tplquad

tplquad is a function in SciPy’s numerical integration toolkit used to compute triple integrals with variable limits. It extends the quad and dblquad routines to three dimensions, enabling an integrand that depends on three variables and limits that can vary with the outer variables.

The typical signature is tplquad(func, a, b, gfun, hfun, qfun, rfun, args=(), epsabs=1.49012e-08, epsrel=1.49012e-08). Here func is the integrand and is called as func(x, y, z, ...). The outermost integral ranges x from a to b. The middle integral over y uses bounds that depend on x, from gfun(x) to hfun(x). The innermost integral over z uses bounds that depend on x and y, from qfun(x, y) to rfun(x, y). The bounds functions must be well-defined for all points in the integration region.

tplquad performs nested one-dimensional quadratures along z, then y, then x, to evaluate the triple integral. It returns a tuple (val, abserr), containing the estimated value of the integral and an estimate of the absolute error. The tolerances epsabs and epsrel control the requested accuracy, and the integration relies on adaptive quadrature routines behind the scenes.

In practice, tplquad is suited for problems with non-rectangular integration regions where the limits are functions of the outer variables. For simpler or fixed bounds, quad or dblquad may be more straightforward, and for fully general multiple integration there are additional options such as nquad. As with other SciPy integrators, performance and accuracy depend on the smoothness of the integrand and the behavior of the limits.