def legendre_simple(order):
"""
Simple Legendre quadrature on the [0, 1] interval.
Use :func:`chaospy.quadrature.legendre` instead.
"""
coefficients = chaospy.construct_recurrence_coefficients(
order=int(order), dist=chaospy.Uniform(-1, 1))
[abscissas], [weights] = chaospy.coefficients_to_quadrature(coefficients)
return abscissas * 0.5 + 0.5, weights
def jacobi(order, alpha, beta, lower=-1, upper=1, physicist=False):
"""
Gauss-Jacobi quadrature rule.
Compute the sample points and weights for Gauss-Jacobi quadrature. The
sample points are the roots of the nth degree Jacobi polynomial. These
sample points and weights correctly integrate polynomials of degree
:math:`2N-1` or less.
Gaussian quadrature come in two variants: physicist and probabilist. For
Gauss-Jacobi physicist means a weight function
:math:`(1-x)^\alpha (1+x)^\beta` and
weights that sum to :math`2^{\alpha+\beta}`, and probabilist means a weight
function is :math:`B(\alpha, \beta) x^{\alpha-1}(1-x)^{\beta-1}` (where
:math:`B` is the beta normalizing constant) which sum to 1.
Args:
order (int):
The quadrature order.
alpha (float):
First Jakobi shape parameter.
beta (float):
Second Jakobi 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.
Returns:
abscissas (numpy.ndarray):
The ``order+1`` quadrature points for where to evaluate the model
function with.
weights (numpy.ndarray):
The quadrature weights associated with each abscissas.
Examples:
>>> abscissas, weights = chaospy.quadrature.jacobi(3, alpha=2, beta=2)
>>> abscissas
array([[-0.69474659, -0.25056281, 0.25056281, 0.69474659]])
>>> weights
array([0.09535261, 0.40464739, 0.40464739, 0.09535261])
See also:
:func:`chaospy.quadrature.gaussian`
"""
order = int(order)
coefficients = chaospy.construct_recurrence_coefficients(
order=order, dist=chaospy.Beta(alpha + 1, beta + 1, lower, upper))
[abscissas], [weights] = chaospy.coefficients_to_quadrature(coefficients)
weights *= 2**(alpha + beta) if physicist else 1
return abscissas[numpy.newaxis], weights
def hermite(order, mu=0.0, sigma=1.0, physicist=False):
r"""
Gauss-Hermite quadrature rule.
Compute the sample points and weights for Gauss-Hermite quadrature. The
sample points are the roots of the nth degree Hermite polynomial. These
sample points and weights correctly integrate polynomials of degree
:math:`2N-1` or less.
Gaussian quadrature come in two variants: physicist and probabilist. For
Gauss-Hermite physicist means a weight function :math:`e^{-x^2}` and
weights that sum to :math`\sqrt(\pi)`, and probabilist means a weight
function is :math:`e^{-x^2/2}` and sum to 1.
Args:
order (int):
The quadrature order.
mu (float):
Non-centrality parameter.
sigma (float):
Scale parameter.
physicist (bool):
Use physicist weights instead of probabilist variant.
Returns:
abscissas (numpy.ndarray):
The ``order+1`` quadrature points for where to evaluate the model
function with.
weights (numpy.ndarray):
The quadrature weights associated with each abscissas.
Examples:
>>> abscissas, weights = chaospy.quadrature.hermite(3)
>>> abscissas
array([[-2.33441422, -0.74196378, 0.74196378, 2.33441422]])
>>> weights
array([0.04587585, 0.45412415, 0.45412415, 0.04587585])
See also:
:func:`chaospy.quadrature.gaussian`
"""
order = int(order)
sigma = float(sigma * 2**-0.5 if physicist else sigma)
coefficients = chaospy.construct_recurrence_coefficients(
order=order, dist=chaospy.Normal(0, sigma))
[abscissas], [weights] = chaospy.coefficients_to_quadrature(coefficients)
weights = weights * numpy.pi**0.5 if physicist else weights
if order % 2 == 0:
abscissas[len(abscissas) // 2] = 0
abscissas += mu
return abscissas[numpy.newaxis], weights
def laguerre(order, alpha=0.0, physicist=False):
r"""
Generalized Gauss-Laguerre quadrature rule.
