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)fromx = a..bandy = 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.
dblquadtries to obtain an accuracy ofabs(i-result) <= max(epsabs, epsrel*abs(i))wherei= inner integral offunc(y, x)fromgfun(x)tohfun(x), andresultis 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 and50 * (machine epsilon). See epsabs above.
- Returns:
- yfloat
The resultant integral.
- abserrfloat
An estimate of the error.
See also
quadsingle integral
tplquadtriple integral
nquadN-dimensional integrals
fixed_quadfixed-order Gaussian quadrature
simpsonintegrator for sampled data
rombintegrator for sampled data
scipy.specialfor 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
quadcalls 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,qagseis used for finite limits orqagieis 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
QAGSis 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**2over the boxxranging from 0 to 2 andyranging 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)