FEE method

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^[1][2] and is so-named because it makes fast computations of the Siegel E-functions possible, in particular of ${\displaystyle e^{x}}$.

A class of functions, which are "similar to the exponential function," was given the name "E-functions" by Carl Ludwig Siegel.^[3] 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 ${\displaystyle y=f(x)}$ 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

${\displaystyle s_{f}(n)=O(M(n)\log ^{2}n).\,}$

Here ${\displaystyle s_{f}(n)}$ is the complexity of computation (bit) of the function ${\displaystyle f(x)}$ with accuracy up to ${\displaystyle n}$ digits, ${\displaystyle M(n)}$ is the complexity of multiplication of two ${\displaystyle n}$-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, ${\displaystyle \pi ,}$ the Euler constant ${\displaystyle \gamma ,}$ the Catalan and the Apéry constants,^[4] such higher transcendental functions as the Euler gamma function and its derivatives, the hypergeometric,^[5] spherical, cylinder (including the Bessel)^[6] functions and some other functions for algebraic values of the argument and parameters, the Riemann zeta function for integer values of the argument^[7][8] and the Hurwitz zeta function for integer argument and algebraic values of the parameter,^[9] and also such special integrals as the integral of probability, the Fresnel integrals, the integral exponential function, the trigonometric integrals, and some other integrals^[10] for algebraic values of the argument with the complexity bound which is close to the optimal one, namely

${\displaystyle s_{f}(n)=O(M(n)\log ^{2}n).\,}$

The FEE makes it possible to calculate fast the values of the functions from the class of higher transcendental functions,^[11] certain special integrals of mathematical physics and such classical constants as Euler's, Catalan's^[12] and Apéry's constants. An additional advantage of the method FEE is the possibility of parallelizing the algorithms based on the FEE.

For fast evaluation of the constant ${\displaystyle \pi ,}$ one can use the Euler formula ${\displaystyle {\frac {\pi }{4}}=\arctan {\frac {1}{2}}+\arctan {\frac {1}{3}},}$ and apply the FEE to sum the Taylor series for

${\displaystyle \arctan {\frac {1}{2}}={\frac {1}{1\cdot 2}}-{\frac {1}{3\cdot 2^{3}}}+\cdots +{\frac {(-1)^{r-1}}{(2r-1)2^{2r-1}}}+R_{1},}$

${\displaystyle \arctan {\frac {1}{3}}={\frac {1}{1\cdot 3}}-{\frac {1}{3\cdot 3^{3}}}+\cdots +{\frac {(-1)^{r-1}}{(2r-1)3^{2r-1}}}+R_{2},}$

with the remainder terms ${\displaystyle R_{1},}$ ${\displaystyle R_{2},}$ which satisfy the bounds

${\displaystyle |R_{1}|\leq {\frac {4}{5}}{\frac {1}{2r+1}}{\frac {1}{2^{2r+1}}};}$

${\displaystyle |R_{2}|\leq {\frac {9}{10}}{\frac {1}{2r+1}}{\frac {1}{3^{2r+1}}};}$

and for

${\displaystyle r=n,\,}$

${\displaystyle 4(|R_{1}|+|R_{2}|)\ <\ 2^{-n}.}$

To calculate ${\displaystyle \pi }$ by the FEE it is possible to use also other approximations^[13] In all cases the complexity is

${\displaystyle s_{\pi }=O(M(n)\log ^{2}n).\,}$

To compute the Euler constant gamma with accuracy up to ${\displaystyle n}$ digits, it is necessary to sum by the FEE two series. Namely, for

${\displaystyle m=6n,\quad k=n,\quad k\geq 1,\,}$

${\displaystyle \gamma =-\log n\sum _{r=0}^{12n}{\frac {(-1)^{r}n^{r+1}}{(r+1)!}}+\sum _{r=0}^{12n}{\frac {(-1)^{r}n^{r+1}}{(r+1)!(r+1)}}+O(2^{-n}).}$

The complexity is

${\displaystyle s_{\gamma }=O(M(n)\log ^{2}n).\,}$

To evaluate fast the constant ${\displaystyle \gamma }$ it is possible to apply the FEE to other approximations.^[14]

By the FEE the two following series are calculated fast:

${\displaystyle f_{1}=f_{1}(z)=\sum _{j=0}^{\infty }{\frac {a(j)}{b(j)}}z^{j},}$

${\displaystyle f_{2}=f_{2}(z)=\sum _{j=0}^{\infty }{\frac {a(j)}{b(j)}}{\frac {z^{j}}{j!}},}$

under the assumption that ${\displaystyle a(j),\quad b(j)}$ are integers,

${\displaystyle |a(j)|+|b(j)|\leq (Cj)^{K};\quad |z|\ <\ 1;\quad K}$

and ${\displaystyle C}$ are constants, and ${\displaystyle z}$ is an algebraic number. The complexity of the evaluation of the series is

