def test_build_sets():
"""
Tests whether the Gaussian quadrature method is properly storing
the orders that have already been computed
"""
x, w = gaussian_quad.gaussxw(3)
x2, w2 = gaussian_quad.gaussxw(12)
np.testing.assert_almost_equal(x, gaussian_quad.x_set[3])
np.testing.assert_almost_equal(w, gaussian_quad.w_set[3])
np.testing.assert_almost_equal(x2, gaussian_quad.x_set[12])
np.testing.assert_almost_equal(w2, gaussian_quad.w_set[12])
def test_build_sets():
"""
Tests whether the Gaussian quadrature method is properly storing
the orders that have already been computed
"""
x, w = gaussian_quad.gaussxw(3)
x2, w2 = gaussian_quad.gaussxw(12)
np.testing.assert_almost_equal(x, gaussian_quad.x_set[3])
np.testing.assert_almost_equal(w, gaussian_quad.w_set[3])
np.testing.assert_almost_equal(x2, gaussian_quad.x_set[12])
np.testing.assert_almost_equal(w2, gaussian_quad.w_set[12])
def test_interp_telles():
# The problem I solve:
# exact is from mathematica
sing_pt = 1.005
denom = lambda x: (sing_pt - x)**2
numer = lambda x: x**3
f = lambda x: numer(x) / denom(x)
exact_f = lambda s: (s * (-4 + 6 * s ** 2 - 3 * s * (-1 + s ** 2) * \
(log((-1 - s) / (1 - s))))) / (-1 + s ** 2)
exact = exact_f(sing_pt)
# Solved with standard Telles quadrature
x_nearest = 1.0
D = sing_pt - 1.0
N = 14
[tx, tw] = telles_quasi_singular(N, x_nearest, D)
est_telles = np.sum(f(tx) * tw)
# Solved with gauss quadrature
[gx, gw] = gaussxw(N)
est_gauss = np.sum(f(gx) * gw)
# Solved with interpolation and Telles quadrature
# X = Interpolation Points
X = gx
# Y = Value of function at the interpolation points
Y = f(gx) * denom(gx)
# WARNING, WARNING, WARNING: This implementation of lagrange interpolation
# is super unstable. I just downloaded it from somewhere online. A
# reimplementation using barycentric Lagrange interpolation is necessary to
# go above N = appx 20.
P = spi.lagrange(X, Y)
est_interp_telles = sum(P(tx) / denom(tx) * tw)
np.testing.assert_almost_equal(est_telles, est_interp_telles)
def test_interp_telles():
# The problem I solve:
# exact is from mathematica
sing_pt = 1.005;
denom = lambda x: (sing_pt - x) ** 2;
numer = lambda x: x ** 3;
f = lambda x: numer(x) / denom(x);
exact_f = lambda s: (s * (-4 + 6 * s ** 2 - 3 * s * (-1 + s ** 2) * \
(log((-1 - s) / (1 - s))))) / (-1 + s ** 2)
exact = exact_f(sing_pt)
# Solved with standard Telles quadrature
x_nearest = 1.0;
D = sing_pt - 1.0;
N = 14;
[tx, tw] = telles_quasi_singular(N, x_nearest, D);
est_telles = np.sum(f(tx) * tw)
# Solved with gauss quadrature
[gx, gw] = gaussxw(N)
est_gauss = np.sum(f(gx) * gw)
# Solved with interpolation and Telles quadrature
# X = Interpolation Points
X = gx;
# Y = Value of function at the interpolation points
Y = f(gx) * denom(gx);
# WARNING, WARNING, WARNING: This implementation of lagrange interpolation
# is super unstable. I just downloaded it from somewhere online. A
# reimplementation using barycentric Lagrange interpolation is necessary to
# go above N = appx 20.
P = spi.lagrange(X, Y)
est_interp_telles = sum(P(tx) / denom(tx) * tw)
np.testing.assert_almost_equal(est_telles, est_interp_telles)
def test_QuadGauss():
f = lambda x: 3 * x ** 2
F = lambda x: x ** 3
exact = F(1) - F(0)
x, w = gaussian_quad.gaussxw(2)
x, w = map_pts_wts(x, w, 0, 1)
est = np.sum(f(x) * w)
np.testing.assert_almost_equal(exact, est)
def test_QuadGauss():
f = lambda x: 3 * x**2
F = lambda x: x**3
exact = F(1) - F(0)
x, w = gaussian_quad.gaussxw(2)
x, w = map_pts_wts(x, w, 0, 1)
est = np.sum(f(x) * w)
np.testing.assert_almost_equal(exact, est)
def test_anotherLogRDouble_G11():
f = lambda x, y: (1 / (3 * np.pi)) *\
np.log(1.0 / np.abs(x - y)) * x * y
exact = 1 / (3 * np.pi)
gx, gw = gaussxw(75)
sum = 0.0
for (pt, wt) in zip(gx, gw):
x, w = telles_singular(76, pt)
g = lambda x: f(x, pt)
sum += np.sum(g(x) * w * wt)
np.testing.assert_almost_equal(exact, sum, 5)
def test_gauss():
"""
Exact values retrieved from the wikipedia page on Gaussian Quadrature
The main function has been tested by the original author for a wide
range of orders. But, this is just to check everything is still working
properly
"""
x, w = gaussian_quad.gaussxw(3)
x, w = map_pts_wts(x, w, -1.0, 1.0)
np.testing.assert_almost_equal(x[0], sqrt(3.0 / 5.0))
np.testing.assert_almost_equal(x[1], 0.0)
np.testing.assert_almost_equal(x[2], -sqrt(3.0 / 5.0))
np.testing.assert_almost_equal(w[0], (5.0 / 9.0))
np.testing.assert_almost_equal(w[1], (8.0 / 9.0))
np.testing.assert_almost_equal(w[2], (5.0 / 9.0))
def test_gauss():
"""
Exact values retrieved from the wikipedia page on Gaussian Quadrature
The main function has been tested by the original author for a wide
range of orders. But, this is just to check everything is still working
properly
"""
x, w = gaussian_quad.gaussxw(3)
x, w = map_pts_wts(x, w, -1.0, 1.0)
np.testing.assert_almost_equal(x[0], sqrt(3.0 / 5.0))
np.testing.assert_almost_equal(x[1], 0.0)
np.testing.assert_almost_equal(x[2], -sqrt(3.0 / 5.0))
np.testing.assert_almost_equal(w[0], (5.0 / 9.0))
np.testing.assert_almost_equal(w[1], (8.0 / 9.0))
np.testing.assert_almost_equal(w[2], (5.0 / 9.0))
def test_quasi_singular():
#test telles quasi-singular quadrature with the example on page 966
# sing_pt = 1.08
sing_pt = 1.004
g = lambda x: (sing_pt - x) ** -2
# for sing_pt = 1.08
# exact = 12.0192
# for sing_pt = 1.004
exact = 249.500998
x_nearest = 1.0
D = sing_pt - 1.0
N = 20
[tx, tw] = telles_quasi_singular(N, x_nearest, D)
est_telles = np.sum(g(tx) * tw)
[gx, gw] = gaussxw(N)
est_gauss = np.sum(g(gx) * gw)
gauss_error = abs(est_gauss - exact) / exact * 100
telles_error = abs(est_telles - exact) / exact * 100
assert(telles_error < gauss_error / 1000)