Compute the sample points and weights for Gauss-Laguerre quadrature. The
sample points are the roots of the nth degree Laguerre polynomial. These
sample points and weights correctly integrate polynomials of degree
:math:`2N-1` or less.
Gaussian quadrature come in two variants: physicist and probabilist. For
Gauss-Laguerre physicist means a weight function :math:`x^\alpha e^{-x}`
and weights that sum to :math`\Gamma(\alpha+1)`, and probabilist means a
weight function is :math:`x^\alpha e^{-x}` and sum to 1.
Args:
order (int):
The quadrature order.
alpha (float):
Shape parameter. Defaults to non-generalized Laguerre if 0.
physicist (bool):
Use physicist weights instead of probabilist.
Returns:
abscissas (numpy.ndarray):
The ``order+1`` quadrature points for where to evaluate the model
function with.
weights (numpy.ndarray):
The quadrature weights associated with each abscissas.
Examples:
>>> abscissas, weights = chaospy.quadrature.laguerre(2)
>>> abscissas
array([[0.41577456, 2.29428036, 6.28994508]])
>>> weights
array([0.71109301, 0.27851773, 0.01038926])
See also:
:func:`chaospy.quadrature.gaussian`
"""
order = int(order)
coefficients = chaospy.construct_recurrence_coefficients(
order=order, dist=chaospy.Gamma(alpha + 1))
[abscissas], [weights] = chaospy.coefficients_to_quadrature(coefficients)
weights *= gamma(alpha + 1) if physicist else 1
return abscissas[numpy.newaxis], weights
def quad_gauss_kronrod(
order,
dist,
recurrence_algorithm="stieltjes",
rule="clenshaw_curtis",
tolerance=1e-10,
scaling=3,
n_max=5000,
):
"""
Generate Gauss-Kronrod quadrature abscissas and weights.
Gauss-Kronrod quadrature is an adaptive method for Gaussian quadrature
rule. It builds on top of other quadrature rules by extending "pure"
Gaussian quadrature rules with extra abscissas and new weights such that
already used abscissas can be reused. For more details, see `Wikipedia
article
<https://en.wikipedia.org/wiki/Gauss%E2%80%93Kronrod_quadrature_formula>`_.
For each order ``N`` taken with ordinary Gaussian quadrature, Gauss-Kronrod
will create ``2N+1`` abscissas where all of the ``N`` "old" abscissas are
all interlaced between the "new" ones.
The algorithm is well suited for any Jacobi scheme, i.e. quadrature
involving Uniform or Beta distribution, and might work on others as well.
However, it will not work everywhere. For example `Kahaner and Monegato
<https://link.springer.com/article/10.1007/BF01590820>`_ showed that higher
order Gauss-Kronrod quadrature for Gauss-Hermite and Gauss-Laguerre does
not exist.
Args:
order (int):
The order of the quadrature.
dist (chaospy.distributions.baseclass.Distribution):
The distribution which density will be used as weight function.
recurrence_algorithm (str):
Name of the algorithm used to generate abscissas and weights. If
omitted, ``analytical`` will be tried first, and ``stieltjes`` used
if that fails.
rule (str):
In the case of ``lanczos`` or ``stieltjes``, defines the
proxy-integration scheme.
tolerance (float):
The allowed relative error in norm between two quadrature orders
before method assumes convergence.
scaling (float):
A multiplier the adaptive order increases with for each step
quadrature order is not converged. Use 0 to indicate unit
increments.
n_max (int):
The allowed number of quadrature points to use in approximation.
Returns:
(numpy.ndarray, numpy.ndarray):
abscissas:
The quadrature points for where to evaluate the model function
with ``abscissas.shape == (len(dist), N)`` where ``N`` is the
number of samples.
weights:
The quadrature weights with ``weights.shape == (N,)``.
