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scipy.integrate.

# nquad

scipy.integrate.nquad(func, ranges, args=None, opts=None, full_output=False)

  Integration over multiple variables.

  Wraps `quad` to enable integration over multiple variables. Various options allow improved integration of discontinuous functions, as well as the use of weighted integration, and generally finer control of the integration process.

  Parameters:
    func{callable, scipy.LowLevelCallable}

      The function to be integrated. Has arguments of `x0, ... xn`, `t0, ... tm`, where integration is carried out over `x0, ... xn`, which must be floats. Where `t0, ... tm` are extra arguments passed in args. Function signature should be `func(x0, x1, ..., xn, t0, t1, ..., tm)`. Integration is carried out in order. That is, integration over `x0` is the innermost integral, and `xn` is the outermost.

      If the user desires improved integration performance, then f may be a `scipy.LowLevelCallable` with one of the signatures:

          double func(int n, double *xx)
          double func(int n, double *xx, void *user_data)

      where `n` is the number of variables and args. The `xx` array contains the coordinates and extra arguments. `user_data` is the data contained in the `scipy.LowLevelCallable`.

    ranges iterable object

      Each element of ranges may be either a sequence of 2 numbers, or else a callable that returns such a sequence. `ranges[0]` corresponds to integration over x0, and so on. If an element of ranges is a callable, then it will be called with all of the integration arguments available, as well as any parametric arguments. e.g., if `func = f(x0, x1, x2, t0, t1)`, then `ranges[0]` may be defined as either `(a, b)` or else as `(a, b) = range0(x1, x2, t0, t1)`.

    args iterable object, optional

      Additional arguments `t0, ... tn`, required by `func`, `ranges`, and `opts`.

    opts iterable object or dict, optional

      Options to be passed to `quad`. May be empty, a dict, or a sequence of dicts or functions that return a dict. If empty, the default options from scipy.integrate.quad are used. If a dict, the same options are used for all levels of integraion. If a sequence, then each element of the sequence corresponds to a particular integration. e.g., `opts[0]` corresponds to integration over `x0`, and so on. If a callable, the signature must be the same as for `ranges`. The available options together with their default values are:

      > - epsabs = 1.49e-08

      > - epsrel = 1.49e-08

      > - limit = 50

      > - points = None

      > - weight = None

      > - wvar = None

      > - wopts = None

      For more information on these options, see `quad`.

    full_output bool, optional

      Partial implementation of `full_output` from scipy.integrate.quad. The number of integrand function evaluations `neval` can be obtained by setting `full_output=True` when calling nquad.

  Returns:
    result float

      The result of the integration.

    abserr float

      The maximum of the estimates of the absolute error in the various integration results.

    out_dict dict, optional

      A dict containing additional information on the integration.

  See also

  `quad`

    1-D numerical integration

  `dblquad`, `tplquad`

    double and triple integrals

  `fixed_quad`

    fixed-order Gaussian quadrature

  Notes

  For valid results, the integral must converge; behavior for divergent integrals is not guaranteed.

  Details of QUADPACK level routines

  `nquad` calls routines from the FORTRAN library QUADPACK. This section provides details on the conditions for each routine to be called and a short description of each routine. The routine called depends on weight, points and the integration limits a and b.

  QUADPACK routine | weight | points | infinite bounds
  --- | --- | --- | ---
  qagse | None | No | No
  qagie | None | No | Yes
  qagpe | None | Yes | No
  qawoe | ‘sin’, ‘cos’ | No | No
  qawfe | ‘sin’, ‘cos’ | No | either a or b
  qawse | ‘alg*’ | No | No
  qawce | ‘cauchy’ | No | No

  The following provides a short description from [1] for each routine.

  qagse

    is an integrator based on globally adaptive interval subdivision in connection with extrapolation, which will eliminate the effects of integrand singularities of several types.

  qagie

    handles integration over infinite intervals. The infinite range is mapped onto a finite interval and subsequently the same strategy as in `QAGS` is applied.

  qagpe

    serves the same purposes as QAGS, but also allows the user to provide explicit information about the location and type of trouble-spots i.e. the abscissae of internal singularities, discontinuities and other difficulties of the integrand function.

  qawoe

    is an integrator for the evaluation of \(\int^b_a \cos(\omega x)f(x)dx\) or \(\int^b_a \sin(\omega x)f(x)dx\) over a finite interval [a,b], where \(\omega\) and \(f\) are specified by the user. The rule evaluation component is based on the modified Clenshaw-Curtis technique

    An adaptive subdivision scheme is used in connection with an extrapolation procedure, which is a modification of that in `QAGS` and allows the algorithm to deal with singularities in \(f(x)\).

  qawfe

    calculates the Fourier transform \(\int^\infty_a \cos(\omega x)f(x)dx\) or \(\int^\infty_a \sin(\omega x)f(x)dx\) for user-provided \(\omega\) and \(f\). The procedure of `QAWO` is applied on successive finite intervals, and convergence acceleration by means of the \(\varepsilon\)-algorithm is applied to the series of integral approximations.

  qawse

    approximate \(\int^b_a w(x)f(x)dx\), with \(a < b\) where \(w(x) = (x-a)^{\alpha}(b-x)^{\beta}v(x)\) with \(\alpha,\beta > -1\), where \(v(x)\) may be one of the following functions: \(1\), \(\log(x-a)\), \(\log(b-x)\), \(\log(x-a)\log(b-x)\).

