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_QuadOneOverR_1():
f = lambda x: 1 / (x - 0.4)
exact = np.log(3.0 / 2.0)
mapped_x0 = map_singular_pt(0.4, 0.0, 1.0)
x, w = piessens(2, mapped_x0, nonsingular_N=10)
qx, qw = map_pts_wts(x, w, 0.0, 1.0)
est = np.sum(f(qx) * qw)
np.testing.assert_almost_equal(exact, est)
def test_quadlogr3():
f = lambda x: x ** 2 * np.log(np.abs(x - 3.9))
exact = -18.459
mapped_x0 = map_singular_pt(3.9, 3, 4)
tx, tw = telles_singular(50, mapped_x0)
tx, tw = map_pts_wts(tx, tw, 3, 4)
est = np.sum(f(tx) * tw)
np.testing.assert_almost_equal(exact, est, 4)
def test_QuadOneOverR_1():
f = lambda x: 1 / (x - 0.4)
exact = np.log(3.0 / 2.0)
mapped_x0 = map_singular_pt(0.4, 0.0, 1.0)
x, w = piessens(2, mapped_x0, nonsingular_N = 10)
qx, qw = map_pts_wts(x, w, 0.0, 1.0)
est = np.sum(f(qx) * qw)
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_QuadOneOverR_4():
f = lambda x: np.exp(x) / (x - 0.8)
exact = -1.13761642399
mapped_x0 = map_singular_pt(0.8, 0.0, 1.0)
x, w = piessens(2, mapped_x0, nonsingular_N = 20)
qx, qw = map_pts_wts(x, w, 0.0, 1.0)
est = np.sum(f(qx) * qw)
np.testing.assert_almost_equal(exact, est)
def test_QuadOneOverR_4():
f = lambda x: np.exp(x) / (x - 0.8)
exact = -1.13761642399
mapped_x0 = map_singular_pt(0.8, 0.0, 1.0)
x, w = piessens(2, mapped_x0, nonsingular_N=20)
qx, qw = map_pts_wts(x, w, 0.0, 1.0)
est = np.sum(f(qx) * qw)
np.testing.assert_almost_equal(exact, est)
def test_lobatto():
f = lambda x: 12 * x ** 11
F = lambda x: x ** 12
exact = F(1) - F(0)
x, w = lobatto_quad(7)
x, w = map_pts_wts(x, w, 0, 1)
assert(len(x) == 7)
est = np.sum(f(x) * w)
np.testing.assert_almost_equal(exact, est)
def test_lobatto():
f = lambda x: 12 * x**11
F = lambda x: x**12
exact = F(1) - F(0)
x, w = lobatto_quad(7)
x, w = map_pts_wts(x, w, 0, 1)
assert (len(x) == 7)
est = np.sum(f(x) * w)
np.testing.assert_almost_equal(exact, est)
def test_quadlogr2():
f = lambda x: x ** 2 * np.log(np.abs(x - 0.9))
exact = -0.764714
mapped_x0 = map_singular_pt(0.9, 0, 1)
tx, tw = telles_singular(40, mapped_x0)
tx, tw = map_pts_wts(tx, tw, 0, 1)
est = np.sum(f(tx) * tw)
np.testing.assert_almost_equal(exact, est, 4)
def test_QuadLogR():
f = lambda x: np.log(np.abs(x - 0.5))
exact = -1.0 - np.log(2.0)
mapped_x0 = map_singular_pt(0.5, 0, 1)
tx, tw = telles_singular(50, mapped_x0)
tx, tw = map_pts_wts(tx, tw, 0, 1)
est = np.sum(f(tx) * tw)
np.testing.assert_almost_equal(exact, est, 4)
def test_piessens_4_5():
f = lambda x: np.exp(x - 4) / (x - 4.2)
exact = 3.139062607254266
mapped_x0 = map_singular_pt(4.2, 4.0, 5.0)
x, w = piessens(20, mapped_x0, nonsingular_N=50)
qx, qw = map_pts_wts(x, w, 4.0, 5.0)
est = np.sum(f(qx) * qw)
np.testing.assert_almost_equal(exact, est)
def test_QuadOneOverR_5():
f = lambda x: np.exp(x) / (x - 0.2)
exact = 3.139062607254266
mapped_x0 = map_singular_pt(0.2, 0.0, 1.0)
x, w = piessens(2, mapped_x0, nonsingular_N = 50)
qx, qw = map_pts_wts(x, w, 0.0, 1.0)
est = np.sum(f(qx) * qw)
np.testing.assert_almost_equal(exact, est)
def test_piessens_4_5():
f = lambda x: np.exp(x - 4) / (x - 4.2)
exact = 3.139062607254266
mapped_x0 = map_singular_pt(4.2, 4.0, 5.0)
x, w = piessens(20, mapped_x0, nonsingular_N = 50)
qx, qw = map_pts_wts(x, w, 4.0, 5.0)
est = np.sum(f(qx) * qw)
np.testing.assert_almost_equal(exact, est)
def test_QuadOneOverR_5():
f = lambda x: np.exp(x) / (x - 0.2)
exact = 3.139062607254266
mapped_x0 = map_singular_pt(0.2, 0.0, 1.0)
x, w = piessens(2, mapped_x0, nonsingular_N=50)
qx, qw = map_pts_wts(x, w, 0.0, 1.0)
est = np.sum(f(qx) * qw)
np.testing.assert_almost_equal(exact, est)
def test_QuadOneOverR_2():
# Example 1 from Piessens
g = lambda x: np.exp(x) / x
f = lambda x: 2 * g((2 * x) - 1)
exact = 2.11450175
mapped_x0 = map_singular_pt(0.5, 0.0, 1.0)
x, w = piessens(8, mapped_x0, nonsingular_N = 10)
qx, qw = map_pts_wts(x, w, 0.0, 1.0)
est = np.sum(f(qx) * qw)
np.testing.assert_almost_equal(exact, est)
def test_QuadOneOverR_2():
# Example 1 from Piessens
g = lambda x: np.exp(x) / x
f = lambda x: 2 * g((2 * x) - 1)
exact = 2.11450175
mapped_x0 = map_singular_pt(0.5, 0.0, 1.0)
x, w = piessens(8, mapped_x0, nonsingular_N=10)
qx, qw = map_pts_wts(x, w, 0.0, 1.0)
est = np.sum(f(qx) * qw)
np.testing.assert_almost_equal(exact, est)
def test_QuadOneOverR_3():
# Example 2 from Piessens
g = lambda x: np.exp(x) / (np.sin(x) - np.cos(x))
f = lambda x: np.pi / 2.0 * g(np.pi / 2.0 * x)
