def test_2d_scalar_1():
print('============== test_2d_scalar_1 ================')
x, y = symbols('x y')
u = Symbol('u')
v = Symbol('v')
a = Lambda((x, y, v, u), Dot(Grad(u), Grad(v)) + u * v)
print('> input := {0}'.format(a))
# ... create a finite element space
p1 = 2
p2 = 2
ne1 = 8
ne2 = 8
print('> Grid :: [{ne1},{ne2}]'.format(ne1=ne1, ne2=ne2))
print('> Degree :: [{p1},{p2}]'.format(p1=p1, p2=p2))
grid_1 = linspace(0., 1., ne1 + 1)
grid_2 = linspace(0., 1., ne2 + 1)
V1 = SplineSpace(p1, grid=grid_1)
V2 = SplineSpace(p2, grid=grid_2)
V = TensorFemSpace(V1, V2)
# ...
# ... create a glt symbol from a string without evaluation
expr = glt_symbol(a, space=V)
print('> glt symbol := {0}'.format(expr))
# ...
# ...
symbol_f90 = compile_symbol('symbol_scalar_1', a, V, backend='fortran')
# ...
# ... example of symbol evaluation
t1 = linspace(-pi, pi, ne1 + 1)
t2 = linspace(-pi, pi, ne2 + 1)
x1 = linspace(0., 1., ne1 + 1)
x2 = linspace(0., 1., ne2 + 1)
e = zeros((ne1 + 1, ne2 + 1), order='F')
symbol_f90(x1, x2, t1, t2, e)
# ...
print('')
def test_3d_scalar_5():
print('============== test_3d_scalar_5 ================')
# ... define the weak formulation
x, y, z = symbols('x y z')
u = Symbol('u')
v = Symbol('v')
a = Lambda((x, y, z, v, u),
dx(dx(u)) * dx(dx(v)) + dy(dy(u)) * dy(dy(v)) +
dz(dz(u)) * dz(dz(v)) + Dot(Grad(u), Grad(v)) + u * v)
# ...
# ... create a finite element space
p1 = 2
p2 = 2
p3 = 2
ne1 = 2
ne2 = 2
ne3 = 2
# ...
print('> Grid :: [{},{},{}]'.format(ne1, ne2, ne3))
print('> Degree :: [{},{},{}]'.format(p1, p2, p3))
grid_1 = linspace(0., 1., ne1 + 1)
grid_2 = linspace(0., 1., ne2 + 1)
grid_3 = linspace(0., 1., ne3 + 1)
V1 = SplineSpace(p1, grid=grid_1, nderiv=2)
V2 = SplineSpace(p2, grid=grid_2, nderiv=2)
V3 = SplineSpace(p3, grid=grid_3, nderiv=2)
V = TensorFemSpace(V1, V2, V3)
# ...
# ...
kernel_py = compile_kernel('kernel_scalar_5', a, V, backend='python')
kernel_f90 = compile_kernel('kernel_scalar_5', a, V, backend='fortran')
M_py = assemble_matrix(V, kernel_py)
M_f90 = assemble_matrix(V, kernel_f90)
# ...
assert_identical_coo(M_py, M_f90)
def test_3d_block_1():
print('============== test_3d_block_1 ================')
x, y, z = symbols('x y z')
u = IndexedBase('u')
v = IndexedBase('v')
a = Lambda((x, y, z, v, u), Dot(Curl(u), Curl(v)) + 0.2 * Dot(u, v))
# ... create a finite element space
p1 = 2
p2 = 2
p3 = 2
ne1 = 2
ne2 = 2
ne3 = 2
print('> Grid :: [{},{},{}]'.format(ne1, ne2, ne3))
print('> Degree :: [{},{},{}]'.format(p1, p2, p3))
grid_1 = linspace(0., 1., ne1 + 1)
grid_2 = linspace(0., 1., ne2 + 1)
grid_3 = linspace(0., 1., ne3 + 1)
V1 = SplineSpace(p1, grid=grid_1)
V2 = SplineSpace(p2, grid=grid_2)
V3 = SplineSpace(p3, grid=grid_3)
W = TensorFemSpace(V1, V2, V3)
# ...
# ... vector space
V = VectorFemSpace(W, W, W)
