def test_even(self):
print('\nChecking Tensors with even grid points...')
for dim, n, fft_form in itertools.product([2, 3], [4, 5], fft_forms):
msg = 'Tensors with: dim={}, n={}, fft_form={}'.format(
dim, n, fft_form)
N = dim * (n, )
M = tuple(2 * np.array(N))
u = Tensor(name='test',
shape=(),
N=N,
Fourier=False,
fft_form=fft_form)
u.randomize()
Fu = u.fourier(copy=True)
FuM = Fu.project(M)
uM = FuM.fourier()
if n % 2 == 0:
self.assertGreaterEqual(u.norm(), FuM.norm(), msg=msg)
self.assertGreaterEqual(u.norm(componentwise=True),
FuM.norm(componentwise=True),
msg=msg)
self.assertGreaterEqual(u.norm(), uM.norm(), msg=msg)
self.assertGreaterEqual(u.norm(componentwise=True),
uM.norm(componentwise=True),
msg=msg)
else:
self.assertAlmostEqual(u.norm(), FuM.norm(), msg=msg)
self.assertAlmostEqual(u.norm(componentwise=True),
FuM.norm(componentwise=True),
msg=msg)
self.assertAlmostEqual(u.norm(), uM.norm(), msg=msg)
self.assertAlmostEqual(u.norm(componentwise=True),
uM.norm(componentwise=True),
msg=msg)
self.assertAlmostEqual(0, u.mean() - FuM.mean(), msg=msg)
self.assertAlmostEqual(u.mean(), uM.mean(), msg=msg)
# testing that that interpolation on double grid have exactly the same values
slc = tuple(u.order * [
slice(None),
] + [slice(0, M[i], 2) for i in range(dim)])
self.assertAlmostEqual(0,
np.linalg.norm(u.val - uM.val[slc]),
msg=msg)
print('...ok')
def test_even(self):
print('\nChecking Tensors with even grid points...')
for dim, n, fft_form in itertools.product([2,3], [4,5], fft_forms):
msg='Tensors with: dim={}, n={}, fft_form={}'.format(dim, n, fft_form)
N=dim*(n,)
M=tuple(2*np.array(N))
u=Tensor(name='test', shape=(), N=N, Fourier=False, fft_form=fft_form)
u.randomize()
Fu=u.fourier(copy=True)
FuM=Fu.project(M)
uM=FuM.fourier()
if n%2 == 0:
self.assertGreaterEqual(u.norm(), FuM.norm(), msg=msg)
self.assertGreaterEqual(u.norm(componentwise=True), FuM.norm(componentwise=True),
msg=msg)
self.assertGreaterEqual(u.norm(), uM.norm(), msg=msg)
self.assertGreaterEqual(u.norm(componentwise=True), uM.norm(componentwise=True),
msg=msg)
else:
self.assertAlmostEqual(u.norm(), FuM.norm(), msg=msg)
self.assertAlmostEqual(u.norm(componentwise=True), FuM.norm(componentwise=True),
msg=msg)
self.assertAlmostEqual(u.norm(), uM.norm(), msg=msg)
self.assertAlmostEqual(u.norm(componentwise=True), uM.norm(componentwise=True),
msg=msg)
self.assertAlmostEqual(0, u.mean()-FuM.mean(), msg=msg)
self.assertAlmostEqual(u.mean(), uM.mean(), msg=msg)
# testing that that interpolation on double grid have exactly the same values
slc=tuple(u.order*[slice(None),]+[slice(0,M[i],2) for i in range(dim)])
self.assertAlmostEqual(0, np.linalg.norm(u.val-uM.val[slc]), msg=msg)
print('...ok')
with representation in Fourier domain (with Fourier coefficients);
FxN = FN*xN = FN(xN) =""")
FxN = FN*xN # Fourier coefficients of xN
print(FxN)
print("""
The forward and inverse DFT are mutually inverse operations that can
be observed by calculation of variable 'xN2':
xN2 = FiN(FxN) = FiN(FN(xN)) =""")
xN2 = FiN(FxN) # values of trigonometric polynomial at grid points
print(xN2)
print("and its comparison with initial trigonometric polynomial 'xN2'")
print("(xN == xN2) = ")
print(xN == xN2)
print("""
The norm of trigonometric polynomial calculated from Fourier
coefficients corresponds to L^2 norm and is the same like for values at grid
points, which is a consequence of Parseval's identity:
xN.norm() = np.linalg.norm(xN.val)/np.prod(xN.N)**0.5 =
= (np.sum(xN.val*xN.val)/np.prod(xN.N))**0.5 = """)
print(xN.norm())
print("""FxN.norm() = np.linalg.norm(FxN.val) =
= np.sum(FxN.val*np.conj(FxN.val)).real**0.5 =""")
print(FxN.norm())
print("""
The trigonometric polynomials can be also multiplied. The standard
multiplication with '*' operations corresponds to scalar product
leading to a square of norm, i.e.
FxN.norm() = xN.norm() = (xN*xN)**0.5 = (FxN*FxN)**0.5 =""")
with representation in Fourier domain (with Fourier coefficients);
FxN = FN*xN = FN(xN) =""")
FxN = FN * xN # Fourier coefficients of xN
print(FxN)
print("""
The forward and inverse DFT are mutually inverse operations that can
be observed by calculation of variable 'xN2':
xN2 = FiN(FxN) = FiN(FN(xN)) =""")
xN2 = FiN(FxN) # values of trigonometric polynomial at grid points
print(xN2)
print("and its comparison with initial trigonometric polynomial 'xN2'")
print("(xN == xN2) = ")
print(xN == xN2)
print("""
The norm of trigonometric polynomial calculated from Fourier
coefficients corresponds to L^2 norm and is the same like for values at grid
points, which is a consequence of Parseval's identity:
xN.norm() = np.linalg.norm(xN.val)/np.prod(xN.N)**0.5 =
= (np.sum(xN.val*xN.val)/np.prod(xN.N))**0.5 = """)
print(xN.norm())
print("""FxN.norm() = np.linalg.norm(FxN.val) =
= np.sum(FxN.val*np.conj(FxN.val)).real**0.5 =""")
print(FxN.norm())
print("""
The trigonometric polynomials can be also multiplied. The standard
multiplication with '*' operations corresponds to scalar product
leading to a square of norm, i.e.
FxN.norm() = xN.norm() = (xN*xN)**0.5 = (FxN*FxN)**0.5 =""")