scipy.integrate.nquad#
- 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.- Parameters:
- func{callable, scipy.LowLevelCallable}
The function to be integrated. Has arguments of
x0, ... xn
,t0, ... tm
, where integration is carried out overx0, ... xn
, which must be floats. Wheret0, ... tm
are extra arguments passed in args. Function signature should befunc(x0, x1, ..., xn, t0, t1, ..., tm)
. Integration is carried out in order. That is, integration overx0
is the innermost integral, andxn
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. Thexx
array contains the coordinates and extra arguments.user_data
is the data contained in thescipy.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., iffunc = f(x0, x1, x2, t0, t1)
, thenranges[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 byfunc
,ranges
, andopts
.- 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 overx0
, and so on. If a callable, the signature must be the same as forranges
. 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 evaluationsneval
can be obtained by settingfull_output=True
when calling nquad.
- Returns:
- resultfloat
The result of the integration.
- abserrfloat
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
quad
1-D numerical integration
dblquad
,tplquad
double and triple integrals
fixed_quad
fixed-order Gaussian quadrature
quadrature
adaptive 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 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)