In mathematics, the FEE method, or fast E-function evaluation method, is the method of fast summation of series of a special form. It was constructed in 1990 by Ekaterina Karatsuba and is so-named because it makes fast computations of the Siegel E-functions.

A class of functions, which are "similar to the exponential function," was given the name "E-functions" by Carl Ludwig 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 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

is the complexity of computation (bit)-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, \pi, the Euler constant 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

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

one can use the Machin-like formula and apply the FEE to sum the Taylor series for

which satisfy the bounds

and for

by the FEE it is possible to use also other approximations In all cases the complexity is

digits, it is necessary to sum by the FEE two series. Namely, for

The complexity is

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:

are integers,

is an algebraic number. The complexity of the evaluation of the series is

FEE calculation of the classical constant e

is determined by the inequalities:

We calculate the sum

steps of the following process.

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

is into

integers of the form

steps of this process are completed.

).

integers of the form

Here

integers.

Etc.

digits. The complexity of all computations is

See also

• Fast algorithms

• AGM method

• Computational complexity

References

External links

• http://www.ccas.ru/personal/karatsuba/divcen.htm

• http://www.ccas.ru/personal/karatsuba/algen.htm