)))
for deg in range(1, k + 1):
m_idx = multiindex(deg, n)
for term in range(n_terms[deg]):
# call to f
if f_args is None:
f_eval = f(*var)
else:
f_eval = f(*(list(var) + list(f_args)))
coeff[deg][:, term] = list(
sympy.diff(sympy.Matrix([f_eval]), *
m_idx.to_var(var)).subs(zip(var, x)) /
m_idx.factorial())
basis[deg][term] = m_idx.to_polynomial(var, x)
m_idx.increment()
for deg in range(k + 1):
poly = list(sympy.Matrix(coeff[deg]) * basis[deg])
m_idx = multiindex(deg, n)
for term in range(n_terms[deg]):
for coord in range(m):
coeff[deg][coord, term] = poly[coord].coeff(pb[deg][term])
m_idx.increment()
self.coeff = coeff
self.update_fun()
#SymPy is a non-commercial alternative to Mathematica and Maple
# SymPy can map variable to a value or a matrix.
# SymPy's Symbolic Statistical Modelling uses scintific computing.
import numpy as np
import sympy as sp
import pandas as pd
import sys
#give input equation
expr = sp.sympify("x_1**2 + x_2*x_3")
#get number of columns from X dataframe and y dataframe
num_of_columns = 10 + 1 #dataframe features len(X.columns) +1
num_of_y_columns = 2 #dataframe features len(Y.columns) +1
#create equation variables as user prefers to enter like x_1 ** 2 + x_2 * x_3
symbols = sp.symarray('x', num_of_columns)
symbols = symbols[1:]
y_symbols = sp.symarray('y', num_of_y_columns)
y_symbols = y_symbols[1:]
df = pd.DataFrame(
[[21, 72, 67.1], [23, 78, 69.5], [32, 74, 56.6], [52, 54, 76.2]],
columns=['feature1', 'feature2', 'feature3'])
#create variables by setattr with dataframe's columns i.e. numpy arrays (series essentially arrays)
mod = sys.modules[__name__]
for i, c in enumerate(df.columns, start=1):
setattr(mod, "x" + str(i), df[c])
#just create variables SymPy way and store it as a stuple for creating f_1(x) -> H_1(x)
variable_tuple = tuple([sp.var(s.name) for s in symbols[0:len(df.columns)]])
#define the R.H.S of equation as user inputs and as parsed
#expr = x_1**2 + x_2*x_3
#use lambdify with symbols that are actually created as variables and expression and
from time import clock
import numpy as np
import sympy as sp
import symengine as se
# Real-life example (ion speciation problem in water chemistry)
x = sp.symarray('x', 14)
p = sp.symarray('p', 14)
args = np.concatenate((x, p))
exp = sp.exp
exprs = [x[0] + x[1] - x[4] + 36.252574322669, x[0] - x[2] + x[3] + 21.3219379611249, x[3] + x[5] - x[6] + 9.9011158998744, 2*x[3] + x[5] - x[7] + 18.190422234653, 3*x[3] + x[5] - x[8] + 24.8679190043357, 4*x[3] + x[5] - x[9] + 29.9336062089226, -x[10] + 5*x[3] + x[5] + 28.5520551531262, 2*x[0] + x[11] - 2*x[4] - 2*x[5] + 32.4401680272417, 3*x[1] - x[12] + x[5] + 34.9992934135095, 4*x[1] - x[13] + x[5] + 37.0716199972041, p[0] - p[1] + 2*p[10] + 2*p[11] - p[12] - 2*p[13] + p[2] + 2*p[5] + 2*p[6] + 2*p[7] + 2*p[8] + 2*p[9] - exp(x[0]) + exp(x[1]) - 2*exp(x[10]) - 2*exp(x[11]) + exp(x[12]) + 2*exp(x[13]) - exp(x[2]) - 2*exp(x[5]) - 2*exp(x[6]) - 2*exp(x[7]) - 2*exp(x[8]) - 2*exp(x[9]), -p[0] - p[1] - 15*p[10] - 2*p[11] - 3*p[12] - 4*p[13] - 4*p[2] - 3*p[3] - 2*p[4] - 3*p[6] - 6*p[7] - 9*p[8] - 12*p[9] + exp(x[0]) + exp(x[1]) + 15*exp(x[10]) + 2*exp(x[11]) + 3*exp(x[12]) + 4*exp(x[13]) + 4*exp(x[2]) + 3*exp(x[3]) + 2*exp(x[4]) + 3*exp(x[6]) + 6*exp(x[7]) + 9*exp(x[8]) + 12*exp(x[9]), -5*p[10] - p[2] - p[3] - p[6] - 2*p[7] - 3*p[8] - 4*p[9] + 5*exp(x[10]) + exp(x[2]) + exp(x[3]) + exp(x[6]) + 2*exp(x[7]) + 3*exp(x[8]) + 4*exp(x[9]), -p[1] - 2*p[11] - 3*p[12] - 4*p[13] - p[4] + exp(x[1]) + 2*exp(x[11]) + 3*exp(x[12]) + 4*exp(x[13]) + exp(x[4]), -p[10] - 2*p[11] - p[12] - p[13] - p[5] - p[6] - p[7] - p[8] - p[9] + exp(x[10]) + 2*exp(x[11]) + exp(x[12]) + exp(x[13]) + exp(x[5]) + exp(x[6]) + exp(x[7]) + exp(x[8]) + exp(x[9])]
lmb_symengine = se.Lambdify(args, exprs)
lmb_sympy = sp.lambdify(args, exprs)
inp = np.ones(28)
tim_symengine = clock()
res_symengine = np.empty(len(exprs))
for i in range(500):
res_symengine = lmb_symengine(inp)
#lmb_symengine.unsafe_real_real(inp, res_symengine)
tim_symengine = clock() - tim_symengine
tim_sympy = clock()
for i in range(500):
res_sympy = lmb_sympy(*inp)
tim_sympy = clock() - tim_sympy
print('symengine speed-up factor (higher is better) vs sympy: %12.5g' %
(tim_sympy/tim_symengine))
def exprToSympy(expr):
if isinstance(expr,Variable):
if expr.type.is_scalar():
return sympy.Symbol(expr.name)
elif expr.type.type in ARRAY_TYPES:
#1D column vector
assert expr.type.size is not None,"Can't convert variable-sized arrays to Sympy Matrix's"
entries = sympy.symarray(expr.name,expr.type.size)
return sympy.Matrix(entries)
else:
raise ValueError("Invalid Variable")
elif isinstance(expr,VariableExpression):
return exprToSympy(expr.var)
elif isinstance(expr,UserDataExpression):
return sympy.Symbol(expr.name)
elif isinstance(expr,ConstantExpression):
return exprToSympy(expr.value)
elif isinstance(expr,OperatorExpression):
fname = expr.functionInfo.name
sname = fname.capitalize()
sargs = [exprToSympy(a) for a in expr.args]
if fname in _sympySpecialConstructors:
return _sympySpecialConstructors[fname](expr.args,sargs)
if fname in _sympyOperators:
try:
return getattr(operator,fname)(*sargs)
except Exception as e:
print("exprToSympy: Error raised while performing operator %s on arguments %s"%(fname,str(expr)))
raise
if hasattr(sympy,sname):
#try capitalized version first
try:
return getattr(sympy,sname)(*sargs)
except Exception:
print("exprToSympy: Error raised while trying sympy.%s on arguments %s"%(sname,str(expr)))
if hasattr(sympy,fname):
#numpy equivalents, like eye
try:
return getattr(sympy,fname)(*sargs)
except Exception:
print("exprToSympy: Error raised while trying sympy.%s on arguments %s"%(fname,str(expr)))
if isinstance(expr.functionInfo.func, collections.Callable):
print("exprToSympy: Function %s does not have Sympy equivalent, returning adaptor "%(fname,))
return _make_sympy_adaptor(expr.functionInfo)(*sargs)
else:
print("exprToSympy: Function %s does not have Sympy equivalent, expanding expression"%(fname,))
assert isinstance(expr.functionInfo.func,Expression)
sfunc = exprToSympy(expr.functionInfo.func)
return sfunc.subs(list(zip(expr.functionInfo.argNames,sargs)))
print("exprToSympy: Function %s does not have Sympy equivalent, returning generic Function"%(fname,))
