def test_l2_h1_2():
eps = 1e-9
func = lambda x: log(x)
mesh1 = Mesh1D((1, 1.5, 2, 2.5, e), (20, 20, 20, 20))
f = Function(func, mesh1)
l2_norm_exact = sqrt(e - 2)
h1_norm_exact = sqrt(e - 1 - exp(-1))
assert abs(f.l2_norm(method="Fekete") - l2_norm_exact) < eps
assert abs(f.l2_norm(method="FE") - l2_norm_exact) < eps
assert abs(f.h1_norm() - h1_norm_exact) < eps
def test_l2_h1_2():
eps = 1e-9
func = lambda x: log(x)
mesh1 = Mesh1D((1, 1.5, 2, 2.5, e), (20, 20, 20, 20))
f = Function(func, mesh1)
l2_norm_exact = sqrt(e-2)
h1_norm_exact = sqrt(e-1-exp(-1))
assert abs(f.l2_norm(method="Fekete")-l2_norm_exact) < eps
assert abs(f.l2_norm(method="FE")-l2_norm_exact) < eps
assert abs(f.h1_norm()-h1_norm_exact) < eps
def test_l2_h1_1():
eps = 1e-12
eps_low = 1e-10
func = lambda x: sin(x)
mesh1 = Mesh1D((0, pi), (20,))
mesh2 = Mesh1D((0, pi/2, pi), (20, 20))
mesh3 = Mesh1D((0, pi/2), (20,))
f = Function(func, mesh1)
g = Function(func, mesh2)
h = Function(func, mesh3)
assert abs(f.l2_norm(method="Fekete")-sqrt(pi/2)) < eps
assert abs(g.l2_norm(method="Fekete")-sqrt(pi/2)) < eps
assert abs(h.l2_norm(method="Fekete")-sqrt(pi/4)) < eps
assert abs(f.l2_norm(method="FE")-sqrt(pi/2)) < eps
assert abs(g.l2_norm(method="FE")-sqrt(pi/2)) < eps
assert abs(h.l2_norm(method="FE")-sqrt(pi/4)) < eps
assert abs(f.h1_norm()-sqrt(pi)) < eps_low
assert abs(g.h1_norm()-sqrt(pi)) < eps_low
assert abs(h.h1_norm()-sqrt(pi/2)) < eps_low
func = lambda x: cos(x)
mesh1 = Mesh1D((0, pi/4, pi/2, 3*pi/4, pi), (20, 20, 20, 20))
f = Function(func, mesh1)
assert abs(f.l2_norm(method="Fekete")-sqrt(pi/2)) < eps
assert abs(f.l2_norm(method="FE")-sqrt(pi/2)) < eps
assert abs(f.h1_norm()-sqrt(pi)) < eps
def test_l2_h1_1():
eps = 1e-12
eps_low = 1e-10
func = lambda x: sin(x)
mesh1 = Mesh1D((0, pi), (20, ))
mesh2 = Mesh1D((0, pi / 2, pi), (20, 20))
mesh3 = Mesh1D((0, pi / 2), (20, ))
f = Function(func, mesh1)
g = Function(func, mesh2)
h = Function(func, mesh3)
assert abs(f.l2_norm(method="Fekete") - sqrt(pi / 2)) < eps
assert abs(g.l2_norm(method="Fekete") - sqrt(pi / 2)) < eps
assert abs(h.l2_norm(method="Fekete") - sqrt(pi / 4)) < eps
assert abs(f.l2_norm(method="FE") - sqrt(pi / 2)) < eps
assert abs(g.l2_norm(method="FE") - sqrt(pi / 2)) < eps
assert abs(h.l2_norm(method="FE") - sqrt(pi / 4)) < eps
assert abs(f.h1_norm() - sqrt(pi)) < eps_low
assert abs(g.h1_norm() - sqrt(pi)) < eps_low
assert abs(h.h1_norm() - sqrt(pi / 2)) < eps_low
func = lambda x: cos(x)
mesh1 = Mesh1D((0, pi / 4, pi / 2, 3 * pi / 4, pi), (20, 20, 20, 20))
f = Function(func, mesh1)
assert abs(f.l2_norm(method="Fekete") - sqrt(pi / 2)) < eps
assert abs(f.l2_norm(method="FE") - sqrt(pi / 2)) < eps
assert abs(f.h1_norm() - sqrt(pi)) < eps
def test_power():
eps = 1e-12
func = lambda x: x
mesh1 = Mesh1D((0, 1), (1,))
mesh2 = Mesh1D((0, 1), (2,))
mesh3 = Mesh1D((0, 1), (3,))
f = Function(func, mesh1)
assert abs(f.l2_norm() - sqrt(1./3)) < eps
assert f**2 != Function(lambda x: x**2, mesh1)
assert f**2 == Function(lambda x: x**2, mesh2)
assert f**2 == Function(lambda x: x**2, mesh3)
func = lambda x: x
mesh1 = Mesh1D((5, 6), (1,))
mesh2 = Mesh1D((5, 6), (2,))
