scipy.integrate.nquad(func, ranges, args=None, opts=None, full_output=False)[source]#

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.

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.

rangesiterable 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).

argsiterable object, optional

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

optsiterable 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 and quad_explain.

full_outputbool, 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.


The result of the integration.


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

out_dictdict, optional

A dict containing additional information on the integration.

See also


1-D numerical integration

dblquad, tplquad

double and triple integrals


fixed-order Gaussian quadrature


adaptive Gaussian quadrature


>>> 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})
>>> scale = .1
>>> def func2(x0, x1, x2, x3, t0, t1):
...     return x0*x1*x3**2 + np.sin(x2) + 1 + (1 if x0+t1*x1-t0>0 else 0)
>>> def lim0(x1, x2, x3, t0, t1):
...     return [scale * (x1**2 + x2 + np.cos(x3)*t0*t1 + 1) - 1,
...             scale * (x1**2 + x2 + np.cos(x3)*t0*t1 + 1) + 1]
>>> def lim1(x2, x3, t0, t1):
...     return [scale * (t0*x2 + t1*x3) - 1,
...             scale * (t0*x2 + t1*x3) + 1]
>>> def lim2(x3, t0, t1):
...     return [scale * (x3 + t0**2*t1**3) - 1,
...             scale * (x3 + t0**2*t1**3) + 1]
>>> def lim3(t0, t1):
...     return [scale * (t0+t1) - 1, scale * (t0+t1) + 1]
>>> def opts0(x1, x2, x3, t0, t1):
...     return {'points' : [t0 - t1*x1]}
>>> def opts1(x2, x3, t0, t1):
...     return {}
>>> def opts2(x3, t0, t1):
...     return {}
>>> def opts3(t0, t1):
...     return {}
>>> integrate.nquad(func2, [lim0, lim1, lim2, lim3], args=(0,0),
...                 opts=[opts0, opts1, opts2, opts3])
(25.066666666666666, 2.7829590483937256e-13)