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GaussKronrodQuadratureaz
  
    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)
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