# ...
# ...
kernel_py = compile_kernel('kernel_block_1', a, V, backend='python')
kernel_f90 = compile_kernel('kernel_block_1', a, V, backend='fortran')
M_py = assemble_matrix(V, kernel_py)
M_f90 = assemble_matrix(V, kernel_f90)
# ...
assert_identical_coo(M_py, M_f90)
def test_2d_block_1():
print('============== test_2d_block_1 ================')
# ... define the weak formulation
x, y = symbols('x y')
u = IndexedBase('u')
v = IndexedBase('v')
a = Lambda((x, y, v, u),
Rot(u) * Rot(v) + Div(u) * Div(v) + 0.2 * Dot(u, v))
# ...
# ... create a finite element space
p1 = 2
p2 = 2
ne1 = 8
ne2 = 8
print('> Grid :: [{ne1},{ne2}]'.format(ne1=ne1, ne2=ne2))
print('> Degree :: [{p1},{p2}]'.format(p1=p1, p2=p2))
grid_1 = linspace(0., 1., ne1 + 1)
grid_2 = linspace(0., 1., ne2 + 1)
V1 = SplineSpace(p1, grid=grid_1)
V2 = SplineSpace(p2, grid=grid_2)
W = TensorFemSpace(V1, V2)
# ...
# ... vector space
V = VectorFemSpace(W, W)
# ...
# ...
kernel_py = compile_kernel('kernel_block_1', a, V, backend='python')
kernel_f90 = compile_kernel('kernel_block_1', a, V, backend='fortran')
M_py = assemble_matrix(V, kernel_py)
M_f90 = assemble_matrix(V, kernel_f90)
# ...
assert_identical_coo(M_py, M_f90)
def test_2d_scalar_5():
print('============== test_2d_scalar_5 ================')
# ... define the weak formulation
x, y = symbols('x y')
u = Symbol('u')
v = Symbol('v')
F = Field('F')
a = Lambda((x, y, v, u), Dot(Grad(F * u), Grad(v)) + u * v)
# ...
# ... create a finite element space
p1 = 2
p2 = 2
ne1 = 8
ne2 = 8
print('> Grid :: [{ne1},{ne2}]'.format(ne1=ne1, ne2=ne2))
print('> Degree :: [{p1},{p2}]'.format(p1=p1, p2=p2))
grid_1 = linspace(0., 1., ne1 + 1)
grid_2 = linspace(0., 1., ne2 + 1)
V1 = SplineSpace(p1, grid=grid_1)
V2 = SplineSpace(p2, grid=grid_2)
V = TensorFemSpace(V1, V2)
# ...
F = Spline(V)
F.coeffs._data[:, :] = 1.
# ...
kernel_py = compile_kernel('kernel_scalar_5', a, V, backend='python')
kernel_f90 = compile_kernel('kernel_scalar_5', a, V, backend='fortran')
M_py = assemble_matrix(V, kernel_py, fields={'F': F})
M_f90 = assemble_matrix(V, kernel_f90, fields={'F': F})
# ...
assert_identical_coo(M_py, M_f90)
def test_2d_scalar_6():
print('============== test_2d_scalar_6 ================')
# ... define the weak formulation
x, y = symbols('x y')
u = Symbol('u')
v = Symbol('v')
a = Lambda((x, y, v, u),
dx(dx(u)) * dx(dx(v)) + dy(dy(u)) * dy(dy(v)) +
Dot(Grad(u), Grad(v)) + u * v)
# ...
# ... create a finite element space
p1 = 2
p2 = 2
ne1 = 8
ne2 = 8
print('> Grid :: [{ne1},{ne2}]'.format(ne1=ne1, ne2=ne2))
print('> Degree :: [{p1},{p2}]'.format(p1=p1, p2=p2))
grid_1 = linspace(0., 1., ne1 + 1)
grid_2 = linspace(0., 1., ne2 + 1)
V1 = SplineSpace(p1, grid=grid_1, nderiv=2)
V2 = SplineSpace(p2, grid=grid_2, nderiv=2)
V = TensorFemSpace(V1, V2)
# ...
# ...
kernel_py = compile_kernel('kernel_scalar_6', a, V, backend='python')
kernel_f90 = compile_kernel('kernel_scalar_6', a, V, backend='fortran')
M_py = assemble_matrix(V, kernel_py)
M_f90 = assemble_matrix(V, kernel_f90)
# ...
assert_identical_coo(M_py, M_f90)
def test_2d_scalar_4():
print('============== test_2d_scalar_4 ================')
# ... define the weak formulation
x, y = symbols('x y')
u = Symbol('u')
v = Symbol('v')
b0 = Function('b0')
b1 = Function('b1')
a = Lambda((x, y, v, u), (b0(x, y) * dx(v) + b1(x, y) * dy(v)) *
(b0(x, y) * dx(u) + b1(x, y) * dy(u)))
