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scipy.integrate.

# cubature

scipy.integrate.cubature(f, a, b, *, rule='gk21', rtol=1e-08, atol=0, max_subdivisions=10000, args=(), workers=1, points=None)

  Adaptive cubature of multidimensional array-valued function.

  Given an arbitrary integration rule, this function returns an estimate of the integral to the requested tolerance over the region defined by the arrays a and b specifying the corners of a hypercube.

  Convergence is not guaranteed for all integrals.

  Parameters:
    f callable

      Function to integrate. f must have the signature:

          f(x : ndarray, *args) -> ndarray

      f should accept arrays `x` of shape:

          (npoints, ndim)

      and output arrays of shape:

          (npoints, output_dim_1, ..., output_dim_n)

      In this case, `cubature` will return arrays of shape:

          (output_dim_1, ..., output_dim_n)

    a, b array_like

      Lower and upper limits of integration as 1D arrays specifying the left and right endpoints of the intervals being integrated over. Limits can be infinite.

    rule str, optional

      Rule used to estimate the integral. If passing a string, the options are “gauss-kronrod” (21 node), or “genz-malik” (degree 7). If a rule like “gauss-kronrod” is specified for an `n`-dim integrand, the corresponding Cartesian product rule is used. “gk21”, “gk15” are also supported for compatibility with `quad_vec`. See Notes.

    rtol, atol float, optional

      Relative and absolute tolerances. Iterations are performed until the error is estimated to be less than `atol + rtol * abs(est)`. Here rtol controls relative accuracy (number of correct digits), while atol controls absolute accuracy (number of correct decimal places). To achieve the desired rtol, set atol to be smaller than the smallest value that can be expected from `rtol * abs(y)` so that rtol dominates the allowable error. If atol is larger than `rtol * abs(y)` the number of correct digits is not guaranteed. Conversely, to achieve the desired atol, set rtol such that `rtol * abs(y)` is always smaller than atol. Default values are 1e-8 for rtol and 0 for atol.

    max_subdivisions int, optional

      Upper bound on the number of subdivisions to perform. Default is 10,000.

    args tuple, optional

      Additional positional args passed to f, if any.

    workers int or map-like callable, optional

      If workers is an integer, part of the computation is done in parallel subdivided to this many tasks (using `multiprocessing.pool.Pool`). Supply -1 to use all cores available to the Process. Alternatively, supply a map-like callable, such as `multiprocessing.pool.Pool.map` for evaluating the population in parallel. This evaluation is carried out as `workers(func, iterable)`.

    points list of array_like, optional

      List of points to avoid evaluating f at, under the condition that the rule being used does not evaluate f on the boundary of a region (which is the case for all Genz-Malik and Gauss-Kronrod rules). This can be useful if f has a singularity at the specified point. This should be a list of array-likes where each element has length `ndim`. Default is empty. See Examples.

  Returns:
    res object

      Object containing the results of the estimation. It has the following attributes:

      estimatendarray

        Estimate of the value of the integral over the overall region specified.

      errorndarray

        Estimate of the error of the approximation over the overall region specified.

      statusstr

        Whether the estimation was successful. Can be either: “converged”, “not_converged”.

      subdivisionsint

        Number of subdivisions performed.

      atol, rtolfloat

        Requested tolerances for the approximation.

      regions: list of object

        List of objects containing the estimates of the integral over smaller regions of the domain.

      Each object in `regions` has the following attributes:

      a, bndarray

        Points describing the corners of the region. If the original integral contained infinite limits or was over a region described by region, then a and b are in the transformed coordinates.

      estimatendarray

        Estimate of the value of the integral over this region.

      errorndarray

        Estimate of the error of the approximation over this region.

  Notes

  The algorithm uses a similar algorithm to `quad_vec`, which itself is based on the implementation of QUADPACK’s DQAG* algorithms, implementing global error control and adaptive subdivision.

  The source of the nodes and weights used for Gauss-Kronrod quadrature can be found in [1], and the algorithm for calculating the nodes and weights in Genz-Malik cubature can be found in [2].

  The rules currently supported via the rule argument are:

  - `"gauss-kronrod"`, 21-node Gauss-Kronrod

  - `"genz-malik"`, n-node Genz-Malik

  If using Gauss-Kronrod for an `n`-dim integrand where `n > 2`, then the corresponding Cartesian product rule will be found by taking the Cartesian product of the nodes in the 1D case. This means that the number of nodes scales exponentially as `21^n` in the Gauss-Kronrod case, which may be problematic in a moderate number of dimensions.

  Genz-Malik is typically less accurate than Gauss-Kronrod but has much fewer nodes, so in this situation using “genz-malik” might be preferable.

  Infinite limits are handled with an appropriate variable transformation. Assuming `a = [a_1, ..., a_n]` and `b = [b_1, ..., b_n]`:

  If \(a_i = -\infty\) and \(b_i = \infty\), the i-th integration variable will use the transformation \(x = \frac{1-|t|}{t}\) and \(t \in (-1, 1)\).

