`scipy.integrate._rules._gauss_kronrod:GaussKronrodQuadrature`

source: /scipy/integrate/_rules/_gauss_kronrod.py :9

Signature

class GaussKronrodQuadrature ( npoints , xp = None )

Members

__init__

Summary

Gauss-Kronrod quadrature.

Extended Summary

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.

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
rule.estimate_error(f, a, b)

✗

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
rule.estimate_error(f, a, b)

✗

Aliases

scipy.integrate._cubature.GaussKronrodQuadrature