return sympy.Function(fname)(*sargs)
else:
if hasattr(expr,'__iter__'):
if hasattr(expr[0],'__iter__'):
#Matrix
assert not hasattr(expr[0][0],'__iter__'),"Sympy can't handle tensors yet"
return sympy.Matrix(expr)
else:
#1-D vector -- treat as column vector
return sympy.Matrix(expr)
if isinstance(expr,(float,int,bool)):
return sympify(expr)
############################################################################################################################################
# if using an integrated environment we recommend restarting the python console after running this script to make sure updates are found
location = "/Users/david/Dropbox/PyTransportDist/PyTransport/" # this should be the location of the PyTransport folder
sys.path.append(location) # we add this location to the python path
import PyTransSetup # the above commands allows python to find the PyTransSetup module and import it
############################################################################################################################################
### Sets potential and compiles PyTransport, users may prefer to do this only once in a separate file (or comment after running below once) ###
### Restart the python kernel after running this file
nF = 2
nP = 5
f = sym.symarray('f', nF)
p = sym.symarray('p', nP)
V = 1. / 2. * p[0]**2 * f[0]**2 + 1 / 2. * p[1]**2 * sym.cos(p[2] / 2.)**2 * (
f[1] -
(f[0] - p[3]) * sym.tan(p[2] / math.pi * sym.atan(p[4] *
(f[0] - p[3]))))**2
PyTransSetup.potential(V, nF, nP) # writes this potential into c file when run
PyTransSetup.compileName(
"LH"
) # this compiles the module with the new potential and places it in the location folder, and adds this folder to the path ready for use
############################################################################################################################################
self.desiredxyzijk = Matrix([x_coord, y_coord, z_coord, xr, yr, zr])
self.start_time = time.perf_counter()
self.damping_coefficient = damping_coefficient
# zero arm thetas to start position
for link in self.link_list:
link.theta = 0
while arm.ee_jacobian(1) == 1:
arm.main_update()
self.desiredxyzijk += Matrix([0, 0, 0, 0, 0, 0])
arm.plot_arm()
fig = plt.figure(1)
ax = fig.add_subplot(111, projection='3d')
sym_theta_list = symarray("theta", 8)
arm = Armature()
arm.add_link((sym_theta_list[0], -pi / 2, 0, 100))
arm.add_link((sym_theta_list[1], pi / 2, 100, 0))
arm.add_link((sym_theta_list[2], pi / 2, 100, 0))
arm.add_link((sym_theta_list[3], -pi / 2, 100, 0))
arm.add_link((sym_theta_list[4], -pi / 2, 100, 0))
arm.add_link((sym_theta_list[5], pi / 2, 100, 0))
arm.add_link((sym_theta_list[6], 0, 100, 0))
arm.add_link((pi / 2, pi / 2, 0, 0))
arm.add_link((sym_theta_list[7], pi / 2, 0, 0))
arm.main_update()
# arm.plot_arm()
def __init__(self,
a=0.0,
b=1.0,
n=5,
bv={},
tag='',
use_std_approach=False,
**kwargs):
# there are two different approaches implemented for evaluating
# the splines which mainly differ in the node that is used in the
# evaluation of the polynomial parts
#
# the reason for this is that in the project thesis from which
# PyTrajectory emerged, the author didn't use the standard approach
# usually used when dealing with spline interpolation
#
# later this standard approach has been implemented and was intended
# to replace the other one, but it turned out that when using this
# standard approach some examples suddendly fail
#
# until this issue is resolved the user is enabled to choose between
# the two approaches be altering the following attribute
self._use_std_approach = use_std_approach
# interval boundaries
assert a < b
self.a = a
self.b = b
# number of polynomial parts
self.n = int(n)
# 'name' of the spline
self.tag = tag
# dictionary with boundary values
# key: order of the spline's derivative to which the values belong
# values: the boundary values the derivative should satisfy
self._boundary_values = bv
# create array of symbolic coefficients
self._coeffs = sp.symarray('c' + tag, (self.n, 4))
self._coeffs_sym = self._coeffs.copy()
# calculate nodes of the spline
self.nodes = get_spline_nodes(self.a,
self.b,
self.n + 1,
nodes_type='equidistant')
self._nodes_type = 'equidistant' #nodes_type
# size of each polynomial part
self._h = (self.b - self.a) / float(self.n)
# the polynomial spline parts
# key: spline part
# value: corresponding polynomial
self._P = dict()
for i in xrange(self.n):
# create polynomials, e.g. for cubic spline:
# P_i(t)= c_i_3*t^3 + c_i_2*t^2 + c_i_1*t + c_i_0
self._P[i] = np.poly1d(self._coeffs[i])
# initialise array for provisionally evaluation of the spline
# if there are no values for its free parameters
#
# they show how the spline coefficients depend on the free coefficients
self._dep_array = None #np.array([])
self._dep_array_abs = None #np.array([])
# steady flag is True if smoothness and boundary conditions are solved
# --> make_steady()
self._steady_flag = False
# provisionally flag is True as long as there are no numerical values
# for the free parameters of the spline
# --> set_coefficients()
self._prov_flag = True
# the free parameters of the spline
self._indep_coeffs = None #np.array([])
"Please, choose the min value of hyperparameter (lambda) (default = 0.0001): "
)
lambda_par = input()
if lambda_par == '':
lambda_par = 0.0001
else:
lambda_par = float(lambda_par)
# max lambda value (starting one)
max_lambda = 0.1
#lambda_par = 0.000001
# just to save your value (e.g. png(s)) under correct prefix
prefix = 'real'
''' Generating Data Set '''
# generating an array of symbolic variables
# based on the desired amount of variables
x_symb = sp.symarray('x', n_vars, real=True)
# making a copy of this array
x_vals = x_symb.copy()
x_vals[0] = np.linspace(0, len(dataset[0]), len(dataset[0]))
x_vals[1] = np.linspace(0, len(dataset), len(dataset))
# library object instantiation
lib = rl.RegressionLibrary(x_symb, x_vals)
# generating output data - first setting-up the proper grid
x, y = np.meshgrid(x_vals[0], x_vals[1])
z = dataset #terrain
# Working with generated (Fake) data
elif value == False:
print('''
#========================#
# Working with Fake Data #
def potential(G,V,nF,nP):
f=sym.symarray('f',nF)