${\displaystyle s_{f_{1}}(n)=O\left(M(n)\log ^{2}n\right),\,}$

${\displaystyle s_{f_{2}}(n)=O\left(M(n)\log n\right).}$

For the evaluation of the constant ${\displaystyle e}$ take ${\displaystyle m=2^{k},\quad k\geq 1}$, terms of the Taylor series for ${\displaystyle e,}$

${\displaystyle e=1+{\frac {1}{1!}}+{\frac {1}{2!}}+\cdots +{\frac {1}{(m-1)!}}+R_{m}.}$

Here we choose ${\displaystyle m}$, requiring that for the remainder ${\displaystyle R_{m}}$ the inequality ${\displaystyle R_{m}\leq 2^{-n-1}}$ is fulfilled. This is the case, for example, when ${\displaystyle m\geq {\frac {4n}{\log n}}.}$ Thus, we take ${\displaystyle m=2^{k}}$ such that the natural number ${\displaystyle k}$ is determined by the inequalities:

${\displaystyle 2^{k}\geq {\frac {4n}{\log n}}>2^{k-1}.}$

We calculate the sum

${\displaystyle S=1+{\frac {1}{1!}}+{\frac {1}{2!}}+\cdots +{\frac {1}{(m-1)!}}=\sum _{j=0}^{m-1}{\frac {1}{(m-1-j)!}},}$

in ${\displaystyle k}$ steps of the following process.

Step 1. Combining in ${\displaystyle S}$ the summands sequentially in pairs we carry out of the brackets the "obvious" common factor and obtain

${\displaystyle {\begin{aligned}S&=\left({\frac {1}{(m-1)!}}+{\frac {1}{(m-2)!}}\right)+\left({\frac {1}{(m-3)!}}+{\frac {1}{(m-4)!}}\right)+\cdots \\&={\frac {1}{(m-1)!}}(1+m-1)+{\frac {1}{(m-3)!}}(1+m-3)+\cdots .\end{aligned}}}$

We shall compute only integer values of the expressions in the parentheses, that is the values

${\displaystyle m,m-2,m-4,\dots .\,}$

Thus, at the first step the sum ${\displaystyle S}$ is into

${\displaystyle S=S(1)=\sum _{j=0}^{m_{1}-1}{\frac {1}{(m-1-2j)!}}\alpha _{m_{1}-j}(1),}$

${\displaystyle m_{1}={\frac {m}{2}},m=2^{k},k\geq 1.}$

At the first step ${\displaystyle {\frac {m}{2}}}$ integers of the form

${\displaystyle \alpha _{m_{1}-j}(1)=m-2j,\quad j=0,1,\dots ,m_{1}-1,}$

are calculated. After that we act in a similar way: combining on each step the summands of the sum ${\displaystyle S}$ 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 ${\displaystyle i}$ steps of this process are completed.

Step ${\displaystyle i+1}$ (${\displaystyle i+1\leq k}$).

${\displaystyle S=S(i+1)=\sum _{j=0}^{m_{i+1}-1}{\frac {1}{(m-1-2^{i+1}j)!}}\alpha _{m_{i+1}-j}(i+1),}$

${\displaystyle m_{i+1}={\frac {m_{i}}{2}}={\frac {m}{2^{i+1}}},}$

we compute only ${\displaystyle {\frac {m}{2^{i+1}}}}$ integers of the form

${\displaystyle \alpha _{m_{i+1}-j}(i+1)=\alpha _{m_{i}-2j}(i)+\alpha _{m_{i}-(2j+1)}(i){\frac {(m-1-2^{i+1}j)!}{(m-1-2^{i}-2^{i+1}j)!}},}$

${\displaystyle j=0,1,\dots ,\quad m_{i+1}-1,\quad m=2^{k},\quad k\geq i+1.}$

Here

${\displaystyle {\frac {(m-1-2^{i+1}j)!}{(m-1-2^{i}-2^{i+1}j)!}}}$

is the product of ${\displaystyle 2^{i}}$ integers.

Etc.

Step ${\displaystyle k}$, the last one. We compute one integer value ${\displaystyle \alpha _{1}(k),}$ we compute, using the fast algorithm described above the value ${\displaystyle (m-1)!,}$ and make one division of the integer ${\displaystyle \alpha _{1}(k)}$ by the integer ${\displaystyle (m-1)!,}$ with accuracy up to ${\displaystyle n}$ digits. The obtained result is the sum ${\displaystyle S,}$ or the constant ${\displaystyle e}$ up to ${\displaystyle n}$ digits. The complexity of all computations is

${\displaystyle O\left(M(m)\log ^{2}m\right)=O\left(M(n)\log n\right).\,}$

• Fast algorithms
• AGM method
• Computational complexity

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