# ...
# ... create a finite element space
p1 = 2
p2 = 2
ne1 = 8
ne2 = 8
print('> Grid :: [{ne1},{ne2}]'.format(ne1=ne1, ne2=ne2))
print('> Degree :: [{p1},{p2}]'.format(p1=p1, p2=p2))
grid_1 = linspace(0., 1., ne1 + 1)
grid_2 = linspace(0., 1., ne2 + 1)
V1 = SplineSpace(p1, grid=grid_1)
V2 = SplineSpace(p2, grid=grid_2)
V = TensorFemSpace(V1, V2)
# ...
# ... user defined function
def b0(x, y):
from numpy import sin
from scipy import pi
r = 1.1659397624413860850012270020670 * (1.0 + 0.1 * sin(2 * pi * y))
return r
def b1(x, y):
from numpy import sin
from scipy import pi
r = 1.0 * (1.0 + 0.1 * sin(2 * pi * y))
return r
# ...
# ... create an interactive pyccel context
from pyccel.epyccel import ContextPyccel
context = ContextPyccel(name='context_4')
context.insert_function(b0, ['double', 'double'],
kind='function',
results=['double'])
context.insert_function(b1, ['double', 'double'],
kind='function',
results=['double'])
context.compile()
# ...
# ...
kernel_py = compile_kernel('kernel_scalar_4',
a,
V,
context=context,
verbose=True,
backend='python')
kernel_f90 = compile_kernel('kernel_scalar_4',
a,
V,
context=context,
verbose=True,
backend='fortran')
# ...
# ...
M_py = assemble_matrix(V, kernel_py)
M_f90 = assemble_matrix(V, kernel_f90)
# ...
assert_identical_coo(M_py, M_f90)
def test_2d_scalar_3():
print('============== test_2d_scalar_3 ================')
# ... define the weak formulation
x, y = symbols('x y')
u = Symbol('u')
v = Symbol('v')
b = Function('b')
a = Lambda((x, y, v, u), Dot(Grad(u), Grad(v)) + b(x, y) * u * v)
# ...
# ... create a finite element space
p1 = 2
p2 = 2
ne1 = 8
ne2 = 8
print('> Grid :: [{ne1},{ne2}]'.format(ne1=ne1, ne2=ne2))
print('> Degree :: [{p1},{p2}]'.format(p1=p1, p2=p2))
grid_1 = linspace(0., 1., ne1 + 1)
grid_2 = linspace(0., 1., ne2 + 1)
V1 = SplineSpace(p1, grid=grid_1)
V2 = SplineSpace(p2, grid=grid_2)
V = TensorFemSpace(V1, V2)
# ...
# ... user defined function
def b(x, y):
r = 1. + x * (1. - x) + y * (1. - y)
return r
# ...
# ... create an interactive pyccel context
from pyccel.epyccel import ContextPyccel
context = ContextPyccel(name='context_3')
context.insert_function(b, ['double', 'double'],
kind='function',
results=['double'])
context.compile()
# ...
# ...
kernel_py = compile_kernel('kernel_scalar_3',
a,
V,
context=context,
verbose=True,
backend='python')
kernel_f90 = compile_kernel('kernel_scalar_3',
a,
V,
context=context,
verbose=True,
backend='fortran')
# ...
# ...
M_py = assemble_matrix(V, kernel_py)
M_f90 = assemble_matrix(V, kernel_f90)
# ...
assert_identical_coo(M_py, M_f90)
# ... numbers of elements and degres
p1 = 2
p2 = 2
ne1 = 32
ne2 = 32
# ...
comm = MPI.COMM_WORLD
rank = comm.Get_rank()
if rank == 0:
print('> Grid :: [{ne1},{ne2}]'.format(ne1=ne1, ne2=ne2))
print('> Degree :: [{p1},{p2}]'.format(p1=p1, p2=p2))
grid_1 = linspace(0., 1., ne1 + 1)
grid_2 = linspace(0., 1., ne2 + 1)
V1 = SplineSpace(p1, grid=grid_1)
V2 = SplineSpace(p2, grid=grid_2)
V = TensorFemSpace(V1, V2, comm=comm)
