Rmpfr (version 0.9-5)

Bernoulli: Bernoulli Numbers in Arbitrary Precision

Description

Computes the Bernoulli numbers in the desired (binary) precision. The computation happens via the zeta function and the formula $$B_k = -k \zeta(1 - k),$$ and hence the only non-zero odd Bernoulli number is \(B_1 = +1/2\). (Another tradition defines it, equally sensibly, as \(-1/2\).)

Usage

Bernoulli(k, precBits = 128)

Value

an mpfr class vector of the same length as

k, with i-th component the k[i]-th Bernoulli number.

Arguments

k

non-negative integer vector

precBits

the precision in bits desired.

Author

Martin Maechler

References

https://en.wikipedia.org/wiki/Bernoulli_number

See Also

zeta is used to compute them.

The next version of package gmp is to contain BernoulliQ(), providing exact Bernoulli numbers as big rationals (class "bigq").

Examples

sessionInfo()
 .libPaths()
 packageDescription("gmp")
Bernoulli(0:10)
plot(as.numeric(Bernoulli(0:15)), type = "h")

curve(-x*zeta(1-x), -.2, 15.03, n=300,
      main = expression(-x %.% zeta(1-x)))
legend("top", paste(c("even","odd  "), "Bernoulli numbers"),
       pch=c(1,3), col=2, pt.cex=2, inset=1/64)
abline(h=0,v=0, lty=3, col="gray")
k <- 0:15; k[1] <- 1e-4
points(k, -k*zeta(1-k), col=2, cex=2, pch=1+2*(k%%2))

## They pretty much explode for larger k :
k2 <- 2*(1:120)
plot(k2, abs(as.numeric(Bernoulli(k2))), log = "y")
title("Bernoulli numbers exponential growth")

Bernoulli(10000)# - 9.0494239636 * 10^27677