def legendre(
order,
lower=-1,
upper=1,
physicist=False,
normed=False,
retall=False,
):
"""
Examples:
>>> polynomials, norms = chaospy.expansion.legendre(3, retall=True)
>>> polynomials.round(5)
polynomial([1.0, q0, q0**2-0.33333, q0**3-0.6*q0])
>>> norms
array([1. , 0.33333333, 0.08888889, 0.02285714])
>>> chaospy.expansion.legendre(3, physicist=True).round(3)
polynomial([1.0, 1.5*q0, 2.5*q0**2-0.556, 4.375*q0**3-1.672*q0])
>>> chaospy.expansion.legendre(3, lower=0, upper=1, normed=True).round(3)
polynomial([1.0, 3.464*q0-1.732, 13.416*q0**2-13.416*q0+2.236,
52.915*q0**3-79.373*q0**2+31.749*q0-2.646])
"""
multiplier = 1.0
if physicist:
multiplier = numpy.arange(1, order + 1)
multiplier = (2 * multiplier + 1) / (multiplier + 1)
_, [polynomials], [norms] = chaospy.recurrence.analytical_stieltjes(
order, chaospy.Uniform(lower, upper), multiplier=multiplier)
if normed:
polynomials = chaospy.true_divide(polynomials, numpy.sqrt(norms))
norms[:] = 1.0
return (polynomials, norms) if retall else polynomials
def laguerre(
order,
alpha=0.0,
physicist=False,
normed=False,
retall=False,
):
"""
Examples:
>>> polynomials, norms = chaospy.expansion.laguerre(3, retall=True)
>>> polynomials
polynomial([1.0, q0-1.0, q0**2-4.0*q0+2.0, q0**3-9.0*q0**2+18.0*q0-6.0])
>>> norms
array([ 1., 1., 4., 36.])
>>> chaospy.expansion.laguerre(3, physicist=True).round(5)
polynomial([1.0, -q0+1.0, 0.5*q0**2-2.0*q0+2.0,
-0.16667*q0**3+1.5*q0**2-5.33333*q0+4.66667])
>>> chaospy.expansion.laguerre(3, alpha=2, normed=True).round(3)
polynomial([1.0, 0.577*q0-1.732, 0.204*q0**2-1.633*q0+2.449,
0.053*q0**3-0.791*q0**2+3.162*q0-3.162])
"""
multiplier = -1.0 / numpy.arange(1, order + 1) if physicist else 1.0
_, [polynomials], [norms] = chaospy.recurrence.analytical_stieltjes(
order, chaospy.Gamma(alpha + 1), multiplier=multiplier)
if normed:
polynomials = chaospy.true_divide(polynomials, numpy.sqrt(norms))
norms[:] = 1.0
return (polynomials, norms) if retall else polynomials
def gegenbauer(
order,
alpha,
lower=-1,
upper=1,
physicist=False,
normed=False,
retall=False,
):
"""
Gegenbauer polynomials.
Args:
order (int):
The polynomial order.
alpha (float):
Gegenbauer shape parameter.
lower (float):
Lower bound for the integration interval.
upper (float):
Upper bound for the integration interval.
physicist (bool):
Use physicist weights instead of probabilist.