Raises:
ValueError:
Error raised if Kronrod algorithm results in negative recurrence
coefficients.
Notes:
Code is adapted from `quadpy <https://github.com/nschloe/quadpy>`_,
which adapted his code from `W. Gautschi
<https://www.cs.purdue.edu/archives/2002/wxg/codes/OPQ.html>`_.
Algorithm for calculating Kronrod-Jacobi matrices was first published
Algorithm as proposed by Laurie :cite:`laurie_calculation_1997`.
Example:
>>> distribution = chaospy.Uniform(-1, 1)
>>> abscissas, weights = quad_gauss_kronrod(3, distribution)
>>> abscissas.round(2)
array([[-0.98, -0.86, -0.64, -0.34, 0. , 0.34, 0.64, 0.86, 0.98]])
>>> weights.round(3)
array([0.031, 0.085, 0.133, 0.163, 0.173, 0.163, 0.133, 0.085, 0.031])
"""
assert not rule.startswith("gauss"), "recursive Gaussian quadrature call"
length = int(numpy.ceil(3 * (order + 1) / 2.0))
coefficients = chaospy.construct_recurrence_coefficients(
order=length,
dist=dist,
recurrence_algorithm=recurrence_algorithm,
rule=rule,
tolerance=tolerance,
scaling=scaling,
n_max=n_max,
)
coefficients = [
kronrod_jacobi(order + 1, coeffs) for coeffs in coefficients
]
abscissas, weights = chaospy.coefficients_to_quadrature(coefficients)
return combine_quadrature(abscissas, weights)
def radau(
order,
dist,
fixed_point=None,
recurrence_algorithm="stieltjes",
rule="clenshaw_curtis",
tolerance=1e-10,
scaling=3,
n_max=5000,
):
"""
Generate the quadrature nodes and weights in Gauss-Radau quadrature.
Gauss-Radau formula for numerical estimation of integrals. It requires
:math:`m+1` points and fits all Polynomials to degree :math:`2m`, so it
effectively fits exactly all Polynomials of degree :math:`2m-1`.
It allows for a single abscissas to be user defined, while the others are
built around this point.
Canonically, Radau is built around Legendre weight function with the fixed
point at the left end. Not all distributions/fixed point combinations
allows for the building of a quadrature scheme.
Args:
order (int):
Quadrature order.
dist (:class:`chaospy.Distribution`):
The distribution weights to be used to create higher order nodes
from.
fixed_point (float):
Fixed point abscissas assumed to be included in the quadrature. If
omitted, use distribution lower bound.
rule (str):
In the case of ``lanczos`` or ``stieltjes``, defines the
proxy-integration scheme.
accuracy (int):
In the case ``rule`` is used, defines the quadrature order of the
scheme used. In practice, must be at least as large as ``order``.
recurrence_algorithm (str):
Name of the algorithm used to generate abscissas and weights. If
omitted, ``analytical`` will be tried first, and ``stieltjes`` used
if that fails.
Returns:
abscissas (numpy.ndarray):
The quadrature points for where to evaluate the model function
with ``abscissas.shape == (len(dist), N)`` where ``N`` is the
number of samples.
weights (numpy.ndarray):
The quadrature weights with ``weights.shape == (N,)``.
Example:
>>> distribution = chaospy.Uniform(-1, 1)
>>> abscissas, weights = chaospy.quadrature.radau(4, distribution)
>>> abscissas.round(3)
array([[-1. , -0.887, -0.64 , -0.295, 0.094, 0.468, 0.771, 0.955]])
>>> weights.round(3)
array([0.016, 0.093, 0.152, 0.188, 0.196, 0.174, 0.125, 0.057])
"""
if fixed_point is None:
fixed_point = dist.lower
else:
fixed_point = numpy.ones(len(dist)) * fixed_point
if order == 0:
return fixed_point.reshape(-1, 1), numpy.ones(1)
coefficients = chaospy.construct_recurrence_coefficients(
order=2 * order - 1,
dist=dist,
recurrence_algorithm=recurrence_algorithm,
rule=rule,
tolerance=tolerance,
scaling=scaling,
n_max=n_max,
)
coefficients = [
radau_jakobi(coeffs, point) for point, coeffs in zip(fixed_point, coefficients)
]
abscissas, weights = chaospy.coefficients_to_quadrature(coefficients)
return combine_quadrature(abscissas, weights)
def quad_gauss_radau(
order,
dist,
fixed_point=None,
recurrence_algorithm="stieltjes",
rule="clenshaw_curtis",
tolerance=1e-10,
scaling=3,
n_max=5000,
):
"""
Generate the quadrature nodes and weights in Gauss-Radau quadrature.
