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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.

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.

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.

Deprecated since version 1.11.0: Parameter full_output is deprecated and will be removed in version 1.13.0. When full_output is unspecified, neval is provided as an attribute of the result object.

Returns:

An object with the following attributes:

integralfloat

The result of the integration.

abserrfloat

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

nevalint

The number of integrand evaluations.

quad

1-D numerical integration

dblquad, tplquad

double and triple integrals

fixed_quad

quadrature

Notes

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

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.

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 desciption 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)