rmation 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 :math:`\int^b_a \cos(\omega x)f(x)dx` or :math:`\int^b_a \sin(\omega x)f(x)dx` over a finite interval [a,b], where :math:`\omega` and :math:`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 :math:`f(x)`. qawfe calculates the Fourier transform :math:`\int^\infty_a \cos(\omega x)f(x)dx` or :math:`\int^\infty_a \sin(\omega x)f(x)dx` for user-provided :math:`\omega` and :math:`f`. The procedure of ``QAWO`` is applied on successive finite intervals, and convergence acceleration by means of the :math:`\varepsilon`-algorithm is applied to the series of integral approximations. qawse approximate :math:`\int^b_a w(x)f(x)dx`, with :math:`a < b` where :math:`w(x) = (x-a)^{\alpha}(b-x)^{\beta}v(x)` with :math:`\alpha,\beta > -1`, where :math:`v(x)` may be one of the following functions: :math:`1`, :math:`\log(x-a)`, :math:`\log(b-x)`, :math:`\log(x-a)\log(b-x)`. The user specifies :math:`\alpha`, :math:`\beta` and the type of the function :math:`v`. A globally adaptive subdivision strategy is applied, with modified Clenshaw-Curtis integration on those subintervals which contain `a` or `b`. qawce compute :math:`\int^b_a f(x) / (x-c)dx` where the integral must be interpreted as a Cauchy principal value integral, for user specified :math:`c` and :math:`f`. The strategy is globally adaptive. Modified Clenshaw-Curtis integration is used on those intervals containing the point :math:`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 .. math:: \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 .. math:: 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} . >>> 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 .. math:: \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 .. math:: 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} and :math:`(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) r