def test_low_grid_basics(npoint):
grid = LegendreGrid(npoint)
assert grid.basis.shape == (npoint, npoint)
assert grid.basis_inv.shape == (npoint, npoint)
fn1 = np.random.uniform(0, 1, npoint)
coeffs = grid.tocoeffs(fn1)
fn2 = grid.tofnvals(coeffs)
assert_allclose(fn1, fn2, atol=1e-14)
fn3 = legval(grid.points, coeffs)
assert_allclose(fn1, fn3, atol=1e-14)
def test_low_grid_sin():
npoint = 101
grid = LegendreGrid(npoint)
assert_allclose(grid.points[npoint // 2], 0.0, atol=1e-10)
fnvals = np.cos(grid.points)
fnvalsa = grid.antiderivative(fnvals)
fnvalsa -= fnvalsa[npoint // 2]
fnvalsd = grid.derivative(fnvals)
assert_allclose(grid.integrate(fnvals),
np.sin(1) - np.sin(-1),
atol=1e-14,
rtol=0)
assert_allclose(fnvalsa, np.sin(grid.points), atol=1e-14, rtol=0)
assert_allclose(fnvalsd, -np.sin(grid.points), atol=1e-10, rtol=0)
def test_low_grid_exp():
npoint = 101
grid = LegendreGrid(npoint)
assert_allclose(grid.points[npoint // 2], 0.0, atol=1e-10)
fnvals = np.exp(grid.points)
fnvalsa = grid.antiderivative(fnvals)
fnvalsa += 1 - fnvalsa[npoint // 2]
fnvalsd = grid.derivative(fnvals)
assert_allclose(grid.integrate(fnvals),
np.exp(1) - np.exp(-1),
atol=1e-14,
rtol=0)
assert_allclose(fnvalsa, fnvals, atol=1e-14, rtol=0)
assert_allclose(fnvalsd, fnvals, atol=1e-10, rtol=0)
def test_low_grid_basics_vectorized():
shape = (5, 4, 8)
npoint = 7
extshape = shape + (npoint, )
grid = LegendreGrid(npoint)
fn1 = np.random.uniform(0, 1, extshape)
coeffs = grid.tocoeffs(fn1)
assert coeffs.shape == extshape
fn2 = grid.tofnvals(coeffs)
assert fn2.shape == extshape
assert_allclose(fn1, fn2, atol=1e-14)
# For legval, the first index of the coefficients array should be for
# different polynomial orders. It returns function values at grid points,
# for which the last index is used.
fn3 = legval(grid.points, coeffs.transpose(3, 0, 1, 2))
assert fn3.shape == extshape
assert_allclose(fn1, fn3, atol=1e-14)
def test_low_grid_sin_vectorized():
npoint = 101
grid = LegendreGrid(npoint)
assert_allclose(grid.points[npoint // 2], 0.0, atol=1e-10)
fnsvals = np.cos(np.outer([1.0, 0.5, 0.2], grid.points))
fnsvalsa = grid.antiderivative(fnsvals)
fnsvalsa -= fnsvalsa[:, npoint // 2, np.newaxis]
fnsvalsd = grid.derivative(fnsvals)
integrals = [
np.sin(1) - np.sin(-1),
2 * np.sin(0.5) - 2 * np.sin(-0.5),
5 * np.sin(0.2) - 5 * np.sin(-0.2),
]
assert_allclose(grid.integrate(fnsvals), integrals, atol=1e-14, rtol=0)
antiderivatives = [
np.sin(grid.points),
2 * np.sin(0.5 * grid.points),
5 * np.sin(0.2 * grid.points),
]
assert_allclose(fnsvalsa, antiderivatives, atol=1e-14, rtol=0)
derivatives = [
-np.sin(grid.points),
-0.5 * np.sin(0.5 * grid.points),
-0.2 * np.sin(0.2 * grid.points),
]
assert_allclose(fnsvalsd, derivatives, atol=1e-10, rtol=0)
fns_other = np.cos(np.outer([1.1, 1.2, 0.8], grid.points))
integrals2 = np.array([[grid.integrate(fn1 * fn2) for fn2 in fns_other]
for fn1 in fnsvals])
assert_allclose(grid.integrate(fnsvals, fns_other),
integrals2,
atol=1e-14,
rtol=0)