Examples:
>>> polynomials, norms = chaospy.expansion.gegenbauer(4, 1, retall=True)
>>> polynomials
polynomial([1.0, q0, q0**2-0.25, q0**3-0.5*q0, q0**4-0.75*q0**2+0.0625])
>>> norms
array([1. , 0.25 , 0.0625 , 0.015625 , 0.00390625])
>>> chaospy.expansion.gegenbauer(3, 1, physicist=True)
polynomial([1.0, 2.0*q0, 4.0*q0**2-0.5, 8.0*q0**3-2.0*q0])
>>> chaospy.expansion.gegenbauer(3, 1, lower=0.5, upper=1.5, normed=True).round(3)
polynomial([1.0, 4.0*q0-4.0, 16.0*q0**2-32.0*q0+15.0,
64.0*q0**3-192.0*q0**2+184.0*q0-56.0])
"""
multiplier = 1
if physicist:
multiplier = numpy.arange(1, order + 1)
multiplier = 2 * (multiplier + alpha - 1) / multiplier
_, [polynomials], [norms] = chaospy.recurrence.analytical_stieltjes(
order,
chaospy.Beta(alpha + 0.5, alpha + 0.5, lower, upper),
multiplier=multiplier,
)
if normed:
polynomials = chaospy.true_divide(polynomials, numpy.sqrt(norms))
norms[:] = 1.0
return (polynomials, norms) if retall else polynomials
def hermite(
order,
mu=0.0,
sigma=1.0,
physicist=False,
normed=False,
retall=False,
):
"""
Hermite orthogonal polynomial expansion.
Args:
order (int):
The quadrature order.
mu (float):
Non-centrality parameter.
sigma (float):
Scale parameter.
physicist (bool):
Use physicist weights instead of probabilist variant.
normed (bool):
If True orthonormal polynomials will be used.
retall (bool):
If true return numerical stabilized norms as well. Roughly the same
as ``cp.E(orth**2, dist)``.
Returns:
(numpoly.ndpoly, numpy.ndarray):
Hermite polynomial expansion. Norms of the orthogonal
expansion on the form ``E(orth**2, dist)``.
Examples:
>>> polynomials, norms = chaospy.expansion.hermite(4, retall=True)
>>> polynomials
polynomial([1.0, q0, q0**2-1.0, q0**3-3.0*q0, q0**4-6.0*q0**2+3.0])
>>> norms
array([ 1., 1., 2., 6., 24.])
>>> chaospy.expansion.hermite(3, physicist=True)
polynomial([1.0, 2.0*q0, 4.0*q0**2-2.0, 8.0*q0**3-12.0*q0])
>>> chaospy.expansion.hermite(3, sigma=2.5, normed=True).round(3)
polynomial([1.0, 0.4*q0, 0.113*q0**2-0.707, 0.026*q0**3-0.49*q0])
"""
multiplier = 2 if physicist else 1
_, [polynomials], [norms] = chaospy.recurrence.analytical_stieltjes(
order, chaospy.Normal(mu, sigma), multiplier=multiplier)
if normed:
polynomials = chaospy.true_divide(polynomials, numpy.sqrt(norms))
norms[:] = 1.0
return (polynomials, norms) if retall else polynomials
def chebyshev_2(
order,
lower=-1,
upper=1,
physicist=False,
normed=False,
retall=False,
):
"""
Chebyshev polynomials of the second kind.
Args:
order (int):
The quadrature order.
lower (float):
Lower bound for the integration interval.
upper (float):
Upper bound for the integration interval.
physicist (bool):
Use physicist weights instead of probabilist.
Returns:
(numpoly.ndpoly, numpy.ndarray):
Chebyshev polynomial expansion. Norms of the orthogonal
expansion on the form ``E(orth**2, dist)``.
Examples:
>>> polynomials, norms = chaospy.expansion.chebyshev_2(4, retall=True)
>>> polynomials
polynomial([1.0, q0, q0**2-0.25, q0**3-0.5*q0, q0**4-0.75*q0**2+0.0625])
>>> norms
array([1. , 0.25 , 0.0625 , 0.015625 , 0.00390625])
>>> chaospy.expansion.chebyshev_2(3, physicist=True)
polynomial([1.0, 2.0*q0, 4.0*q0**2-0.5, 8.0*q0**3-2.0*q0])
>>> chaospy.expansion.chebyshev_2(3, lower=0.5, upper=1.5, normed=True).round(3)
polynomial([1.0, 4.0*q0-4.0, 16.0*q0**2-32.0*q0+15.0,
64.0*q0**3-192.0*q0**2+184.0*q0-56.0])
"""
multiplier = 2 if physicist else 1
_, [polynomials], [norms] = chaospy.recurrence.analytical_stieltjes(
order, chaospy.Beta(1.5, 1.5, lower, upper), multiplier=multiplier
)
if normed:
polynomials = chaospy.true_divide(polynomials, numpy.sqrt(norms))
norms[:] = 1.0
return (polynomials, norms) if retall else polynomials
def chebyshev_1(
order,
lower=-1,
upper=1,
physicist=False,
normed=False,
retall=False,
):
"""
Chebyshev polynomials of the first kind.