# ...
wt = MPI.Wtime()
assembly(V, kernel)
wt = MPI.Wtime() - wt
print('rank: ', rank, '> Elapsed time: {}'.format(wt))
# ...
def test_2d_block_2():
print('============== test_2d_block_2 ================')
x, y = symbols('x y')
u = Symbol('u')
v = Symbol('v')
epsilon = Constant('epsilon')
Laplace = lambda v, u: Dot(Grad(v), Grad(u))
Mass = lambda v, u: v * u
u1, u2, p = symbols('u1 u2 p')
v1, v2, q = symbols('v1 v2 q')
a = Lambda((x, y, v1, v2, q, u1, u2, p),
Laplace(v1, u1) - dx(v1) * p + Laplace(v2, u2) - dy(v2) * p +
q * (dx(u1) + dy(u2)) + epsilon * Mass(q, p))
print('> input := {0}'.format(a))
# ... create a finite element space
p1 = 2
p2 = 2
ne1 = 8
ne2 = 8
print('> Grid :: [{ne1},{ne2}]'.format(ne1=ne1, ne2=ne2))
print('> Degree :: [{p1},{p2}]'.format(p1=p1, p2=p2))
grid_1 = linspace(0., 1., ne1 + 1)
grid_2 = linspace(0., 1., ne2 + 1)
V1 = SplineSpace(p1, grid=grid_1)
V2 = SplineSpace(p2, grid=grid_2)
V = TensorFemSpace(V1, V2)
V = VectorFemSpace(V, V, V)
# ...
# ... create a glt symbol from a string without evaluation
expr = glt_symbol(a, space=V)
print('> glt symbol := {0}'.format(expr))
# ...
# TODO not working yet => need complex numbers
# # ...
# symbol_f90 = compile_symbol('symbol_block_2', a, V,
# d_constants={'epsilon': 0.1},
# backend='fortran')
# # ...
#
# # ... example of symbol evaluation
# t1 = linspace(-pi,pi, ne1+1)
# t2 = linspace(-pi,pi, ne2+1)
# x1 = linspace(0.,1., ne1+1)
# x2 = linspace(0.,1., ne2+1)
# e = zeros((2, 2, ne1+1, ne2+1), order='F')
# symbol_f90(x1,x2,t1,t2, e)
# # ...
print('')
def test_2d_scalar_3():
print('============== test_2d_scalar_3 ================')
x, y = symbols('x y')
u = Symbol('u')
v = Symbol('v')
b = Function('b')
a = Lambda((x, y, v, u), Dot(Grad(u), Grad(v)) + b(x, y) * u * v)
print('> input := {0}'.format(a))
# ... create a finite element space
p1 = 2
p2 = 2
ne1 = 8
ne2 = 8
print('> Grid :: [{ne1},{ne2}]'.format(ne1=ne1, ne2=ne2))
print('> Degree :: [{p1},{p2}]'.format(p1=p1, p2=p2))
grid_1 = linspace(0., 1., ne1 + 1)
grid_2 = linspace(0., 1., ne2 + 1)
V1 = SplineSpace(p1, grid=grid_1)
V2 = SplineSpace(p2, grid=grid_2)
V = TensorFemSpace(V1, V2)
# ...
# ... create a glt symbol from a string without evaluation
expr = glt_symbol(a, space=V)
print('> glt symbol := {0}'.format(expr))
# ...
# ... user defined function
def b(x, y):
r = 1. + x * y
return r
# ...
# ... create an interactive pyccel context
from pyccel.epyccel import ContextPyccel
context = ContextPyccel(name='context_scalar_3')
context.insert_function(b, ['double', 'double'],
kind='function',
results=['double'])
context.compile()
# ...
# ...
symbol_f90 = compile_symbol('symbol_scalar_3',
a,
V,
context=context,
backend='fortran')