exact = 2.61398312
# Piessens estimate derived with a two pt rule.
piessens_est = 2.61398135
mapped_x0 = map_singular_pt(0.5, 0.0, 1.0)
x, w = piessens(2, mapped_x0)
qx, qw = map_pts_wts(x, w, 0.0, 1.0)
est = np.sum(f(qx) * qw)
np.testing.assert_almost_equal(piessens_est, est)
def test_QuadOneOverR_3():
# Example 2 from Piessens
g = lambda x: np.exp(x) / (np.sin(x) - np.cos(x))
f = lambda x: np.pi / 2.0 * g(np.pi / 2.0 * x)
exact = 2.61398312
# Piessens estimate derived with a two pt rule.
piessens_est = 2.61398135
mapped_x0 = map_singular_pt(0.5, 0.0, 1.0)
x, w = piessens(2, mapped_x0)
qx, qw = map_pts_wts(x, w, 0.0, 1.0)
est = np.sum(f(qx) * qw)
np.testing.assert_almost_equal(piessens_est, est)
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_mapped_recursive_quad2():
mapped_ay = map_singular_pt(0.2, 0.0, 1.0)
mapped_by = map_distance_to_interval(0.3, 0.0, 1.0)
moments = modified_moments(rl1, 10, mapped_ay, mapped_by)
x, w = recursive_quad(moments)
x, w = map_pts_wts(x, w, 0.0, 1.0)
w = map_weights_by_inv_power(w, 1.0, 0.0, 1.0)
exact = [2.332556553293539,
0.9603565576440610,
0.5636909785950152, \
0.3894662160471265,
0.2949533988361775,
0.2365588120191993, \
0.1971850086215070,
0.1689379319389373,
0.1477219765628722, \
0.1312177383160300]
for i in range(len(exact)):
est = np.sum(w * x ** i)
np.testing.assert_almost_equal(exact[i], est, 9)
def test_mapped_recursive_quad2():
mapped_ay = map_singular_pt(0.2, 0.0, 1.0)
mapped_by = map_distance_to_interval(0.3, 0.0, 1.0)
moments = modified_moments(rl1, 10, mapped_ay, mapped_by)
x, w = recursive_quad(moments)
x, w = map_pts_wts(x, w, 0.0, 1.0)
w = map_weights_by_inv_power(w, 1.0, 0.0, 1.0)
exact = [2.332556553293539,
0.9603565576440610,
0.5636909785950152, \
0.3894662160471265,
0.2949533988361775,
0.2365588120191993, \
0.1971850086215070,
0.1689379319389373,
0.1477219765628722, \
0.1312177383160300]
for i in range(len(exact)):
est = np.sum(w * x**i)
np.testing.assert_almost_equal(exact[i], est, 9)
def test_mapped_recursive_quad():
mapped_ay = map_singular_pt(0.5, 1.0, 2.0)
mapped_by = map_distance_to_interval(0.5, 1.0, 2.0)
moments = modified_moments(rl1, 10, mapped_ay, mapped_by)
x, w = recursive_quad(moments)
x, w = map_pts_wts(x, w, 1.0, 2.0)
w = map_weights_by_inv_power(w, 1.0, 1.0, 2.0)
exact = [0.9370728722124352,
1.342568485003826,
2.000243585190683, \
3.091829672374373,
4.940760583752816,
8.128175556459884, \
13.70706664710914,
23.59959784014733,
41.33761425072659, \
73.44841816566141]
for i in range(10):
est = np.sum(w * x ** i)
np.testing.assert_almost_equal(exact[i], est, 10)
def test_mapped_recursive_quad():
mapped_ay = map_singular_pt(0.5, 1.0, 2.0)
mapped_by = map_distance_to_interval(0.5, 1.0, 2.0)
moments = modified_moments(rl1, 10, mapped_ay, mapped_by)
x, w = recursive_quad(moments)
x, w = map_pts_wts(x, w, 1.0, 2.0)
w = map_weights_by_inv_power(w, 1.0, 1.0, 2.0)
exact = [0.9370728722124352,
1.342568485003826,
2.000243585190683, \
3.091829672374373,
4.940760583752816,
8.128175556459884, \
13.70706664710914,
23.59959784014733,
41.33761425072659, \
73.44841816566141]
for i in range(10):
est = np.sum(w * x**i)
np.testing.assert_almost_equal(exact[i], est, 10)