p=sym.symarray('p',nP)
vd=sym.symarray('vd',nF)
vdd=sym.symarray('vdd',nF*nF)
vddd=sym.symarray('vddd',nF*nF*nF)
g, Ga, Ri, Rm =fieldmetric(G,nF)
FMP=0
for i in range(nF):
vd[i] = sym.simplify(V.diff(f[i]))
for i in range(nF):
for j in range(nF):
for l in range(nF):
FMP=FMP-Ga(-(l+1),i+1,j+1) * vd[l]
vdd[i+j*nF] = sym.simplify(V.diff(f[i]).diff(f[j]))+FMP
FMP=0
for i in range(nF):
for j in range(nF):
for k in range(nF):
for l in range(nF):
FMP=FMP-Ga(-(l+1),k+1,i+1)*vdd[l+j*nF] - Ga(-(l+1),k+1,j+1)*vdd[i+l*nF]
vddd[i+j*nF+k*nF*nF] = sym.simplify(V.diff(f[i]).diff(f[j]).diff(f[k]))
FMP=0
import os
dir = os.path.dirname(__file__)
filename1 = os.path.join(dir, 'MTeasyC/potentialProto.h')
filename2 = os.path.join(dir, 'MTeasyC/potential.h')
f = open(filename1, 'r')
g = open(filename2, 'w')
for line in f:
g.write(line)
if line == "// #FP\n":
g.write('nF='+str(nF)+';\n'+'nP='+str(nP)+';\n')
#if line == "// pdef\n":
#for i in range(nP):
# g.write('p['+str(i)+']='+str(pp[i])+';\n')
if line == "// Pot\n":
expr=str(sym.ccode(V))
for l in range(max(nP,nF)):
expr=expr.replace("_"+str(l),"["+str(l)+"]")
g.write('\n sum='+str(expr)+';\n')
if line == "// dPot\n":
for i in range(nF):
expr=str(sym.ccode(vd[i]))
for l in range(max(nF,nP)):
expr=expr.replace("_"+str(l),"["+str(l)+"]")
g.write('\n sum['+str(i)+']='+str(expr)+';\n')
if line == "// ddPot\n":
for i in range(nF):
for j in range(nF):
expr=str(sym.ccode(vdd[i+nF*j]))
for l in range(max(nF,nP)):
expr=expr.replace("_"+str(l),"["+str(l)+"]")
g.write('\n sum['+str(i)+'+nF*'+str(j)+']='+str(expr)+';\n')
if line == "// dddPot\n":
for i in range(nF):
for j in range(nF):
for k in range(nF):
expr=str(sym.ccode(vddd[i+nF*j+nF*nF*k]))
for l in range(max(nF,nP)):
expr=expr.replace("_"+str(l),"["+str(l)+"]")
g.write('\n sum['+str(i)+'+nF*'+str(j)+'+nF*nF*'+str(k)+']='+str(expr)+';\n')
g.close()
f.close()
def n_bar_pendulum(N=1, param_values=dict()):
'''
Returns the mass matrix :math:`M` and right hand site :math:`B` of motion equations
.. math::
M * (d^2/dt^2) x = B
for the :math:`N`\ -bar pendulum.
Parameters
----------
N : int
Number of bars.
param_values : dict
Numeric values for the system parameters,
such as lengths, masses and gravitational acceleration.
Returns
-------
sympy.Matrix
The mass matrix `M`
sympy.Matrix
The right hand site `B`
list
List of symbols for state variables
list
List with symbol for input variable
'''
# first we have to create some symbols
F = sp.Symbol('F') # the force that acts on the car
g = sp.Symbol('g') # the gravitational acceleration
m = sp.symarray('m', N+1) # masses of the car (`m0`) and the bars
l = sp.symarray('l', N+1)#[1:] # length of the bars (`l0` is not needed nor used)
phi = sp.symarray('phi', N+1)#[1:] # deflaction angles of the bars (`phi0` is not needed nor used)
dphi = sp.symarray('dphi', N+1)#[1:] # 1st derivative of the deflaction angles (`dphi0` is not needed nor used)
if param_values.has_key('F'):
F = param_values['F']
elif param_values.has_key(F):
F = param_values[F]
if param_values.has_key('g'):
g = param_values['g']
elif param_values.has_key(g):
g = param_values[g]
else:
g = 9.81
for i, mi in enumerate(m):
if param_values.has_key(mi.name):
m[i] = param_values[mi.name]
elif param_values.has_key(mi):
m[i] = param_values[mi]
for i, li in enumerate(l):
if param_values.has_key(li.name):
l[i] = param_values[li.name]
elif param_values.has_key(li):
l[i] = param_values[li]
C = np.empty((N,N), dtype=object)
S = np.empty((N,N), dtype=object)
I = np.empty((N), dtype=object)
for i in xrange(1,N+1):
for j in xrange(1,N+1):
C[i-1,j-1] = cos(phi[i] - phi[j])
S[i-1,j-1] = sin(phi[i] - phi[j])
for i in xrange(1,N+1):
if param_values.has_key('I_%d'%i):
I[i-1] = param_values['I_%d'%i]
#elif param_values.has_key(Ii):
# I[i] = param_values[Ii]
else:
I[i-1] = 4.0/3.0 * m[i] * l[i]**2
#-------------#
# Mass matrix #
#-------------#
M = np.empty((N+1, N+1), dtype=object)
# 1st row
M[0,0] = m.sum()
for j in xrange(1,N):
M[0,j] = (m[j] + 2*m[j+1:].sum()) * l[j] * cos(phi[j])
M[0,N] = m[N] * l[N] * cos(phi[N])
# rest of upper triangular part, except last column
for i in xrange(1,N):
M[i,i] = I[i-1] + (m[i] + 4.0*m[i+1:].sum()) * l[i]**2
for j in xrange(i+1,N):
M[i,j] = 2.0*(m[j] + 2.0*m[j+1:].sum())*l[i]*l[j]*C[j-1,i-1]
# the last column
for i in xrange(1,N):
M[i,N] = 2.0*(m[N]*l[i]*l[N]*C[N-1,i-1])
M[N,N] = I[N-1] + m[N]*l[N]**2
# the rest (lower triangular part)
for i in xrange(N+1):
for j in xrange(i,N+1):
M[j,i] = 1 * M[i,j]
#-----------------#
# Right hand site #
#-----------------#
B = np.empty((N+1), dtype=object)
# first row
B[0] = F
for j in xrange(1,N):
B[0] += (m[j] + 2.0*m[j+1:].sum())*l[j]*sin(phi[j]) * dphi[j]**2
B[0] += (m[N]*l[N]*sin(phi[N])) * dphi[N]**2
# rest except for last row
for i in xrange(1,N):
B[i] = (m[i] + 2.0*m[i+1:].sum())*g*l[i]*sin(phi[i])
for j in xrange(1,N):
B[i] += (2.0*(m[j] + 2.0*m[j+1:].sum())*l[j]*l[i]*S[j-1,i-1]) * dphi[j]**2
B[i] += (2.0*m[N]*l[N]*l[N]*S[N-1,i-1]) * dphi[N]**2
# last row
B[N] = m[N]*g*l[N]*sin(phi[N])
for j in xrange(1,N+1):
B[N] += (2.0*m[N]*l[j]*l[N]*S[j-1,N-1]) * dphi[j]**2
# build lists of state and input variables
x, dx = sp.symbols('x, dx')
state_vars = [x, dx]
for i in xrange(1,N+1):
state_vars.append(phi[i])
state_vars.append(dphi[i])
input_vars = [F]
# return stuff
return sp.Matrix(M), sp.Matrix(B), state_vars, input_vars
def fieldmetric(G,nF,nP,simple=False,silent=True):
f=sym.symarray('f',nF)
p=sym.symarray('p',nP)
COR = Coordinates('\chi', f)
g = MetricTensor('g',COR , G)
Ga = Christoffel('Ga', g)
Ri = Ricci('Ri', g)
Rm = Riemann('Rm',g)
import os
dir = os.path.dirname(__file__)
filename1 = os.path.join(dir, 'CppTrans', 'fieldmetricProto.h')
filename2 = os.path.join(dir, 'CppTrans', 'fieldmetric.h')
e = open(filename1, 'r')
h = open(filename2, 'w')
G_array = sym.symarray('G', 2*nF * 2*nF)
Gamma_array = sym.symarray('Gamma', 2*nF * 2*nF * 2*nF)
R_array = sym.symarray('Riemann', nF*nF*nF*nF)
gradR_array = sym.symarray('gradRiemann', nF*nF*nF*nF*nF)
# populate Riemann matrix
for i in range(2 * nF):
for j in range(2 * nF):
if i < nF:
ii = -i - 1
else:
ii = i - (nF - 1)
if j < nF:
jj = -j - 1
else:
jj = j - (nF - 1)
if simple is True:
G_array[(2 * nF) * i + j] = sym.simplify(g(ii, jj))