# ...
# ... example of symbol evaluation
t1 = linspace(-pi, pi, ne1 + 1)
t2 = linspace(-pi, pi, ne2 + 1)
x1 = linspace(0., 1., ne1 + 1)
x2 = linspace(0., 1., ne2 + 1)
e = zeros((ne1 + 1, ne2 + 1), order='F')
symbol_f90(x1, x2, t1, t2, e)
# ...
print('')
def test_3d_scalar_2():
print('============== test_3d_scalar_2 ================')
x, y, z = symbols('x y z')
u = Symbol('u')
v = Symbol('v')
c = Constant('c')
b0 = Constant('b0')
b1 = Constant('b1')
b2 = Constant('b2')
b = Tuple(b0, b1, b2)
a = Lambda((x, y, z, v, u),
c * u * v + Dot(b, Grad(v)) * u + Dot(b, Grad(u)) * v)
print('> input := {0}'.format(a))
# ... create a finite element space
p1 = 2
p2 = 2
p3 = 2
ne1 = 2
ne2 = 2
ne3 = 2
# ...
print('> Grid :: [{},{},{}]'.format(ne1, ne2, ne3))
print('> Degree :: [{},{},{}]'.format(p1, p2, p3))
grid_1 = linspace(0., 1., ne1 + 1)
grid_2 = linspace(0., 1., ne2 + 1)
grid_3 = linspace(0., 1., ne3 + 1)
V1 = SplineSpace(p1, grid=grid_1)
V2 = SplineSpace(p2, grid=grid_2)
V3 = SplineSpace(p3, grid=grid_3)
V = TensorFemSpace(V1, V2, V3)
# ...
# ... create a glt symbol from a string without evaluation
expr = glt_symbol(a, space=V)
print('> glt symbol := {0}'.format(expr))
# ...
# ...
symbol_f90 = compile_symbol('symbol_scalar_2',
a,
V,
d_constants={
'b0': 0.1,
'b1': 1.,
'b2': 1.,
'c': 0.2
},
backend='fortran')
# ...
# ... example of symbol evaluation
t1 = linspace(-pi, pi, ne1 + 1)
t2 = linspace(-pi, pi, ne2 + 1)
t3 = linspace(-pi, pi, ne3 + 1)
x1 = linspace(0., 1., ne1 + 1)
x2 = linspace(0., 1., ne2 + 1)
x3 = linspace(0., 1., ne3 + 1)
e = zeros((ne1 + 1, ne2 + 1, ne3 + 1), order='F')
symbol_f90(x1, x2, x3, t1, t2, t3, e)
# ...
print('')
def test_3d_block_4():
print('============== test_3d_block_4 ================')
"""Alfven operator."""
x, y, z = symbols('x y z')
u = IndexedBase('u')
v = IndexedBase('v')
bx = Constant('bx')
by = Constant('by')
bz = Constant('bz')
b = Tuple(bx, by, bz)
c0 = Constant('c0')
c1 = Constant('c1')
c2 = Constant('c2')
a = Lambda((x, y, z, v, u),
(c0 * Dot(u, v) + c1 * Div(u) * Div(v) +
c2 * Dot(Curl(Cross(b, u)), Curl(Cross(b, v)))))
print('> input := {0}'.format(a))
# ... create a finite element space
p1 = 2
p2 = 2
p3 = 2
ne1 = 2
ne2 = 2
ne3 = 2
# ...
print('> Grid :: [{},{},{}]'.format(ne1, ne2, ne3))
print('> Degree :: [{},{},{}]'.format(p1, p2, p3))
grid_1 = linspace(0., 1., ne1 + 1)
grid_2 = linspace(0., 1., ne2 + 1)
grid_3 = linspace(0., 1., ne3 + 1)
V1 = SplineSpace(p1, grid=grid_1)
V2 = SplineSpace(p2, grid=grid_2)
V3 = SplineSpace(p3, grid=grid_3)
Vx = TensorFemSpace(V1, V2, V3)
Vy = TensorFemSpace(V1, V2, V3)
Vz = TensorFemSpace(V1, V2, V3)
V = VectorFemSpace(Vx, Vy, Vz)
# ...
# ... create a glt symbol from a string without evaluation
expr = glt_symbol(a, space=V)
print('> glt symbol := {0}'.format(expr))
# ...
# ...
symbol_f90 = compile_symbol('symbol_block_4',
a,
V,
d_constants={
'bx': 0.1,
'by': 1.,
'bz': 0.2,
'c0': 0.1,
'c1': 1.,
'c2': 1.
