`scipy.integrate._quadpack_py:tplquad`

source: /scipy/integrate/_quadpack_py.py :832

Signature

   def     tplquad (    func    ,    a    ,    b    ,    gfun    ,    hfun    ,    qfun    ,    rfun    ,    args    =  ()   ,    epsabs    =  1.49e-08   ,    epsrel    =  1.49e-08     )

Summary

Compute a triple (definite) integral.

Extended Summary

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

func : function: A Python function or method of at least three variables in the order (z, y, x).
a, b : float: The limits of integration in x: a < b
gfun : function 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.
hfun : function or float: The upper boundary curve in y (same requirements as gfun).
qfun : function 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.
rfun : function or float: The upper boundary surface in z. (Same requirements as qfun.)
args : tuple, optional: Extra arguments to pass to func.
epsabs : float, optional: Absolute tolerance passed directly to the innermost 1-D quadrature integration. Default is 1.49e-8.
epsrel : float, optional: Relative tolerance of the innermost 1-D integrals. Default is 1.49e-8.

Returns

y : float: The resultant integral.
abserr : float: An estimate of the error.

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. The integration is is performed using a 21-point Gauss-Kronrod quadrature within each subinterval.
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.

Array API Standard Support

tplquad has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

====================  ====================  ====================
Library               CPU                   GPU
====================  ====================  ====================
NumPy                 ✅                     n/a                 
CuPy                  n/a                   ⛔                   
PyTorch               ⛔                     ⛔                   
JAX                   ⛔                     ⛔                   
Dask                  ⛔                     n/a                 
====================  ====================  ====================

See dev-arrayapi for more information.

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, :math:`\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)

✗

Calculate :math:`\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)

✗

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

f = lambda z, y, x, a: a*x*y*z

✓

integrate.tplquad(f, 0, 1, 0, 1, 0, 1, args=(1,))
integrate.tplquad(f, 0, 1, 0, 1, 0, 1, args=(3,))

✗

Compute the three-dimensional Gaussian Integral, which is the integral of the Gaussian function :math:`f(x,y,z) = e^{-(x^{2} + y^{2} + z^{2})}`, over :math:`(-\infty,+\infty)`. That is, compute the integral :math:`\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)

✗

Aliases

scipy.integrate.tplquad