from scipy._lib._array_api import np_compat, array_namespace
from functools import cached_property
from ._base import NestedFixedRule
from ._gauss_legendre import GaussLegendreQuadrature
class GaussKronrodQuadrature(NestedFixedRule):
"""
Gauss-Kronrod quadrature.
Gauss-Kronrod rules consist of two quadrature rules, one higher-order and one
lower-order. The higher-order rule is used as the estimate of the integral and the
difference between them is used as an estimate for the error.
Gauss-Kronrod is a 1D rule. To use it for multidimensional integrals, it will be
necessary to use ProductNestedFixed and multiple Gauss-Kronrod rules. See Examples.
For n-node Gauss-Kronrod, the lower-order rule has ``n//2`` nodes, which are the
ordinary Gauss-Legendre nodes with corresponding weights. The higher-order rule has
``n`` nodes, ``n//2`` of which are the same as the lower-order rule and the
remaining nodes are the Kronrod extension of those nodes.
Parameters
----------
npoints : int
Number of nodes for the higher-order rule.
xp : array_namespace, optional
The namespace for the node and weight arrays. Default is None, where NumPy is
used.
Attributes
----------
lower : Rule
Lower-order rule.
References
----------
.. [1] R. Piessens, E. de Doncker, Quadpack: A Subroutine Package for Automatic
Integration, files: dqk21.f, dqk15.f (1983).
Examples
--------
Evaluate a 1D integral. Note in this example that ``f`` returns an array, so the
estimates will also be arrays, despite the fact that this is a 1D problem.
>>> import numpy as np
>>> from scipy.integrate import cubature
>>> from scipy.integrate._rules import GaussKronrodQuadrature
>>> def f(x):
... return np.cos(x)
>>> rule = GaussKronrodQuadrature(21) # Use 21-point GaussKronrod
>>> a, b = np.array([0]), np.array([1])
>>> rule.estimate(f, a, b) # True value sin(1), approximately 0.84147
array([0.84147098])
>>> rule.estimate_error(f, a, b)
array([1.11022302e-16])
Evaluate a 2D integral. Note that in this example ``f`` returns a float, so the
estimates will also be floats.
>>> import numpy as np
>>> from scipy.integrate import cubature
>>> from scipy.integrate._rules import (
... ProductNestedFixed, GaussKronrodQuadrature
... )
>>> def f(x):
... # f(x) = cos(x_1) + cos(x_2)
... return np.sum(np.cos(x), axis=-1)
>>> rule = ProductNestedFixed(
... [GaussKronrodQuadrature(15), GaussKronrodQuadrature(15)]
... ) # Use 15-point Gauss-Kronrod
>>> a, b = np.array([0, 0]), np.array([1, 1])
>>> rule.estimate(f, a, b) # True value 2*sin(1), approximately 1.6829
np.float64(1.682941969615793)
>>> rule.estimate_error(f, a, b)
np.float64(2.220446049250313e-16)
"""
def __init__(self, npoints, xp=None):
# TODO: nodes and weights are currently hard-coded for values 15 and 21, but in
# the future it would be best to compute the Kronrod extension of the lower rule
if npoints != 15 and npoints != 21:
raise NotImplementedError("Gauss-Kronrod quadrature is currently only"
"supported for 15 or 21 nodes")
self.npoints = npoints
if xp is None:
xp = np_compat
self.xp = array_namespace(xp.empty(0))
self.gauss = GaussLegendreQuadrature(npoints//2, xp=self.xp)
@cached_property
def nodes_and_weights(self):
# These values are from QUADPACK's `dqk21.f` and `dqk15.f` (1983).
if self.npoints == 21:
nodes = self.xp.asarray(
[
0.995657163025808080735527280689003,
0.973906528517171720077964012084452,
0.930157491355708226001207180059508,
0.865063366688984510732096688423493,
0.780817726586416897063717578345042,
0.679409568299024406234327365114874,
0.562757134668604683339000099272694,
0.433395394129247190799265943165784,
0.294392862701460198131126603103866,
0.148874338981631210884826001129720,
0,
-0.148874338981631210884826001129720,
-0.294392862701460198131126603103866,
-0.433395394129247190799265943165784,
-0.562757134668604683339000099272694,
-0.679409568299024406234327365114874,
-0.780817726586416897063717578345042,
-0.865063366688984510732096688423493,
-0.930157491355708226001207180059508,
-0.973906528517171720077964012084452,
-0.995657163025808080735527280689003,
],
dtype=self.xp.float64,
)
weights = self.xp.asarray(
[
0.011694638867371874278064396062192,
0.032558162307964727478818972459390,
0.054755896574351996031381300244580,
0.075039674810919952767043140916190,
0.093125454583697605535065465083366,
0.109387158802297641899210590325805,
0.123491976262065851077958109831074,
0.134709217311473325928054001771707,
0.142775938577060080797094273138717,
0.147739104901338491374841515972068,
0.149445554002916905664936468389821,
0.147739104901338491374841515972068,
0.142775938577060080797094273138717,
0.134709217311473325928054001771707,
0.123491976262065851077958109831074,
0.109387158802297641899210590325805,
0.093125454583697605535065465083366,
0.075039674810919952767043140916190,
0.054755896574351996031381300244580,
0.032558162307964727478818972459390,
0.011694638867371874278064396062192,
],
dtype=self.xp.float64,
)
elif self.npoints == 15:
nodes = self.xp.asarray(
[
0.991455371120812639206854697526329,
0.949107912342758524526189684047851,
0.864864423359769072789712788640926,
0.741531185599394439863864773280788,
0.586087235467691130294144838258730,
0.405845151377397166906606412076961,
0.207784955007898467600689403773245,
0.000000000000000000000000000000000,
-0.207784955007898467600689403773245,
-0.405845151377397166906606412076961,
-0.586087235467691130294144838258730,
-0.741531185599394439863864773280788,
-0.864864423359769072789712788640926,
-0.949107912342758524526189684047851,
-0.991455371120812639206854697526329,
],
dtype=self.xp.float64,
)
weights = self.xp.asarray(
[
0.022935322010529224963732008058970,
0.063092092629978553290700663189204,
0.104790010322250183839876322541518,
0.140653259715525918745189590510238,
0.169004726639267902826583426598550,
0.190350578064785409913256402421014,
0.204432940075298892414161999234649,
0.209482141084727828012999174891714,
0.204432940075298892414161999234649,
0.190350578064785409913256402421014,
0.169004726639267902826583426598550,
0.140653259715525918745189590510238,
0.104790010322250183839876322541518,
0.063092092629978553290700663189204,
0.022935322010529224963732008058970,
],
dtype=self.xp.float64,
)
return nodes, weights
@property
def lower_nodes_and_weights(self):
return self.gauss.nodes_and_weights