Module integrals 

Incomplete gamma and Kambe integrals used by the Ewald lattice sums. Recurrences follow DLMF 8.8 and Kambe (1968), as in treams.

Upstream: treams.special.incgamma and intkambe (_integrals.pyx).

incgamma and intkambe evaluate one integral after checking its inputs. incgamma_array and intkambe_array evaluate them over arrays and return an IncgammaResidual or IntkambeResidual, which keeps the arguments for the gradient. The private submodules hold the numerics:

• gamma: the upper incomplete gamma function and its scaled ladder;
• kambe: the Kambe integrals;
• double: double-double arithmetic for the small-split Kambe series.

The Kambe integrals have several entry points, one per kind of caller:

Entry point	Returns	Caller
intkambe	I_n(z, eta) after checking the inputs	treams_rs.special.intkambe, intkambe_array
kambe	I_n(z, eta) without checks	intkambe, the IntkambeResidual pullback (order n + 2), real-space terms of the Ewald sums (lattice::real)
kambe_with_bound	I_n and a bound on its rounding error	integer-order reduced integrals of the 1D spherical sums (lattice::reduced)
kambe_series, kambe_series_with	the series over scaled incomplete gamma functions	kambe, kambe_with_bound and SmallSplitKambe only
EvenKambe	the even orders of one argument from one recurrence	half-integer reduced integrals of the 2D spherical and 1D cylindrical sums (lattice::reduced)
SmallSplitKambe	the orders of real-space terms at a split below every automatic one, with bounds	real-space terms at small splits (lattice::real)

Modules

double 🔒

Double-double arithmetic: reals hi + lo with |lo| at most half an ulp of hi, and complex numbers with such parts.

gamma 🔒

The upper incomplete gamma function Gamma(n, z) of integer and half-integer degree.

kambe 🔒

Kambe integrals I_n(z, eta) = integral_eta^infinity t^n exp(-z^2 t^2/2 + 1/(2t^2)) dt with first-order bounds on their rounding errors.

Structs

IncgammaResidual

What incgamma_array saves for its pullback: the degrees and the arguments. Only the arguments get gradients.

IntkambeResidual

What intkambe_array saves for its pullback: the orders and the arguments. The pullback uses the adjacent-order recurrence and the endpoint integrand.

Constants

BROADCAST 🔒

COTANGENT 🔒

PARALLEL 🔒

Elementwise integral evaluations and pullbacks run in parallel from this many elements.

Functions

incgamma

Upper incomplete gamma function Gamma(n, z) = integral_z^infinity t^(n-1) e^-t dt.

incgamma_array

incgamma over arrays, each input one value or one value per element, with the residual of their argument gradient.

intkambe

The Kambe integral I_n(z, eta) = integral_eta^infinity t^n exp(-z^2 t^2/2 + 1/(2t^2)) dt.

intkambe_array

intkambe over arrays, each input one value or one value per element, with the residual of their z and eta gradients; the integer order has none.

kambe_eta_derivative 🔒

kambe_z_derivative 🔒