def test_laguerre_scheme(scheme):
def integrate_exact(k):
# \int_0^\infty x^k * exp(-x)
return math.gamma(k + 1)
degree = check_degree_1d(
lambda poly: quadpy.line_segment.integrate(
poly, numpy.array([[-1.0], [1.0]]), scheme), integrate_exact,
scheme.degree + 1)
assert degree == scheme.degree
return
def test_hermite_scheme(scheme):
def integrate_exact(k):
# \int_-\infty^\infty x^k * exp(-x^2)
return 0.5 * ((-1)**k + 1) * math.gamma(0.5 * (k + 1))
degree = check_degree_1d(
lambda poly: quadpy.line_segment.integrate(
poly, numpy.array([[-1.0], [1.0]]), scheme), integrate_exact,
scheme.degree + 1)
assert degree == scheme.degree
return
def test_cheb1_scheme(scheme):
# \int_-1^1 x^k / sqrt(1 - x^2)
# <https://github.com/nschloe/note-no-gamma>
def integrate_exact(k):
if k == 0:
return numpy.pi
elif k == 1:
return 0
return integrate_exact(k - 2) * (k - 1) / k
degree = check_degree_1d(
lambda poly: scheme.integrate(poly, numpy.array([[-1.0], [1.0]])),
integrate_exact,
scheme.degree + 1,
)
assert degree >= scheme.degree
def test_cheb2_scheme(scheme):
def integrate_exact(k):
# \int_-1^1 x^k * sqrt(1 - x^2)
if k == 0:
return 0.5 * numpy.pi
if k % 2 == 1:
return 0.0
return (numpy.sqrt(numpy.pi) * ((-1)**k + 1) *
math.gamma(0.5 * (k + 1)) / math.gamma(0.5 * k + 2) / 4)
degree = check_degree_1d(
lambda poly: scheme.integrate(poly, numpy.array([[-1.0], [1.0]])),
integrate_exact,
scheme.degree + 1,
)
assert degree >= scheme.degree
def test_cheb1_scheme(scheme):
def integrate_exact(k):
# \int_-1^1 x^k / sqrt(1 - x^2)
if k == 0:
return numpy.pi
if k % 2 == 1:
return 0.0
return numpy.sqrt(numpy.pi) * ((-1)**k + 1) \
* math.gamma(0.5*(k+1)) / math.gamma(0.5*k) \
/ k
degree = check_degree_1d(
lambda poly: quadpy.line_segment.integrate(
poly, numpy.array([[-1.0], [1.0]]), scheme), integrate_exact,
scheme.degree + 1)
assert degree >= scheme.degree
return