else:
G_array[(2 * nF) * i + j] = g(ii, jj)
# populate connexion matrix
for i in range(2 * nF):
for j in range(2 * nF):
for k in range(2 * nF):
if i < nF:
ii = -i - 1
else:
ii = i - (nF - 1)
if j < nF:
jj = -j - 1
else:
jj = j - (nF - 1)
if k < nF:
kk = -k - 1
else:
kk = k - (nF - 1)
if kk < 0 or jj < 0 or ii > 0:
Gamma_array[(2*nF)*(2*nF)*i+(2*nF)*j+k] = sym.simplify(0)
else:
if simple is True:
Gamma_array[(2*nF)*(2*nF)*i+(2*nF)*j+k] = sym.simplify(Ga(ii, jj, kk))
else:
Gamma_array[(2*nF)*(2*nF)*i+(2*nF)*j+k] = Ga(ii, jj, kk)
# populate Riemann matrix
for i in range(nF):
for j in range(nF):
for k in range(nF):
for l in range(nF):
ii=i+1
jj=j+1
kk=k+1
ll=l+1
if simple is True:
R_array[(nF)*(nF)*(nF)*i+(nF)*(nF)*j+(nF)*k+l] = sym.simplify(Rm(ii,jj,kk,ll))
else:
R_array[(nF)*(nF)*(nF)*i+(nF)*(nF)*j+(nF)*k+l] = Rm(ii,jj,kk,ll)
# populate covariant-derivative of Riemann matrix
for i in range(nF):
for j in range(nF):
for k in range(nF):
for l in range(nF):
for m in range(nF):
ii = i + 1
jj = j + 1
kk = k + 1
ll = l + 1
mm = m + 1
if simple is True:
gradR_array[(nF)*(nF)*(nF)*(nF)*i+(nF)*(nF)*(nF)*j+(nF)*(nF)*k+(nF)*l+m] = sym.simplify(Rm.covariantD(ii, jj, kk, ll, mm))
else:
gradR_array[(nF)*(nF)*(nF)*(nF)*i+(nF)*(nF)*(nF)*j+(nF)*(nF)*k+(nF)*l+m] = Rm.covariantD(ii, jj, kk, ll, mm)
for line in e:
h.write(line)
if line == "// #FP\n":
#h.write('nF='+str(nF)+';\n')
h.write('nF='+str(nF)+';\n'+'nP='+str(nP)+';\n')
if line == "// metric\n":
if not silent:
timer_cse = time.clock()
print(' [{time}] performing CSE for field metric'.format(time=time.ctime()))
decls, new_expr = sym.cse(G_array, order='none')
if not silent:
print(' [{time}] complete in {x} sec'.format(time=time.ctime(), x=time.clock() - timer_cse))
# emit declarations for CSE variables
write_cse_decls(decls, h, nF, nP)
for i in range(2*nF):
for j in range(2*nF):
# emit main expression
emit_expr = sym.printing.cxxcode(new_expr[(2*nF)*i+j])
rw_expr = rewrite_indices(emit_expr, nF, nP)
h.write('\n FM['+str((2*nF)*i+j)+']=' + str(rw_expr) + ';\n')
if line == "// Christoffel\n":
if not silent:
timer_cse = time.clock()
print(' [{time}] performing CSE for Christoffel symbols'.format(time=time.ctime()))
decls, new_expr = sym.cse(Gamma_array, order='none')
if not silent:
print(' [{time}] complete in {x} sec'.format(time=time.ctime(), x=time.clock() - timer_cse))
# emit declarations for CSE variables
write_cse_decls(decls, h, nF, nP)
for i in range(2 * nF):
for j in range(2 * nF):
for k in range(2 * nF):
# emit main expression
emit_expr = sym.printing.cxxcode(new_expr[(2*nF)*(2*nF)*i+(2*nF)*j+k])
rw_expr = rewrite_indices(emit_expr, nF, nP)
h.write('\n CS['+str((2*nF)*(2*nF)*i+(2*nF)*j+k)+']=' + str(rw_expr) + ';\n')
if line == "// Riemann\n":
if not silent:
timer_cse = time.clock()
print(' [{time}] performing CSE for Riemann tensor'.format(time=time.ctime()))
decls, new_expr = sym.cse(R_array, order='none')
if not silent:
print(' [{time}] complete in {x} sec'.format(time=time.ctime(), x=time.clock() - timer_cse))
# emit declarations for CSE variables
write_cse_decls(decls, h, nF, nP)
for i in range(nF):
for j in range(nF):
for k in range(nF):
for l in range(nF):
# emit main expression
emit_expr = sym.printing.cxxcode(new_expr[(nF)*(nF)*(nF)*i+(nF)*(nF)*j+(nF)*k+l])
rw_expr = rewrite_indices(emit_expr, nF, nP)
h.write('\n RM['+str((nF)*(nF)*(nF)*i+(nF)*(nF)*j+(nF)*k+l)+']=' + str(rw_expr) + ';\n')
if line == "// Riemanncd\n":
if not silent:
timer_cse = time.clock()
print(' [{time}] performing CSE for Riemann tensor'.format(time=time.ctime()))
decls, new_expr = sym.cse(gradR_array, order='none')
if not silent:
print(' [{time}] complete in {x} sec'.format(time=time.ctime(), x=time.clock() - timer_cse))
# emit declarations for CSE variables
write_cse_decls(decls, h, nF, nP)
for i in range(nF):
for j in range(nF):
for k in range(nF):
for l in range(nF):
for m in range(nF):
# emit main expression
emit_expr = sym.printing.cxxcode(new_expr[(nF)*(nF)*(nF)*(nF)*i+(nF)*(nF)*(nF)*j+(nF)*(nF)*k+(nF)*l+m])
rw_expr = rewrite_indices(emit_expr, nF, nP)
h.write('\n RMcd['+str((nF)*(nF)*(nF)*(nF)*i+(nF)*(nF)*(nF)*j+(nF)*(nF)*k+(nF)*l+m)+']=' + str(rw_expr) + ';\n')
h.close()
e.close()
return g, Ga, Ri, Rm
def __init__(self,
a=0.0,
b=1.0,
n=5,
bv=None,
tag='',
use_std_approach=False,
**kwargs):
# there are two different approaches implemented for evaluating
# the splines which mainly differ in the node that is used in the
# evaluation of the polynomial parts
#
# the reason for this is that in the project thesis from which
# PyTrajectory emerged, the author didn't use the standard approach
# usually used when dealing with spline interpolation
#
# later this standard approach has been implemented and was intended
# to replace the other one, but it turned out that
# convergence properties of the whole algorithm for some examples got worse
#
# until this issue is resolved the user is enabled to choose between
# the two approaches be altering the following attribute
self.masterobject = kwargs.get("masterobject")
self.init_logger(self.masterobject)
self._init_bc(bv)
self._use_std_approach = use_std_approach
# interval boundaries
assert a < b
self.a = a
self.b = b
# number of polynomial parts
self.n = int(n)
# 'name' of the spline
self.tag = tag
# dictionary with boundary values
# key: order of the spline's derivative to which the values belong
# values: the boundary values the derivative should satisfy
# create array of symbolic coefficients
self._coeffs = sp.symarray(
'c' + tag, (self.n, 4)
) ##:: e.g.: array([[cxi_0_0, cxi_0_1, cxi_0_2, cxi_0_3],...,[cxi_9_0, cxi_9_1, cxi_9_2, cxi_9_3]])
self._coeffs_sym = self._coeffs.copy()
# calculate nodes of the spline
self.nodes = get_spline_nodes(self.a,
self.b,
self.n + 1,
nodes_type='equidistant')
self._nodes_type = 'equidistant'
# size of each polynomial part
self._h = (self.b - self.a) / float(self.n)
# the polynomial spline parts
# key: spline part
# value: corresponding polynomial
self._P = dict()
for i in xrange(self.n):
# create polynomials, e.g. for cubic spline:
# P_i(t)= c_i_3*t^3 + c_i_2*t^2 + c_i_1*t + c_i_0
# poly1d expects coeffs in decreasing order -> [::-1]
self._P[i] = np.poly1d(self._coeffs[i][::-1])