Args:
order (int):
Quadrature order.
dist (chaospy.distributions.baseclass.Distribution):
The distribution weights to be used to create higher order nodes
from.
fixed_point (float):
Fixed point abscissas assumed to be included in the quadrature. If
imitted, use distribution lower point ``dist.range()[0]``.
rule (str):
In the case of ``lanczos`` or ``stieltjes``, defines the
proxy-integration scheme.
accuracy (int):
In the case ``rule`` is used, defines the quadrature order of the
scheme used. In practice, must be at least as large as ``order``.
recurrence_algorithm (str):
Name of the algorithm used to generate abscissas and weights. If
omitted, ``analytical`` will be tried first, and ``stieltjes`` used
if that fails.
Returns:
(numpy.ndarray, numpy.ndarray):
abscissas:
The quadrature points for where to evaluate the model function
with ``abscissas.shape == (len(dist), N)`` where ``N`` is the
number of samples.
weights:
The quadrature weights with ``weights.shape == (N,)``.
Example:
>>> abscissas, weights = quad_gauss_radau(4, chaospy.Uniform(-1, 1))
>>> abscissas.round(3)
array([[-1. , -0.887, -0.64 , -0.295, 0.094, 0.468, 0.771, 0.955]])
>>> weights.round(3)
array([0.016, 0.093, 0.152, 0.188, 0.196, 0.174, 0.125, 0.057])
"""
assert not rule.startswith("gauss"), "recursive Gaussian quadrature call"
if fixed_point is None:
fixed_point = dist.lower
else:
fixed_point = numpy.ones(len(dist))*fixed_point
if order == 0:
return fixed_point.reshape(-1, 1), numpy.ones(1)
coefficients = chaospy.construct_recurrence_coefficients(
order=2*order-1,
dist=dist,
recurrence_algorithm=recurrence_algorithm,
rule=rule,
tolerance=tolerance,
scaling=scaling,
n_max=n_max,
)
coefficients = [radau_jakobi(coeffs, point)
for point, coeffs in zip(fixed_point, coefficients)]
abscissas, weights = chaospy.coefficients_to_quadrature(coefficients)
return combine_quadrature(abscissas, weights)
def quad_gauss_lobatto(
order,
dist,
recurrence_algorithm="stieltjes",
rule="fejer",
tolerance=1e-10,
scaling=3,
n_max=5000,
):
"""
Generate the abscissas and weights in Gauss-Loboto quadrature.
Args:
order (int):
Quadrature order.
dist (chaospy.distributions.baseclass.Distribution):
The distribution weights to be used to create higher order nodes
from.
recurrence_algorithm (str):
Name of the algorithm used to generate abscissas and weights.
rule (str):
In the case of ``lanczos`` or ``stieltjes``, defines the
proxy-integration scheme.
tolerance (float):
The allowed relative error in norm between two quadrature orders
before method assumes convergence.
scaling (float):
A multiplier the adaptive order increases with for each step
quadrature order is not converged. Use 0 to indicate unit
increments.
n_max (int):
The allowed number of quadrature points to use in approximation.
Returns:
(numpy.ndarray, numpy.ndarray):
abscissas:
The quadrature points for where to evaluate the model function
with ``abscissas.shape == (len(dist), N)`` where ``N`` is the
number of samples.
weights:
The quadrature weights with ``weights.shape == (N,)``.
