Computes the Boys function, $F_m (T) = \int_0^1 u^{2m} \exp(-T u^2) \, {\rm d}u$ , using single algorithm (asymptotic expansion). More...

#include <boys.h>

Static Public Member Functions
static Real	eval (Real T, size_t m, Real absolute_precision)
	computes a single value of using MacLaurin series.

static void	eval (Real *Fm, Real T, size_t mmax, Real absolute_precision)
	fills up an array of Fm(T) for m in [0,mmax] More...

Detailed Description

template<typename Real>
struct libint2::FmEval_Reference< Real >

Computes the Boys function, $F_m (T) = \int_0^1 u^{2m} \exp(-T u^2) \, {\rm d}u$ , using single algorithm (asymptotic expansion).

Slow for the sake of precision control. Useful in two cases:

for reference purposes, if Real supports high/arbitrary precision, and
for moderate values of , if Real is a low-precision floating-point type. N.B. FmEval_Reference2 , which can compute for all practical values of and , is recommended with standard Real types (double and float).

Note: Precision is controlled heuristically, i.e. cannot be guaranteed mathematically; will stop if absolute precision is reached, or precision of Real is exhausted. It is important that std::numeric_limits<Real> is defined appropriately.

Template Parameters

Real	the type to use for all floating-point computations. Must be able to compute logarithm and exponential, i.e. log(x) and exp(x), where x is Real, must be valid expressions.

Member Function Documentation

template<typename Real >

static void libint2::FmEval_Reference< Real >::eval	(	Real *	Fm,
		Real	T,
		size_t	mmax,
		Real	absolute_precision
	)

inlinestatic

fills up an array of Fm(T) for m in [0,mmax]

Parameters

[out]	Fm	array to be filled in with the Boys function values, must be at least mmax+1 elements long
[in]	T	the Boys function argument
[in]	mmax	the maximum value of m for which Boys function will be computed;
[in]	absolute_precision	the absolute precision to which to compute the result

downward recursion does not maintain absolute precision, only relative precision, and cannot be used for T > 10 if (mmax > 0) { const Real T2 = 2.0 * T; const Real exp_T = exp(-T); for (int m = mmax - 1; m >= 0; m–) Fm[m] = (Fm[m + 1] * T2 + exp_T) / (2 * m + 1); }

The documentation for this struct was generated from the following file:

boys.h

Static Public Member Functions

Detailed Description

template<typename Real> struct libint2::FmEval_Reference< Real >

Member Function Documentation

template<typename Real>
struct libint2::FmEval_Reference< Real >