## note: here _P is only for one state/input-component!!
# initialise array for provisionally evaluation of the spline
# if there are no values for its free parameters
# they show how the spline coefficients depend on the free coefficients
self._dep_array = None #np.array([])
self._dep_array_abs = None #np.array([])
# steady flag is True if smoothness and boundary conditions are solved
# --> make_steady()
self._steady_flag = False
# provisionally flag is True as long as there are no numerical values
# for the free parameters of the spline
# --> set_coefficients()
self._prov_flag = True
# the free parameters of the spline
self._indep_coeffs = None
self._indep_coeffs_sym = None
# cache for a frequently used part of a block-matrix
self._node_eval_block = None
self._steady_flag = False
def linear_equation_solution():
from sympy import Matrix, symarray, solve
a = Matrix([[2, 3, 1], [4, 2, 3], [7, 1, -1]])
b = Matrix([[4], [17], [1]])
x = symarray('x', 3).reshape(3, 1)
print(solve(a * x - b))
v = sym.simplify(udot + R*Omega_skew*s)
print('v = ')
display(v)
# Define acceleration of element endpoints (nodes)
a = sym.simplify(uddot + R*Omega_skew*Omega_skew*s + R*Alpha_skew*s)
print('\na = ')
display(a)
# ### Compute the Mass Matrix
# In[ ]:
# Define shape function for element with one node at each end
h = sym.symarray('h', 2)
h[0] = sym.Rational(1,2)*(1 - x)
h[1] = sym.Rational(1,2)*(1 + x)
# Compute shape function matrix
H = sym.expand(sym.Matrix([h[0]*sym.eye(3), h[1]*sym.eye(3)])).T
print('\nH = ')
display(H.T)
# Define velocity of any point
Vp = H*v
print('\nV = ')
display(Vp)
# Define velocity of any point
import sys # import the sys module used below
############################################################################################################################################
# if using an integrated environment we recommend restarting the python console after running this script to make sure updates are found
location = "/home/jwr/Code/June/PyTransport2Dist/PyTransport/" # this should be the location of the PyTransport folder
sys.path.append(location) # we add this location to the python path
import PyTransSetup # the above commands allows python to find the PyTransSetup module and import it
############################################################################################################################################
nF = 2 # number of fields needed to define the double quadratic potential
nP = 2 # number of parameters needed to define the double quadtartic potential
f = sym.symarray(
'f', nF) # an array representing the nF fields present for this model
p = sym.symarray(
'p', nP
) # an array representing the nP parameters needed to define this model (that we might wish to change) if we don't
# wish to change them they could be typed explicitly in the potential below
V = 1. / 2. * p[0]**2.0 * f[0]**2.0 + 1. / 2. * p[1]**2.0 * f[
1]**2.0 # this is the potential written in sympy notation
#The last argument is for whether the sympy's simplify is used to on derivatives of the potential and field geometric quantites.
#Caution is recomended as sympy's simplify is known to have bugs. Simplification can increase the speed of numerically evolutions, but at the cost of compling more slowly.
PyTransSetup.potential(
V, nF, nP, True
) # writes this potential and its derivatives into C++ file potential.h when run
PyTransSetup.compileName(
def __init__(self, *args, **kwargs):
"""
Initialises a new `Symbol` object in one of three ways::
u=Symbol('u')
returns a scalar symbol by the name 'u'.
alpha=Symbol('alpha', (4,3))
returns a rank 2 symbol with the shape (4,3), whose elements are
named '[alpha]_i_j' (with i=0..3, j=0..2).
a,b,c=symbols('a,b,c')
x=Symbol([[a+b,0,0],[0,b-c,0],[0,0,c-a]])
returns a rank 2 symbol with the shape (3,3) whose elements are
explicitly specified by numeric values and other symbols/expressions
within a list or numpy array.
The dimensionality of the symbol can be specified through the ``dim``
keyword. All other keywords are passed to the underlying symbolic
library (currently sympy).
