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

Compute a triple (definite) integral.

Return the triple integral of func(z, y, x) from x = a..b, y = gfun(x)..hfun(x), and z = qfun(x,y)..rfun(x,y).

Parameters:
funcfunction

A Python function or method of at least three variables in the order (z, y, x).

a, bfloat

The limits of integration in x: a < b

gfunfunction 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.

hfunfunction or float

The upper boundary curve in y (same requirements as gfun).

qfunfunction or float

The lower boundary surface in z. It must be a function that takes two floats in the order (x, y) and returns a float or a float indicating a constant boundary surface.

rfunfunction or float

The upper boundary surface in z. (Same requirements as qfun.)

argstuple, optional

Extra arguments to pass to func.

epsabsfloat, optional

Absolute tolerance passed directly to the innermost 1-D quadrature integration. Default is 1.49e-8.

epsrelfloat, optional

Relative tolerance of the innermost 1-D integrals. Default is 1.49e-8.

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 three integration results.

nevalint

The number of integrand evaluations.

Double integrals

N-dimensional integrals

romb

Integrators for sampled data

simpson

Integrators for sampled data

ode

ODE integrators

odeint

ODE integrators

scipy.special

For coefficients and roots of orthogonal polynomials

Notes

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

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 triple integral of x * y * z, over x ranging from 1 to 2, y ranging from 2 to 3, z ranging from 0 to 1. That is, $$\int^{x=2}_{x=1} \int^{y=3}_{y=2} \int^{z=1}_{z=0} x y z \,dz \,dy \,dx$$.

>>> import numpy as np
>>> from scipy import integrate
>>> f = lambda z, y, x: x*y*z
>>> integrate.tplquad(f, 1, 2, 2, 3, 0, 1)
(1.8749999999999998, 3.3246447942574074e-14)

Calculate $$\int^{x=1}_{x=0} \int^{y=1-2x}_{y=0} \int^{z=1-x-2y}_{z=0} x y z \,dz \,dy \,dx$$. Note: qfun/rfun takes arguments in the order (x, y), even though f takes arguments in the order (z, y, x).

>>> f = lambda z, y, x: x*y*z
>>> integrate.tplquad(f, 0, 1, 0, lambda x: 1-2*x, 0, lambda x, y: 1-x-2*y)
(0.05416666666666668, 2.1774196738157757e-14)

Calculate $$\int^{x=1}_{x=0} \int^{y=1}_{y=0} \int^{z=1}_{z=0} a x y z \,dz \,dy \,dx$$ for $$a=1, 3$$.

>>> f = lambda z, y, x, a: a*x*y*z
>>> integrate.tplquad(f, 0, 1, 0, 1, 0, 1, args=(1,))
(0.125, 5.527033708952211e-15)
>>> integrate.tplquad(f, 0, 1, 0, 1, 0, 1, args=(3,))
(0.375, 1.6581101126856635e-14)

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

>>> f = lambda x, y, z: np.exp(-(x ** 2 + y ** 2 + z ** 2))
>>> integrate.tplquad(f, -np.inf, np.inf, -np.inf, np.inf, -np.inf, np.inf)
(5.568327996830833, 4.4619078828029765e-08)