    The user specifies \(\alpha\), \(\beta\) and the type of the function \(v\). A globally adaptive subdivision strategy is applied, with modified Clenshaw-Curtis integration on those subintervals which contain a or b.

  qawce

    compute \(\int^b_a f(x) / (x-c)dx\) where the integral must be interpreted as a Cauchy principal value integral, for user specified \(c\) and \(f\). The strategy is globally adaptive. Modified Clenshaw-Curtis integration is used on those intervals containing the point \(x = c\).

  References

  [1]

  Piessens, Robert; de Doncker-Kapenga, Elise; Überhuber, Christoph W.; Kahaner, David (1983). QUADPACK: A subroutine package for automatic integration. Springer-Verlag. ISBN 978-3-540-12553-2.

  Examples

  Calculate

  \[\int^{1}_{-0.15} \int^{0.8}_{0.13} \int^{1}_{-1} \int^{1}_{0} f(x_0, x_1, x_2, x_3) \,dx_0 \,dx_1 \,dx_2 \,dx_3 ,\]

  where

  \[\begin{split}f(x_0, x_1, x_2, x_3) = \begin{cases} x_0^2+x_1 x_2-x_3^3+ \sin{x_0}+1 & (x_0-0.2 x_3-0.5-0.25 x_1 > 0) \\ x_0^2+x_1 x_2-x_3^3+ \sin{x_0}+0 & (x_0-0.2 x_3-0.5-0.25 x_1 \leq 0) \end{cases} .\end{split}\]

      >>> import numpy as np
      >>> from scipy import integrate
      >>> func = lambda x0,x1,x2,x3 : x0**2 + x1*x2 - x3**3 + np.sin(x0) + (
      ...                                 1 if (x0-.2*x3-.5-.25*x1>0) else 0)
      >>> def opts0(*args, **kwargs):
      ...     return {'points':[0.2*args[2] + 0.5 + 0.25*args[0]]}
      >>> integrate.nquad(func, [[0,1], [-1,1], [.13,.8], [-.15,1]],
      ...                 opts=[opts0,{},{},{}], full_output=True)
      (1.5267454070738633, 2.9437360001402324e-14, {'neval': 388962})

  Calculate

  \[\int^{t_0+t_1+1}_{t_0+t_1-1} \int^{x_2+t_0^2 t_1^3+1}_{x_2+t_0^2 t_1^3-1} \int^{t_0 x_1+t_1 x_2+1}_{t_0 x_1+t_1 x_2-1} f(x_0,x_1, x_2,t_0,t_1) \,dx_0 \,dx_1 \,dx_2,\]

  where

  \[\begin{split}f(x_0, x_1, x_2, t_0, t_1) = \begin{cases} x_0 x_2^2 + \sin{x_1}+2 & (x_0+t_1 x_1-t_0 > 0) \\ x_0 x_2^2 +\sin{x_1}+1 & (x_0+t_1 x_1-t_0 \leq 0) \end{cases}\end{split}\]

  and \((t_0, t_1) = (0, 1)\) .

      >>> def func2(x0, x1, x2, t0, t1):
      ...     return x0*x2**2 + np.sin(x1) + 1 + (1 if x0+t1*x1-t0>0 else 0)
      >>> def lim0(x1, x2, t0, t1):
      ...     return [t0*x1 + t1*x2 - 1, t0*x1 + t1*x2 + 1]
      >>> def lim1(x2, t0, t1):
      ...     return [x2 + t0**2*t1**3 - 1, x2 + t0**2*t1**3 + 1]
      >>> def lim2(t0, t1):
      ...     return [t0 + t1 - 1, t0 + t1 + 1]
      >>> def opts0(x1, x2, t0, t1):
      ...     return {'points' : [t0 - t1*x1]}
      >>> def opts1(x2, t0, t1):
      ...     return {}
      >>> def opts2(t0, t1):
      ...     return {}
      >>> integrate.nquad(func2, [lim0, lim1, lim2], args=(0,1),
      ...                 opts=[opts0, opts1, opts2])
      (36.099919226771625, 1.8546948553373528e-07)