:param args: initialisation arguments as described above
:keyword dim: dimensionality of the new Symbol (default: 2)
:type dim: ``int``
"""
if not HAVE_SYMBOLS:
raise RuntimeError("Trying to instantiate a Symbol but sympy not available")
if 'dim' in kwargs:
self._dim=kwargs.pop('dim')
else:
self._dim=-1 # undefined
if 'subs' in kwargs:
self._subs=kwargs.pop('subs')
else:
self._subs={}
if len(args)==1:
arg=args[0]
if isinstance(arg, str):
if arg.find('[')>=0 or arg.find(']')>=0:
raise ValueError("Name must not contain '[' or ']'")
self._arr=numpy.array(sympy.Symbol(arg, **kwargs))
elif hasattr(arg, "__array__") or isinstance(arg, list):
if isinstance(arg, list): arg=numpy.array(arg)
arr=arg.__array__()
if len(arr.shape)>4:
raise ValueError("Symbol only supports tensors up to order 4")
res=numpy.empty(arr.shape, dtype=object)
for idx in numpy.ndindex(arr.shape):
if hasattr(arr[idx], "item"):
res[idx]=arr[idx].item()
else:
res[idx]=arr[idx]
self._arr=res
if isinstance(arg, Symbol):
self._subs.update(arg._subs)
if self._dim==-1:
self._dim=arg._dim
elif isinstance(arg, sympy.Basic):
self._arr=numpy.array(arg)
else:
raise TypeError("Unsupported argument type %s"%str(type(arg)))
elif len(args)==2:
if not isinstance(args[0], str):
raise TypeError("First argument must be a string")
if not isinstance(args[1], tuple):
raise TypeError("Second argument must be a tuple")
name=args[0]
shape=args[1]
if name.find('[')>=0 or name.find(']')>=0:
raise ValueError("Name must not contain '[' or ']'")
if len(shape)>4:
raise ValueError("Symbol only supports tensors up to order 4")
if len(shape)==0:
self._arr=numpy.array(sympy.Symbol(name, **kwargs))
else:
try:
self._arr=sympy.symarray(shape, '['+name+']')
except TypeError:
self._arr=sympy.symarray('['+name+']', shape)
else:
raise TypeError("Unsupported number of arguments")
if self._arr.ndim==0:
self.name=str(self._arr.item())
else:
self.name=str(self._arr.tolist())
#sample script: calculate covariant derivatives
#----------by Zhiqi Huang--for the course ''General Relativity''-------------
import sympy as sym
sym.init_printing()
dim = 2
#coordinates theta and phi
u = sym.symarray('u', dim)
#r = sym.symbols('r')
#gdown = sym.diag( r**2, r**2*sym.sin(u[0])**2 )
gdown = sym.diag(1, sym.sin(u[0])**2)
#gdown = sym.diag(1, sym.exp(2*u[0]))
#gdown = sym.diag(1/(1+u[0]**2+u[1]**2),1/(1+u[0]**2+u[1]**2))
#gdown = sym.diag(1/(1+u[0]**2+u[1]**2)**2,1/(1+u[0]**2+u[1]**2)**2)
#gdown = sym.Matrix([[1/(u[0]**2+u[1]**2+1), 1/(u[0]**2+u[1]**2+1)/2],[1/(u[0]**2+u[1]**2+1)/2, 1/(u[0]**2+u[1]**2+1)]])
gup = gdown**-1
detg = gdown.det()
def connection_down(i, j, k):
return (sym.diff(gdown[i, j], u[k]) + sym.diff(gdown[i, k], u[j]) -
sym.diff(gdown[j, k], u[i])) / 2
##allocate space to save connections
#!/usr/bin/env python
from time import clock
import numpy as np
import sympy as sp
import symengine as se
import warnings
# Real-life example (ion speciation problem in water chemistry)
x = sp.symarray('x', 14)
p = sp.symarray('p', 14)
args = np.concatenate((x, p))
exp = sp.exp
exprs = [
x[0] + x[1] - x[4] + 36.252574322669, x[0] - x[2] + x[3] +
21.3219379611249, x[3] + x[5] - x[6] + 9.9011158998744,
2 * x[3] + x[5] - x[7] + 18.190422234653,
3 * x[3] + x[5] - x[8] + 24.8679190043357,
4 * x[3] + x[5] - x[9] + 29.9336062089226,
-x[10] + 5 * x[3] + x[5] + 28.5520551531262,
2 * x[0] + x[11] - 2 * x[4] - 2 * x[5] + 32.4401680272417,
3 * x[1] - x[12] + x[5] + 34.9992934135095,
4 * x[1] - x[13] + x[5] + 37.0716199972041, p[0] - p[1] + 2 * p[10] +
2 * p[11] - p[12] - 2 * p[13] + p[2] + 2 * p[5] + 2 * p[6] + 2 * p[7] +
2 * p[8] + 2 * p[9] - exp(x[0]) + exp(x[1]) - 2 * exp(x[10]) -
2 * exp(x[11]) + exp(x[12]) + 2 * exp(x[13]) - exp(x[2]) - 2 * exp(x[5]) -
2 * exp(x[6]) - 2 * exp(x[7]) - 2 * exp(x[8]) - 2 * exp(x[9]),
-p[0] - p[1] - 15 * p[10] - 2 * p[11] - 3 * p[12] - 4 * p[13] - 4 * p[2] -
3 * p[3] - 2 * p[4] - 3 * p[6] - 6 * p[7] - 9 * p[8] - 12 * p[9] +
exp(x[0]) + exp(x[1]) + 15 * exp(x[10]) + 2 * exp(x[11]) + 3 * exp(x[12]) +
4 * exp(x[13]) + 4 * exp(x[2]) + 3 * exp(x[3]) + 2 * exp(x[4]) +
def createNullSpaceFunction(SciPyModel):
'''
Generate the NullSpace of the kinetic parameters
(parameters which enter the derivative matrix
linearly). The nullspace in this case can be used
to solve the system for the steady-state condition.
This operation happens using the SymPy module to
solve the homogenous equation A*x = 0.
To Do
-----
1. Implement a method for manually setting
parameters to be considered kinetic parameters.
2. Investigate method for sampling using restriction
given by N*g = k > 0.
Parameters
----------
SciPyModel : internal object instance
Returns
-------
SciPyModel : internal object instance
SciPyModel with NullSpaceFunction based on
model parameterization.
See Also
--------
Notes
-----
The anonymous function can be called as follows:
NullSpaceFunction( E, X )
This produces a SymPy matrix of the nullspace
given the desired steady state condition as specified
by the user choice in E and X.
'''
import sympy, datetime, numpy
# Automatically flag kinetic parameters.
for rxn in SciPyModel.Reactions.Formulas:
if SciPyModel.Reactions.Formulas[0][0] == 'p':
SciPyModel.Parameters.KineticFlag[int(
rxn.split(']')[0].split('[')[1])] = True
# Write derivative function to file
TempName = (SciPyModel.MetaData.Name +
datetime.datetime.now().strftime("%I%M%p%B%d%Y") + '.py')
open(TempName, 'w+').write(SciPyModel.ToolboxFunctions.DerivativeFunction)
# Import derivative function from temporary file.
TempModule = __import__(TempName[:-3])
# Create symbolic species and parameter vectors
y = sympy.symarray('y', len(SciPyModel.Species.Names))
p = sympy.symarray('p', len(SciPyModel.Parameters.Names))
# Create symbolic reaction and stoichiometry matrices
R = sympy.Matrix(TempModule.rxn_fun(y, 0, p))
S = sympy.Matrix(SciPyModel.Reactions.Stoichiometry)
# Create symbolic derivative matrix
DerivativeMatrix = sympy.Matrix([S[:, i] * R[i] for i in range(len(R))
]).reshape(S.shape[1],
S.shape[0]).transpose()
# Extract kinetic parameters
M = sympy.Matrix(
sympy.lambdify(
(p[SciPyModel.Parameters.KineticFlag]), DerivativeMatrix, 'sympy')(
*sympy.ones(len(p[SciPyModel.Parameters.KineticFlag]), 1)))
# Obtain basis for nullspace
NullBasis = M.nullspace()
SciPyModel.Parameters.NullSpaceDimension = len(NullBasis)
# Assemble nullspace matrix from basis
NullSpace = sympy.Matrix([NullBasis[i] for i in range(len(NullBasis))])
NullSpace = NullSpace.reshape(len(NullBasis),
len(NullSpace) / len(NullBasis)).transpose()
# Create anonymous sampling function
SciPyModel.ToolboxFunctions.NullSpaceFunction = sympy.lambdify(
(p[numpy.invert(SciPyModel.Parameters.KineticFlag)], y),
NullSpace,
'numpy',
dummify=False)
return SciPyModel
def check_input(self, inputs):
"""check_input
Checks the input and create a list of function handle strings with T as
argument. Inputs can be strings, floats, ints, or pint quantaties.
