FEE method

In mathematics, the FEE method is the method of fast summation of series of a special form. It was constructed in 1990 by Ekaterina A. Karatsuba and was called FEE – fast E-function evaluation – because it makes fast computations of the Siegel formula_1 -functions possible, in particular of formula_2.
A class of functions, which are 'similar to the exponential function' was given the name 'E-functions' by Siegel. Among these functions are such special functions as the hypergeometric function, cylinder, spherical functions and so on. 
Using the FEE, it is possible to prove the following theorem:
Theorem: Let formula_3 be an elementary transcendental function, that is the exponential function, or a 
trigonometric function, or an elementary algebraic function, or their superposition, or their inverse, or a superposition of the inverses. Then
Here formula_5 is the complexity of computation (bit) of the function formula_6 with accuracy up to formula_7 digits, formula_8 is the complexity of multiplication of two formula_7-digit integers.
The algorithms based on the method FEE include the algorithms for fast calculation of any elementary transcendental function for any value of the argument, the classical constants e, formula_10 the Euler constant formula_11 the Catalan and the Apéry constants, such higher transcendental functions as the Euler gamma function and its derivatives, the hypergeometric, spherical, cylinder (including the Bessel) functions and some other functions for
algebraic values of the argument and parameters, the Riemann zeta function for integer values of the argument and the Hurwitz zeta function for integer argument and algebraic values of the parameter, and also such special integrals as the integral of probability, the Fresnel integrals, the integral exponential function, the trigonometric integrals, and some other integrals for algebraic values of the argument with the complexity bound which is close to the optimal one, namely
At present, only the FEE makes it possible to calculate fast the values of the functions from the class of higher transcendental functions, certain special integrals of mathematical physics and such classical constants as Euler's, Catalan's and Apéry's constants. An additional advantage of the method FEE is the possibility of parallelizing the algorithms based on the FEE.
FEE-computation of classical constants.
For fast evaluation of the
constant formula_10 one can use the Euler formula
formula_14
and apply the FEE to sum the Taylor series for
with the remainder terms formula_17 formula_18 which satisfy the bounds
and for
To calculate formula_23 by the
FEE it is possible to use also other approximations In all cases the complexity is
To compute the Euler constant gamma with accuracy up to formula_7
digits, it is necessary to sum by the FEE two series. Namely, for
The complexity is
To evaluate fast the constant formula_29 
it is possible to apply the
FEE to other approximations.
FEE-computation of certain power series.
By the FEE the two following series are calculated fast:
under the assumption that formula_32 are
integers,
and formula_34 are constants, and formula_35 is an algebraic number. The complexity of the evaluation of the series is
The FEE details on the example of fast calculation of the classical constant "e".
For the evaluation of the constant formula_38 take formula_39, terms of the Taylor series for formula_40
Here we choose formula_42, requiring that for the remainder formula_43 the
inequality formula_44 is fulfilled. This is the case, for
example, when formula_45 Thus, we take formula_46
such that the natural number formula_47 is determined by the
inequalities:
We calculate the sum
in formula_47 steps of the following process.
Step 1. Combining in formula_51 the summands sequentially in pairs we
carry out of the brackets the "obvious" common factor and obtain
We shall compute only integer values of the expressions in the
parentheses, that is the values
Thus, at the first step the sum formula_51 is into
At the first step formula_57 integers of the form
are calculated. After that we act in a similar way: combining on
each step the summands of the sum formula_51 sequentially in pairs, we
take out of the brackets the 'obvious' common factor and compute
only the integer values of the expressions in the brackets. Assume
that the first formula_60 steps of this process are completed.
Step formula_61 (formula_62).
we compute only formula_65 integers of the form
Here
is the product of formula_69 integers.
Etc.
Step formula_47, the last one. We compute one integer value
formula_71 we compute, using the fast algorithm described
above the value formula_72 and make one division of the integer
formula_73 by the integer formula_72 
with accuracy up to formula_7
digits. The obtained result is the sum formula_76 or the constant formula_38 up
to formula_7 digits. The complexity of all computations is