Example:
>>> abscissas, weights = quad_gauss_lobatto(
... 4, chaospy.Uniform(-1, 1))
>>> abscissas.round(3)
array([[-1. , -0.872, -0.592, -0.209, 0.209, 0.592, 0.872, 1. ]])
>>> weights.round(3)
array([0.018, 0.105, 0.171, 0.206, 0.206, 0.171, 0.105, 0.018])
"""
assert not rule.startswith("gauss"), "recursive Gaussian quadrature call"
if order == 0:
return dist.lower.reshape(1, -1), numpy.array([1.])
coefficients = chaospy.construct_recurrence_coefficients(
order=2 * order - 1,
dist=dist,
recurrence_algorithm=recurrence_algorithm,
rule=rule,
tolerance=tolerance,
scaling=scaling,
n_max=n_max,
)
coefficients = [
_lobatto(coeffs, (lo, up))
for coeffs, lo, up in zip(coefficients, dist.lower, dist.upper)
]
abscissas, weights = chaospy.coefficients_to_quadrature(coefficients)
return combine_quadrature(abscissas, weights)
def quad_gauss_legendre(
order,
domain=(0, 1),
recurrence_algorithm="stieltjes",
rule="clenshaw_curtis",
tolerance=1e-10,
scaling=3,
n_max=5000,
):
r"""
Generate the quadrature nodes and weights in Gauss-Legendre quadrature.
Note that this rule exists to allow for integrating functions with weight
functions without actually adding the quadrature. Like:
.. math:
\int_a^b p(x) f(x) dx \approx \sum_i p(X_i) f(X_i) W_i
instead of the more traditional:
.. math:
\int_a^b p(x) f(x) dx \approx \sum_i f(X_i) W_i
To get the behavior where the weight function is taken into consideration,
use :func:`chaospy.quad_gaussian`.
Args:
order (int, numpy.ndarray):
Quadrature order.
domain (chaospy.distributions.baseclass.Distribution, numpy.ndarray):
Either distribution or bounding of interval to integrate over.
recurrence_algorithm (str):
Name of the algorithm used to generate abscissas and weights.
rule (str):
In the case of ``lanczos`` or ``stieltjes``, defines the
proxy-integration scheme.
tolerance (float):
The allowed relative error in norm between two quadrature orders
before method assumes convergence.
scaling (float):
A multiplier the adaptive order increases with for each step
quadrature order is not converged. Use 0 to indicate unit
increments.
n_max (int):
The allowed number of quadrature points to use in approximation.
Returns:
(numpy.ndarray, numpy.ndarray):
abscissas:
The quadrature points for where to evaluate the model function
with ``abscissas.shape == (len(dist), N)`` where ``N`` is the
number of samples.
weights:
The quadrature weights with ``weights.shape == (N,)``.
Example:
>>> abscissas, weights = quad_gauss_legendre(3)
>>> abscissas.round(4)
array([[0.0694, 0.33 , 0.67 , 0.9306]])
>>> weights.round(4)
array([0.1739, 0.3261, 0.3261, 0.1739])
"""
from ..distributions.baseclass import Distribution
from ..distributions.collection import Uniform
if isinstance(domain, Distribution):
abscissas, weights = quad_gauss_legendre(
order=order,
domain=(domain.lower, domain.upper),
recurrence_algorithm=recurrence_algorithm,
rule=rule,
tolerance=tolerance,
scaling=scaling,
n_max=n_max,
)
eps = 1e-14 * (domain.upper - domain.lower)
abscissas_ = numpy.clip(abscissas.T, domain.lower + eps,
domain.upper - eps).T
weights *= domain.pdf(abscissas_).flatten()
weights /= numpy.sum(weights)
return abscissas, weights
order = numpy.asarray(order, dtype=int).flatten()
lower, upper = numpy.array(domain)
lower = numpy.asarray(lower).flatten()
upper = numpy.asarray(upper).flatten()
dim = max(lower.size, upper.size, order.size)
order = numpy.ones(dim, dtype=int) * order
lower = numpy.ones(dim) * lower
upper = numpy.ones(dim) * upper
coefficients = chaospy.construct_recurrence_coefficients(
order=numpy.max(order),
dist=Uniform(0, 1),
recurrence_algorithm=recurrence_algorithm,
rule=rule,
tolerance=tolerance,
scaling=scaling,
n_max=n_max,
)
abscissas, weights = zip(*[
chaospy.coefficients_to_quadrature(coefficients[:order_ + 1])
for order_ in order
])
abscissas = list(numpy.asarray(abscissas).reshape(dim, -1))
weights = list(numpy.asarray(weights).reshape(dim, -1))
return combine_quadrature(abscissas, weights, (lower, upper))
def quad_gaussian(
order,
dist,
recurrence_algorithm="stieltjes",
rule="clenshaw_curtis",
tolerance=1e-10,
scaling=3,
n_max=5000,
):
"""
Create Gaussian quadrature nodes and weights.
