tplquad — SciPy v1.15.3 Manual (original) (raw)
scipy.integrate.
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:
yfloat
The resultant integral.
abserrfloat
An estimate of the error.
See also
Adaptive quadrature using QUADPACK
Fixed-order Gaussian quadrature
Double integrals
N-dimensional integrals
Integrators for sampled data
Integrators for sampled data
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, qagseis 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: xyz 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 ftakes arguments in the order (z, y, x).
f = lambda z, y, x: xyz integrate.tplquad(f, 0, 1, 0, lambda x: 1-2x, 0, lambda x, y: 1-x-2y) (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: axy*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)