Args:
order (int):
The polynomial order.
lower (float):
Lower bound for the integration interval.
upper (float):
Upper bound for the integration interval.
physicist (bool):
Use physicist weights instead of probabilist.
Returns:
(numpoly.ndpoly, numpy.ndarray):
Chebyshev polynomial expansion. Norms of the orthogonal
expansion on the form ``E(orth**2, dist)``.
Examples:
>>> polynomials, norms = chaospy.expansion.chebyshev_1(4, retall=True)
>>> polynomials
polynomial([1.0, q0, q0**2-0.5, q0**3-0.75*q0, q0**4-q0**2+0.125])
>>> norms
array([1. , 0.5 , 0.125 , 0.03125 , 0.0078125])
>>> chaospy.expansion.chebyshev_1(3, physicist=True)
polynomial([1.0, q0, 2.0*q0**2-1.0, 4.0*q0**3-2.5*q0])
>>> chaospy.expansion.chebyshev_1(3, lower=0.5, upper=1.5, normed=True).round(3)
polynomial([1.0, 2.828*q0-2.828, 11.314*q0**2-22.627*q0+9.899,
45.255*q0**3-135.765*q0**2+127.279*q0-36.77])
"""
multiplier = 1 + numpy.arange(order).astype(bool) if physicist else 1
_, [polynomials], [norms] = chaospy.recurrence.analytical_stieltjes(
order, chaospy.Beta(0.5, 0.5, lower, upper), multiplier=multiplier
)
if normed:
polynomials = chaospy.true_divide(polynomials, numpy.sqrt(norms))
norms[:] = 1.0
return (polynomials, norms) if retall else polynomials
def jacobi(
order,
alpha,
beta,
lower=-1,
upper=1,
physicist=False,
normed=False,
retall=False,
):
"""
Jacobi polynomial expansion.
Examples:
>>> polynomials, norms = chaospy.expansion.jacobi(4, 0.5, 0.5, retall=True)
>>> polynomials
polynomial([1.0, q0, q0**2-0.5, q0**3-0.75*q0, q0**4-q0**2+0.125])
>>> norms
array([1. , 0.5 , 0.125 , 0.03125 , 0.0078125])
>>> chaospy.expansion.jacobi(3, 0.5, 0.5, physicist=True).round(4)
polynomial([1.0, 1.5*q0, 2.5*q0**2-0.8333, 4.375*q0**3-2.1146*q0])
>>> chaospy.expansion.jacobi(3, 1.5, 0.5, normed=True)
polynomial([1.0, 2.0*q0, 4.0*q0**2-1.0, 8.0*q0**3-4.0*q0])
"""
multiplier = 1
if physicist:
multiplier = numpy.arange(1, order + 1)
multiplier = ((2 * multiplier + alpha + beta - 1) *
(2 * multiplier + alpha + beta) /
(2 * multiplier * (multiplier + alpha + beta)))
_, [polynomials], [norms] = chaospy.recurrence.analytical_stieltjes(
order,
chaospy.Beta(alpha, beta, lower=lower, upper=upper),
multiplier=multiplier,
)
if normed:
polynomials = chaospy.true_divide(polynomials, numpy.sqrt(norms))
norms[:] = 1.0
return (polynomials, norms) if retall else polynomials