Generating Gaussian quadrature by first generating so called *three terms
recurrence* coefficients using one various different algorithms. The
coefficients are them converted to abscissas and weights by constructing
lower banded Jacobi matrix and calculating eigenvalues and eigenvectors,
which can be directly translated to abscissas and weights. Construction of
the coefficients is potentially numerically unstable, there for multiple
algorithms exists.
Args:
dist (chaospy.Distribution):
The distribution which density will be used as weight function.
order (int):
The order of the quadrature.
recurrence_algorithm (str):
Name of the algorithm used to generate abscissas and weights.
rule (str):
In the case of ``lanczos`` or ``stieltjes``, defines the
proxy-integration scheme.
tolerance (float):
The allowed relative error in norm between two quadrature orders
before method assumes convergence.
scaling (float):
A multiplier the adaptive order increases with for each step
quadrature order is not converged. Use 0 to indicate unit
increments.
n_max (int):
The allowed number of quadrature points to use in approximation.
Returns:
(numpy.ndarray, numpy.ndarray):
Gaussian quadrature abscissas and weights with shapes
``(len(dist), order+1)`` and ``(order+1,)`` respectively.
Raises:
NotImplementedError:
In the case of recurrence algorithm ``analytical``, error is raised
if the distribution does not implement the three terms recurrence
algorithm analytically.
numpy.linalg.LinAlgError:
For non-canonical random variables, the construction might fail
because of illegal numerical operations.
Notes:
Quadrature constructed as outlined by Walter Gautschi
:cite:`gautschi_construction_1968`.
Examples:
>>> distribution = chaospy.Normal(0, 1)
>>> abscissas, weights = chaospy.quad_gaussian(
... 5, distribution, recurrence_algorithm="stieltjes")
>>> abscissas.round(4)
array([[-3.3243, -1.8892, -0.6167, 0.6167, 1.8892, 3.3243]])
>>> weights.round(4)
array([0.0026, 0.0886, 0.4088, 0.4088, 0.0886, 0.0026])
>>> distribution = chaospy.J(chaospy.Uniform(), chaospy.Normal())
>>> abscissas, weights = chaospy.quad_gaussian(
... 2, distribution, recurrence_algorithm="chebyshev")
>>> abscissas.round(2)
array([[ 0.11, 0.11, 0.11, 0.5 , 0.5 , 0.5 , 0.89, 0.89, 0.89],
[-1.73, 0. , 1.73, -1.73, 0. , 1.73, -1.73, 0. , 1.73]])
>>> weights.round(3)
array([0.046, 0.185, 0.046, 0.074, 0.296, 0.074, 0.046, 0.185, 0.046])
"""
coefficients = chaospy.construct_recurrence_coefficients(
order=order,
dist=dist,
recurrence_algorithm=recurrence_algorithm,
rule=rule,
tolerance=tolerance,
scaling=scaling,
n_max=n_max,
)
abscissas, weights = chaospy.coefficients_to_quadrature(coefficients)
return combine_quadrature(abscissas, weights)