An efficient implementation of the zeta-function algorithm for accurate computing Bernoulli numbers [1]. The code was written in Python using math and decimallibs.
The Bernoulli numbers are a sequence of rational numbers which are widely used in mathematics. The explicit definition is given by Louis Saalschütz in 1893 and could be written as
But the explicit computation of the large Bernoulli numbers is irrelevant and co uld be inaccurate. The more accurate way is to define the numerator and the denominator of the accurate fraction. Thereby the program prints the numerator, the denominator and the decimal fraction after n for the n-th Bernoulli number.
The general procedure for n > 1.
One of the most important application of the Bernoulli numbers is in the numerical computation of integrals by the Euler–Maclaurin formula
Here the first Bernoulli number is +1/2.
Another essential application could be found in the computation of Faulhaber's formula
In 1713, Jacob Bernoulli showed that such sum could be expressed as
where the falling factorial is
We have computed the following sum
The result is
322580645161290322580645161790322580645161290322580645411290322580645161290322580645161290322580645161290322546811827956989247311827956989247311827956989247311827962644247311827956989247311827956989247311827956989247310979706989247311827956989247311827956989247311827956989355699327956989247311827956989247311827956989247311827945466822311827956989247311827956989247311827956989247312825955739247311827956989247311827956989247311827956989178554466456989247311827956989247311827956989247311827960646658436827956989247311827956989247311827956989247311683433845606402737047898338220918866080156402737047898342247811378580156402737047898338220918866080156402737047824896521215294651584974165619326909649490294651584974166400652868699490294651584974165619326909649490294651584970207375822076316156961318251640832285993576316156961318257656641025000000000000000000000000000
The total CPU time was less than 2s for 1 core of Intel(R) Core(TM) i5-2520M CPU @ 2.50GHz.
[1] Kevin J. McGown, Computing Bernoulli Numbers Quickly, 2005.