mesh3 = Mesh1D((5, 6), (3,))
f = Function(func, mesh1)
assert f**2 != Function(lambda x: x**2, mesh1)
assert f**2 == Function(lambda x: x**2, mesh2)
assert f**2 == Function(lambda x: x**2, mesh3)
func = lambda x: x**3+x
mesh1 = Mesh1D((5, 6), (3,))
mesh2 = Mesh1D((5, 6), (5,))
mesh3 = Mesh1D((5, 6), (6,))
mesh4 = Mesh1D((5, 6), (9,))
f = Function(func, mesh1)
assert f**2 != Function(lambda x: x**6+2*x**4+x**2, mesh1)
assert f**2 != Function(lambda x: x**6+2*x**4+x**2, mesh2)
assert f**2 == Function(lambda x: x**6+2*x**4+x**2, mesh3)
assert f**3 == Function(lambda x: x**3 + 3*x**5 + 3*x**7 + x**9, mesh4)
func = lambda x: x**3+x
mesh1 = Mesh1D((5, 5.1, 6), (3, 3))
mesh2 = Mesh1D((5, 5.1, 6), (5, 5))
mesh3 = Mesh1D((5, 5.1, 6), (6, 6))
mesh4 = Mesh1D((5, 5.1, 6), (9, 9))
mesh5 = Mesh1D((5, 5.1, 6), (9, 7))
mesh6 = Mesh1D((5, 5.1, 6), (6, 5))
mesh7 = Mesh1D((5, 6), (6,))
mesh8 = Mesh1D((5, 6), (9,))
f = Function(func, mesh1)
assert f**2 != Function(lambda x: x**6+2*x**4+x**2, mesh1)
assert f**2 != Function(lambda x: x**6+2*x**4+x**2, mesh2)
assert f**2 != Function(lambda x: x**6+2*x**4+x**2, mesh6)
assert f**2 == Function(lambda x: x**6+2*x**4+x**2, mesh3)
assert f**3 == Function(lambda x: x**3 + 3*x**5 + 3*x**7 + x**9, mesh4)
assert f**3 != Function(lambda x: x**3 + 3*x**5 + 3*x**7 + x**9, mesh5)
assert f**2 == Function(lambda x: x**6+2*x**4+x**2, mesh7)
assert f**3 == Function(lambda x: x**3 + 3*x**5 + 3*x**7 + x**9, mesh8)
def test_power():
eps = 1e-12
func = lambda x: x
mesh1 = Mesh1D((0, 1), (1, ))
mesh2 = Mesh1D((0, 1), (2, ))
mesh3 = Mesh1D((0, 1), (3, ))
f = Function(func, mesh1)
assert abs(f.l2_norm() - sqrt(1. / 3)) < eps
assert f**2 != Function(lambda x: x**2, mesh1)
assert f**2 == Function(lambda x: x**2, mesh2)
assert f**2 == Function(lambda x: x**2, mesh3)
func = lambda x: x
mesh1 = Mesh1D((5, 6), (1, ))
mesh2 = Mesh1D((5, 6), (2, ))
mesh3 = Mesh1D((5, 6), (3, ))
f = Function(func, mesh1)
assert f**2 != Function(lambda x: x**2, mesh1)
assert f**2 == Function(lambda x: x**2, mesh2)
assert f**2 == Function(lambda x: x**2, mesh3)
func = lambda x: x**3 + x
mesh1 = Mesh1D((5, 6), (3, ))
mesh2 = Mesh1D((5, 6), (5, ))
mesh3 = Mesh1D((5, 6), (6, ))
mesh4 = Mesh1D((5, 6), (9, ))
f = Function(func, mesh1)
assert f**2 != Function(lambda x: x**6 + 2 * x**4 + x**2, mesh1)
assert f**2 != Function(lambda x: x**6 + 2 * x**4 + x**2, mesh2)
assert f**2 == Function(lambda x: x**6 + 2 * x**4 + x**2, mesh3)
assert f**3 == Function(lambda x: x**3 + 3 * x**5 + 3 * x**7 + x**9, mesh4)
func = lambda x: x**3 + x
mesh1 = Mesh1D((5, 5.1, 6), (3, 3))
mesh2 = Mesh1D((5, 5.1, 6), (5, 5))
mesh3 = Mesh1D((5, 5.1, 6), (6, 6))
mesh4 = Mesh1D((5, 5.1, 6), (9, 9))
mesh5 = Mesh1D((5, 5.1, 6), (9, 7))
mesh6 = Mesh1D((5, 5.1, 6), (6, 5))
mesh7 = Mesh1D((5, 6), (6, ))
mesh8 = Mesh1D((5, 6), (9, ))
f = Function(func, mesh1)
assert f**2 != Function(lambda x: x**6 + 2 * x**4 + x**2, mesh1)
assert f**2 != Function(lambda x: x**6 + 2 * x**4 + x**2, mesh2)
assert f**2 != Function(lambda x: x**6 + 2 * x**4 + x**2, mesh6)
assert f**2 == Function(lambda x: x**6 + 2 * x**4 + x**2, mesh3)
assert f**3 == Function(lambda x: x**3 + 3 * x**5 + 3 * x**7 + x**9, mesh4)