  If \(a_i \ne \pm\infty\) and \(b_i = \infty\), the i-th integration variable will use the transformation \(x = a_i + \frac{1-t}{t}\) and \(t \in (0, 1)\).

  If \(a_i = -\infty\) and \(b_i \ne \pm\infty\), the i-th integration variable will use the transformation \(x = b_i - \frac{1-t}{t}\) and \(t \in (0, 1)\).

  References

  [1]

  R. Piessens, E. de Doncker, Quadpack: A Subroutine Package for Automatic Integration, files: dqk21.f, dqk15.f (1983).

  [2]

  A.C. Genz, A.A. Malik, Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region, Journal of Computational and Applied Mathematics, Volume 6, Issue 4, 1980, Pages 295-302, ISSN 0377-0427 DOI:10.1016/0771-050X(80)90039-X

  Examples

  1D integral with vector output:

  \[\int^1_0 \mathbf f(x) \text dx\]

  Where `f(x) = x^n` and `n = np.arange(10)` is a vector. Since no rule is specified, the default “gk21” is used, which corresponds to Gauss-Kronrod integration with 21 nodes.

      >>> import numpy as np
      >>> from scipy.integrate import cubature
      >>> def f(x, n):
      ...    # Make sure x and n are broadcastable
      ...    return x[:, np.newaxis]**n[np.newaxis, :]
      >>> res = cubature(
      ...     f,
      ...     a=[0],
      ...     b=[1],
      ...     args=(np.arange(10),),
      ... )
      >>> res.estimate
       array([1.        , 0.5       , 0.33333333, 0.25      , 0.2       ,
              0.16666667, 0.14285714, 0.125     , 0.11111111, 0.1       ])

  7D integral with arbitrary-shaped array output:

      f(x) = cos(2*pi*r + alphas @ x)

  for some `r` and `alphas`, and the integral is performed over the unit hybercube, \([0, 1]^7\). Since the integral is in a moderate number of dimensions, “genz-malik” is used rather than the default “gauss-kronrod” to avoid constructing a product rule with \(21^7 \approx 2 \times 10^9\) nodes.

      >>> import numpy as np
      >>> from scipy.integrate import cubature
      >>> def f(x, r, alphas):
      ...     # f(x) = cos(2*pi*r + alphas @ x)
      ...     # Need to allow r and alphas to be arbitrary shape
      ...     npoints, ndim = x.shape[0], x.shape[-1]
      ...     alphas = alphas[np.newaxis, ...]
      ...     x = x.reshape(npoints, *([1]*(len(alphas.shape) - 1)), ndim)
      ...     return np.cos(2*np.pi*r + np.sum(alphas * x, axis=-1))
      >>> rng = np.random.default_rng()
      >>> r, alphas = rng.random((2, 3)), rng.random((2, 3, 7))
      >>> res = cubature(
      ...     f=f,
      ...     a=np.array([0, 0, 0, 0, 0, 0, 0]),
      ...     b=np.array([1, 1, 1, 1, 1, 1, 1]),
      ...     rtol=1e-5,
      ...     rule="genz-malik",
      ...     args=(r, alphas),
      ... )
      >>> res.estimate
       array([[-0.79812452,  0.35246913, -0.52273628],
              [ 0.88392779,  0.59139899,  0.41895111]])

  Parallel computation with workers:

      >>> from concurrent.futures import ThreadPoolExecutor
      >>> with ThreadPoolExecutor() as executor:
      ...     res = cubature(
      ...         f=f,
      ...         a=np.array([0, 0, 0, 0, 0, 0, 0]),
      ...         b=np.array([1, 1, 1, 1, 1, 1, 1]),
      ...         rtol=1e-5,
      ...         rule="genz-malik",
      ...         args=(r, alphas),
      ...         workers=executor.map,
      ...      )
      >>> res.estimate
       array([[-0.79812452,  0.35246913, -0.52273628],
              [ 0.88392779,  0.59139899,  0.41895111]])

  2D integral with infinite limits:

  \[\int^{ \infty }_{ -\infty } \int^{ \infty }_{ -\infty } e^{-x^2-y^2} \text dy \text dx\]

      >>> def gaussian(x):
      ...     return np.exp(-np.sum(x**2, axis=-1))
      >>> res = cubature(gaussian, [-np.inf, -np.inf], [np.inf, np.inf])
      >>> res.estimate
       3.1415926

  1D integral with singularities avoided using points:

  \[\int^{ 1 }_{ -1 } \frac{\sin(x)}{x} \text dx\]

  It is necessary to use the points parameter to avoid evaluating f at the origin.

      >>> def sinc(x):
      ...     return np.sin(x)/x
      >>> res = cubature(sinc, [-1], [1], points=[[0]])
      >>> res.estimate
       1.8921661