Args:
inputs (list[str, int, float, Quantity]): list of strings, int, floats,
or Pint quantities.
Returns:
(tuple):
- *output (list[@lambda])* - list of lambda functions.
- *output_strs (list[str])* - list of string-representations.
"""
output = []
output_strs = []
# if the input is not a list, we convert it to one
if not isinstance(inputs, list):
inputs = [inputs]
# update number of subsystems
K = self.num_sub_systems
k = len(inputs)
if k != K:
print('Number of subsystems changed from {:d} to {:d}.'.format(
K, k))
self.num_sub_systems = k
T = symbols('T')
# traverse each list element and convert it to a function handle
for input in inputs:
if isfunction(input):
raise ValueError(
'Please use string representation of function!')
elif isinstance(input, str):
try:
# backwards compatibility for direct lambda definition
if ':' in input:
# strip lambda prefix
input = input.split(':')[1]
# backwards compatibility for []-indexing
input = input.replace('[', '_').replace(']', '')
# check for presence of indexing and use symarray as argument
if '_' in input:
T = symarray('T', k)
output.append(lambdify([T], input, modules='numpy'))
else:
output.append(lambdify(T, input, modules='numpy'))
output_strs.append(input.strip())
except Exception as e:
print('String input for layer property ' + input + ' \
cannot be converted to function handle!')
print(e)
elif isinstance(input, (int, float)):
output.append(lambdify(T, input, modules='numpy'))
output_strs.append(str(input))
elif isinstance(input, object):
output.append(
lambdify(T,
input.to_base_units().magnitude,
modules='numpy'))
output_strs.append(str(input.to_base_units().magnitude))
else:
raise ValueError(
'Layer property input has to be a single or '
'list of numerics, Quantities, or function handle strings '
'which can be converted into a lambda function!')
return output, output_strs
def _bondgraph_to_residuals(model, control_vars=None):
dX = sp.IndexedBase('dX')
X = sp.IndexedBase('X')
U = sp.IndexedBase('U')
x_subs = []
dx_subs = []
u_subs = []
u_func = []
n = len(model.state_vars)
m = 0
for i, x in enumerate(model.state_vars):
x_subs.append((x, X[i]))
dx_subs.append((sp.S(f'dx_{i}'), dX[i]))
if len(model.control_vars) > 0:
u_func_dict = {}
u_constants = {}
if isinstance(control_vars, list):
u_func_dict.update({i: f for i, f in enumerate(control_vars)})
elif isinstance(control_vars, dict):
u_func_dict.update(
{int(v[2:]): f
for v, f in control_vars.items()})
elif len(model.control_vars) == 1:
u_func_dict[0] = control_vars
else:
raise TypeError(f"Control argument {control_vars} is invalid")
test_x = np.zeros(shape=(n, ), dtype=np.float32)
for idx, f in u_func_dict.items():
try:
if isinstance(f, str):
f = _to_function(f, X, dX, dx_subs + x_subs)
u_func_dict[idx] = f
if isinstance(f, (float, int, sp.Number)):
u_constants[idx] = f
if n == 1:
r = f(0, 0, 0)
else:
r = f(0, test_x, test_x)
assert isinstance(
r, (float, int, sp.Number)), "Invalid output from control"
except Exception as ex:
message = f"Invalid control function for var: {idx}.\n " \
"Control functions should be of the form:\n" \
f"{idx} = f(t, x, dx/dt)"
raise ModelException(message)
for i, u in enumerate(model.control_vars):
if i in u_constants:
u_subs.append((u, u_constants[i]))
continue
u_subs.append((u, U[m]))
try:
u_func.append(u_func_dict[i])
except KeyError:
raise ModelException(f"Control variable {u} must be specified")
m += 1
rels = [
r.subs(dx_subs).subs(x_subs).subs(u_subs)
for r in model.constitutive_relations
]
if len(rels) != n:
raise ModelException(
"Model simplification error: system is under-determined")
Fsym = sp.symarray('F', shape=n)
for i, r in enumerate(rels):
Fsym[i] = r
t = sp.S('t')
_r = np.empty(shape=(n, ), dtype=np.float64)
if not u_func:
F = sp.lambdify((t, X, dX), Fsym)
def residual(_t, _x, _dx, _res):
_r = F(_t, _x, _dx)
for i in range(n):
_res[i] = _r[i]
else:
_u = np.empty(shape=(m, ), dtype=np.float64)
Fsym_u = sp.lambdify((t, X, dX, U), Fsym)
def residual(_t, _x, _dx, _res):
_u = [u_f(_t, _x, _dx) for u_f in u_func]
_r = Fsym_u(_t, _x, _dx, _u)
for i in range(n):
_res[i] = _r[i]
return residual, X
def __init__(self, a=0.0, b=1.0, n=5, bv={},
tag='', use_std_approach=False, **kwargs):
# there are two different approaches implemented for evaluating
# the splines which mainly differ in the node that is used in the
# evaluation of the polynomial parts
#
# the reason for this is that in the project thesis from which
# PyTrajectory emerged, the author didn't use the standard approach
# usually used when dealing with spline interpolation
#
# later this standard approach has been implemented and was intended
# to replace the other one, but it turned out that when using this
# standard approach some examples suddendly fail
#
# until this issue is resolved the user is enabled to choose between
# the two approaches be altering the following attribute
self._use_std_approach = use_std_approach
# interval boundaries
assert a < b
self.a = a
self.b = b
# number of polynomial parts
self.n = int(n)
# 'name' of the spline
self.tag = tag
# dictionary with boundary values
# key: order of the spline's derivative to which the values belong
# values: the boundary values the derivative should satisfy
self._boundary_values = bv
# create array of symbolic coefficients
self._coeffs = sp.symarray('c'+tag, (self.n, 4))
self._coeffs_sym = self._coeffs.copy()
# calculate nodes of the spline
self.nodes = get_spline_nodes(self.a, self.b, self.n+1, nodes_type='equidistant')
self._nodes_type = 'equidistant' #nodes_type
# size of each polynomial part
self._h = (self.b - self.a) / float(self.n)
# the polynomial spline parts
# key: spline part
# value: corresponding polynomial
self._P = dict()
for i in xrange(self.n):
# create polynomials, e.g. for cubic spline:
# P_i(t)= c_i_3*t^3 + c_i_2*t^2 + c_i_1*t + c_i_0
self._P[i] = np.poly1d(self._coeffs[i])
# initialise array for provisionally evaluation of the spline
# if there are no values for its free parameters
#
# they show how the spline coefficients depend on the free coefficients
self._dep_array = None #np.array([])
self._dep_array_abs = None #np.array([])
# steady flag is True if smoothness and boundary conditions are solved
# --> make_steady()
self._steady_flag = False
# provisionally flag is True as long as there are no numerical values
# for the free parameters of the spline
# --> set_coefficients()
self._prov_flag = True
# the free parameters of the spline
self._indep_coeffs = None #np.array([])
import math
import sys
