scipy.integrate.dblquad#

scipy.integrate.dblquad(func, a, b, gfun, hfun, args=(), epsabs=1.49e-08, epsrel=1.49e-08)[source]#

Compute a double integral.

Return the double (definite) integral of func(y, x) from x = a..b and y = gfun(x)..hfun(x).

Parameters:

funccallable: A Python function or method of at least two variables: y must be the first argument and x the second argument.
a, bfloat: The limits of integration in x: a < b
gfuncallable or float: The lower boundary curve in y which is a function taking a single floating point argument (x) and returning a floating point result or a float indicating a constant boundary curve.
hfuncallable or float: The upper boundary curve in y (same requirements as gfun).
argssequence, optional: Extra arguments to pass to func.
epsabsfloat, optional: Absolute tolerance passed directly to the inner 1-D quadrature integration. Default is 1.49e-8. dblquad tries to obtain an accuracy of abs(i-result) <= max(epsabs, epsrel*abs(i)) where i = inner integral of func(y, x) from gfun(x) to hfun(x), and result is the numerical approximation. See epsrel below.
epsrelfloat, optional: Relative tolerance of the inner 1-D integrals. Default is 1.49e-8. If epsabs <= 0, epsrel must be greater than both 5e-29 and 50 * (machine epsilon). See epsabs above.

Returns:

yfloat: The resultant integral.
abserrfloat: An estimate of the error.

See also

quad: single integral
tplquad: triple integral
nquad: N-dimensional integrals
fixed_quad: fixed-order Gaussian quadrature
quadrature: adaptive Gaussian quadrature
odeint: ODE integrator
ode: ODE integrator
simpson: integrator for sampled data
romb: integrator for sampled data
scipy.special: for coefficients and roots of orthogonal polynomials

Notes

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

Details of QUADPACK level routines

quad 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. For each level of integration, qagse is used for finite limits or qagie is used if either limit (or both!) are infinite. 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.

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

Compute the double integral of x * y**2 over the box x ranging from 0 to 2 and y ranging from 0 to 1. That is, \(\int^{x=2}_{x=0} \int^{y=1}_{y=0} x y^2 \,dy \,dx\).

>>> import numpy as np
>>> from scipy import integrate
>>> f = lambda y, x: x*y**2
>>> integrate.dblquad(f, 0, 2, 0, 1)
    (0.6666666666666667, 7.401486830834377e-15)

Calculate \(\int^{x=\pi/4}_{x=0} \int^{y=\cos(x)}_{y=\sin(x)} 1 \,dy \,dx\).

>>> f = lambda y, x: 1
>>> integrate.dblquad(f, 0, np.pi/4, np.sin, np.cos)
    (0.41421356237309503, 1.1083280054755938e-14)

Calculate \(\int^{x=1}_{x=0} \int^{y=2-x}_{y=x} a x y \,dy \,dx\) for \(a=1, 3\).

>>> f = lambda y, x, a: a*x*y
>>> integrate.dblquad(f, 0, 1, lambda x: x, lambda x: 2-x, args=(1,))
    (0.33333333333333337, 5.551115123125783e-15)
>>> integrate.dblquad(f, 0, 1, lambda x: x, lambda x: 2-x, args=(3,))
    (0.9999999999999999, 1.6653345369377348e-14)

Compute the two-dimensional Gaussian Integral, which is the integral of the Gaussian function \(f(x,y) = e^{-(x^{2} + y^{2})}\), over \((-\infty,+\infty)\). That is, compute the integral \(\iint^{+\infty}_{-\infty} e^{-(x^{2} + y^{2})} \,dy\,dx\).

>>> f = lambda x, y: np.exp(-(x ** 2 + y ** 2))
>>> integrate.dblquad(f, -np.inf, np.inf, -np.inf, np.inf)
    (3.141592653589777, 2.5173086737433208e-08)