},
backend='fortran')
# ...
# ... example of symbol evaluation
t1 = linspace(-pi, pi, ne1 + 1)
t2 = linspace(-pi, pi, ne2 + 1)
t3 = linspace(-pi, pi, ne3 + 1)
x1 = linspace(0., 1., ne1 + 1)
x2 = linspace(0., 1., ne2 + 1)
x3 = linspace(0., 1., ne3 + 1)
e = zeros((3, 3, ne1 + 1, ne2 + 1, ne3 + 1), order='F')
symbol_f90(x1, x2, x3, t1, t2, t3, e)
# ...
print('')
def test_3d_block_3():
print('============== test_3d_block_3 ================')
x, y, z = symbols('x y z')
u = IndexedBase('u')
v = IndexedBase('v')
b = Tuple(1.0, 0., 0.)
a = Lambda((x, y, z, v, u),
Dot(Curl(Cross(b, u)), Curl(Cross(b, v))) + 0.2 * Dot(u, v))
print('> input := {0}'.format(a))
# ... create a finite element space
p1 = 2
p2 = 2
p3 = 2
ne1 = 2
ne2 = 2
ne3 = 2
# ...
print('> Grid :: [{},{},{}]'.format(ne1, ne2, ne3))
print('> Degree :: [{},{},{}]'.format(p1, p2, p3))
grid_1 = linspace(0., 1., ne1 + 1)
grid_2 = linspace(0., 1., ne2 + 1)
grid_3 = linspace(0., 1., ne3 + 1)
V1 = SplineSpace(p1, grid=grid_1)
V2 = SplineSpace(p2, grid=grid_2)
V3 = SplineSpace(p3, grid=grid_3)
Vx = TensorFemSpace(V1, V2, V3)
Vy = TensorFemSpace(V1, V2, V3)
Vz = TensorFemSpace(V1, V2, V3)
V = VectorFemSpace(Vx, Vy, Vz)
# ...
# ... create a glt symbol from a string without evaluation
expr = glt_symbol(a, space=V)
print('> glt symbol := {0}'.format(expr))
# ...
# ...
symbol_f90 = compile_symbol('symbol_block_3', a, V, backend='fortran')
# ...
# ... example of symbol evaluation
t1 = linspace(-pi, pi, ne1 + 1)
t2 = linspace(-pi, pi, ne2 + 1)
t3 = linspace(-pi, pi, ne3 + 1)
x1 = linspace(0., 1., ne1 + 1)
x2 = linspace(0., 1., ne2 + 1)
x3 = linspace(0., 1., ne3 + 1)
e = zeros((3, 3, ne1 + 1, ne2 + 1, ne3 + 1), order='F')
symbol_f90(x1, x2, x3, t1, t2, t3, e)
# ...
print('')
def test_pdes_3d_2():
print('============ test_pdes_3d_2 =============')
# ... abstract model
V = H1Space('V', ldim=3)
v = TestFunction(V, name='v')
u = TestFunction(V, name='u')
c = Constant('c', real=True, label='mass stabilization')
a = BilinearForm((v, u), dot(grad(v), grad(u)) + c * v * u)
# ...
# ... discretization
# Input data: degree, number of elements
p1 = 1
p2 = 1
p3 = 1
ne1 = 4
ne2 = 4
ne3 = 4
# Create uniform grid
grid_1 = linspace(0., 1., num=ne1 + 1)
grid_2 = linspace(0., 1., num=ne2 + 1)
grid_3 = linspace(0., 1., num=ne3 + 1)
# Create 1D finite element spaces and precompute quadrature data
V1 = SplineSpace(p1, grid=grid_1)
V1.init_fem()
V2 = SplineSpace(p2, grid=grid_2)
V2.init_fem()
V3 = SplineSpace(p3, grid=grid_3)
V3.init_fem()
# Create 2D tensor product finite element space
V = TensorFemSpace(V1, V2, V3)
# ...
# ...
discretize_symbol(a, [V, V])
# print(mass.symbol.__doc__)
# ...
# ...
n1 = 21
n2 = 21
n3 = 21
t1 = linspace(-pi, pi, n1)
t2 = linspace(-pi, pi, n2)
t3 = linspace(-pi, pi, n3)
x1 = linspace(0., 1., n1)
x2 = linspace(0., 1., n2)
x3 = linspace(0., 1., n3)
xs = [x1, x2, x3]
ts = [t1, t2, t3]
e = a.symbol(*xs, *ts, 0.25)
print('> c = 0.25 :: ', e.min(), e.max())
e = a.symbol(*xs, *ts, 0.6)
print('> c = 0.6 :: ', e.min(), e.max())