assert f**3 != Function(lambda x: x**3 + 3 * x**5 + 3 * x**7 + x**9, mesh5)
assert f**2 == Function(lambda x: x**6 + 2 * x**4 + x**2, mesh7)
assert f**3 == Function(lambda x: x**3 + 3 * x**5 + 3 * x**7 + x**9, mesh8)
def test8():
eps = 1e-12
func = lambda x: x**2
mesh1 = Mesh1D((-5, -4, 3, 10), (2, 5, 2))
mesh2 = Mesh1D((-5, -4, 3, 10), (2, 2, 2))
mesh3 = Mesh1D((-5, -4, 3, 10), (2, 2, 1))
mesh4 = Mesh1D((-5, 10), (2,))
mesh5 = Mesh1D((-5, 10), (3,))
mesh6 = Mesh1D((-5, 10), (1,))
f = Function(func, mesh1)
g = Function(func, mesh2)
h = Function(func, mesh3)
l = Function(func, mesh4)
zero = Function(lambda x: 0., Mesh1D((-5, 10), (1,)))
assert zero.l2_norm() < eps
assert Function(lambda x: 0., mesh1) == zero
assert Function(lambda x: 0., mesh2) == zero
assert Function(lambda x: 0., mesh3) == zero
assert Function(lambda x: 0., mesh4) == zero
assert Function(lambda x: 0., mesh5) == zero
assert Function(lambda x: 0., mesh6) == zero
assert f - g == zero
assert (f-g).l2_norm() < eps
assert g - f == zero
assert (g - f).l2_norm() < eps
assert f - l == zero
assert (f - l).l2_norm() < eps
assert g - l == zero
assert (g - l).l2_norm() < eps
assert f - h != zero
assert (f - h).l2_norm() > eps
assert h - f != zero
assert (h - f).l2_norm() > eps
assert g - h != zero
assert (g - h).l2_norm() > eps
assert h - g != zero
assert (h - g).l2_norm() > eps
assert f - Function(lambda x: x**2, mesh1) == zero
assert f - Function(lambda x: x**3, mesh1) != zero
assert f - Function(lambda x: x**2, mesh2) == zero
assert f - Function(lambda x: x**2, mesh4) == zero
assert f - Function(lambda x: x**2, mesh5) == zero
assert f - Function(lambda x: x**2, mesh6) != zero
def test8():
eps = 1e-12
func = lambda x: x**2
mesh1 = Mesh1D((-5, -4, 3, 10), (2, 5, 2))
mesh2 = Mesh1D((-5, -4, 3, 10), (2, 2, 2))
mesh3 = Mesh1D((-5, -4, 3, 10), (2, 2, 1))
mesh4 = Mesh1D((-5, 10), (2, ))
mesh5 = Mesh1D((-5, 10), (3, ))
mesh6 = Mesh1D((-5, 10), (1, ))
f = Function(func, mesh1)
g = Function(func, mesh2)
h = Function(func, mesh3)
l = Function(func, mesh4)
zero = Function(lambda x: 0., Mesh1D((-5, 10), (1, )))
assert zero.l2_norm() < eps
assert Function(lambda x: 0., mesh1) == zero
assert Function(lambda x: 0., mesh2) == zero
assert Function(lambda x: 0., mesh3) == zero
assert Function(lambda x: 0., mesh4) == zero
assert Function(lambda x: 0., mesh5) == zero
assert Function(lambda x: 0., mesh6) == zero
assert f - g == zero
assert (f - g).l2_norm() < eps
assert g - f == zero
assert (g - f).l2_norm() < eps
assert f - l == zero
assert (f - l).l2_norm() < eps
assert g - l == zero
assert (g - l).l2_norm() < eps
assert f - h != zero
assert (f - h).l2_norm() > eps
assert h - f != zero
assert (h - f).l2_norm() > eps
assert g - h != zero
assert (g - h).l2_norm() > eps
assert h - g != zero
assert (h - g).l2_norm() > eps
assert f - Function(lambda x: x**2, mesh1) == zero
assert f - Function(lambda x: x**3, mesh1) != zero
assert f - Function(lambda x: x**2, mesh2) == zero
assert f - Function(lambda x: x**2, mesh4) == zero
assert f - Function(lambda x: x**2, mesh5) == zero
assert f - Function(lambda x: x**2, mesh6) != zero