############################################################################################################################################
############################################################################################################################################
# if using an integrated environment we recommend restarting the python console after running this script to make sure updates are found
location = "/Users/david/Dropbox/PyTransportDist/PyTransport/" # this should be the location of the PyTransport folder
sys.path.append(location) # we add this location to the python path
import PyTransSetup # the above commands allows python to find the PyTransSetup module and import it
############################################################################################################################################
### Sets potential and compiles PyTransport, users may prefer to do this only once in a separate file (or comment after running below once) ###
### Restart the python kernel after running this file
nF=2
nP=5
f=sym.symarray('f',nF)
p=sym.symarray('p',nP)
V= 1./2. * p[0]**2 * f[0]**2 + 1/2.*p[1]**2 * sym.cos(p[2]/2.)**2*(f[1] - (f[0]-p[3])*sym.tan(p[2]/math.pi*sym.atan(p[4]*(f[0]-p[3]))))**2
PyTransSetup.potential(V,nF,nP) # writes this potential into c file when run
PyTransSetup.compileName("LH") # this compiles the module with the new potential and places it in the location folder, and adds this folder to the path ready for use
############################################################################################################################################
def potential(V,nF,nP,simple=False,G=0,silent=True):
f=sym.symarray('f',nF)
p=sym.symarray('p',nP)
vd=sym.symarray('vd',nF)
vdd=sym.symarray('vdd',nF*nF)
vddd=sym.symarray('vddd',nF*nF*nF)
if not silent:
timer = time.clock()
print('[{time}] computing symbolic potential derivatives'.format(time=time.ctime()))
if G!=0:
if not silent:
timer2 = time.clock()
print(' [{time}] computing curvature quantities'.format(time=time.ctime()))
g, Ga, Ri, Rm =fieldmetric(G,nF,nP,simple=simple,silent=silent)
if not silent:
print(' [{time}] complete in {x} sec'.format(time=time.ctime(), x=time.clock() - timer2))
FMP=0
for i in range(nF):
if simple==True:
vd[i] = sym.simplify(V.diff(f[i]))
else:
vd[i] = V.diff(f[i])
for i in range(nF):
for j in range(nF):
for l in range(nF):
FMP=FMP+Ga(-(l+1),i+1,j+1) * vd[l]
if simple==True:
vdd[i+j*nF] = sym.simplify(V.diff(f[i]).diff(f[j])-FMP)
else:
vdd[i+j*nF] = V.diff(f[i]).diff(f[j])-FMP
FMP=0
for i in range(nF):
for j in range(nF):
for k in range(nF):
for l in range(nF):
FMP=FMP+Ga(-(l+1),i+1,k+1)*vdd[l+j*nF] + Ga(-(l+1),j+1,k+1)*vdd[i+l*nF] +sym.expand(Ga(-(1+l),1+i,1+j)).diff(f[k])*vd[l]+sym.expand(Ga(-(1+l),1+i,j+1))*vd[l].diff(f[k])# +sym.expand(Ga(-(l+1),i+1,j+1)).diff(f[k]) * vd[l] +Ga(-(l+1),i+1,j+1)* (sym.expand(vd[l]).diff(f[k]))
if simple==True:
vddd[i+j*nF+k*nF*nF] =sym.simplify(V.diff(f[i]).diff(f[j]).diff(f[k]) -FMP)
else:
vddd[i+j*nF+k*nF*nF] =V.diff(f[i]).diff(f[j]).diff(f[k]) -FMP
FMP=0
else:
for i in range(nF):
if simple==True:
vd[i] = sym.simplify(V.diff(f[i]))
else:
vd[i] = V.diff(f[i])
for j in range(nF):
if simple==True:
vdd[i+j*nF] = sym.simplify(V.diff(f[i]).diff(f[j]) )
else:
vdd[i+j*nF] = V.diff(f[i]).diff(f[j])
for k in range(nF):
if simple==True:
vddd[i+j*nF+k*nF*nF] = sym.simplify(V.diff(f[i]).diff(f[j]).diff(f[k]))
else:
vddd[i+j*nF+k*nF*nF] = V.diff(f[i]).diff(f[j]).diff(f[k])
if not silent:
print('[{time}] complete in {x} sec'.format(time=time.ctime(), x=time.clock() - timer))
import os
dir = os.path.dirname(__file__)
filename1 = os.path.join(dir, 'CppTrans', 'potentialProto.h')
filename2 = os.path.join(dir, 'CppTrans', 'potential.h')
f = open(filename1, 'r')
g = open(filename2, 'w')
if not silent:
timer = time.clock()
print('[{time}] writing to potential.h'.format(time=time.ctime()))
for line in f:
g.write(line)
if line == "// #Rewrite\n":
g.write('// Potential file rewriten at' + ' ' + time.strftime("%c") +'\n')
if line == "// #FP\n":
g.write('nF='+str(nF)+';\n'+'nP='+str(nP)+';\n')
if line == "// Pot\n":
# extract common subexpressions from V
if not silent:
timer_cse = time.clock()
print(' [{time}] performing CSE for V'.format(time=time.ctime()))
decls, new_expr = sym.cse(V, order='none')
if not silent:
print(' [{time}] complete in {x} sec'.format(time=time.ctime(), x=time.clock() - timer_cse))
# emit declarations for CSE variables
write_cse_decls(decls, g, nF, nP)
# emit main expression
emit_expr = sym.printing.cxxcode(new_expr[0])
rw_expr = rewrite_indices(emit_expr, nF, nP)
g.write(' sum='+str(rw_expr)+';\n')
if line == "// dPot\n":
if not silent:
timer_cse = time.clock()
print(' [{time}] performing CSE for dV'.format(time=time.ctime()))
decls, new_exprs = sym.cse(vd, order='none')
if not silent:
print(' [{time}] complete in {x} sec'.format(time=time.ctime(), x=time.clock() - timer_cse))
# emit declarations for CSE variables
write_cse_decls(decls, g, nF, nP)
for i in range(nF):
emit_expr = sym.printing.cxxcode(new_exprs[i])
rw_expr = rewrite_indices(emit_expr, nF, nP)
g.write('\n sum[' + str(i) + ']=' + str(rw_expr) + ';\n')
if line == "// ddPot\n":
if not silent:
timer_cse = time.clock()
print(' [{time}] performing CSE for ddV'.format(time=time.ctime()))
decls, new_exprs = sym.cse(vdd, order='none')
if not silent:
print(' [{time}] complete in {x} sec'.format(time=time.ctime(), x=time.clock() - timer_cse))
# emit declarations for CSE variables
write_cse_decls(decls, g, nF, nP)
for i in range(nF):
for j in range(nF):
emit_expr = sym.printing.cxxcode(new_exprs[i + nF * j])
rw_expr = rewrite_indices(emit_expr, nF, nP)
g.write('\n sum[' + str(i + nF * j) + ']=' + str(rw_expr) + ';\n')
if line == "// dddPot\n":
if not silent:
timer_cse = time.clock()
print(' [{time}] performing CSE for dddV'.format(time=time.ctime()))
decls, new_exprs = sym.cse(vddd, order='none')
if not silent:
print(' [{time}] complete in {x} sec'.format(time=time.ctime(), x=time.clock() - timer_cse))
# emit declarations for CSE variables
write_cse_decls(decls, g, nF, nP)
for i in range(nF):
for j in range(nF):
for k in range(nF):
emit_expr = sym.printing.cxxcode(new_exprs[i + nF * j + nF * nF * k])
rw_expr = rewrite_indices(emit_expr, nF, nP)
g.write('\n sum[' + str(i + nF * j + nF * nF * k) + ']=' + str(rw_expr) + ';\n')
if not silent:
print('[{time}] complete in {x} sec'.format(time=time.ctime(), x=time.clock() - timer))
g.close()
f.close()