copula (version 0.999-15)

Bernoulli: Compute Bernoulli Numbers

Description

Compute the $n$th Bernoulli number, or generate all Bernoulli numbers up to the $n$th, using diverse methods, that is, algorithms.

NOTE the current default methods will be changed -- to get better accuracy!

Usage

Bernoulli    (n, method = c("sumBin", "sumRamanujan", "asymptotic"), verbose = FALSE)
Bernoulli.all(n, method = c("A-T", "sumBin", "sumRamanujan", "asymptotic"), precBits = NULL, verbose = getOption("verbose"))

Arguments

n

positive integer, indicating the index of the largest (and last) of the Bernoulli numbers needed.

method

character string, specifying which method should be applied. The default for Bernoulli.all(), "A-T" stands for the Akiyama-Tanigawa algorithm which is nice and simple but has bad numerical properties. It can however work with high precision "mpfr"-numbers, see precBits. "sumRamanujan" is somewhat more efficient but not yet implemented.

precBits

currently only for method = "A-T" -- NULL or a positive integer indicating the precision of the initial numbrs in bits, using "Rmpfr"'s package multiprecision arithmetic.

verbose

(for "A-T":) logical indicating if the intermediate results of the algorithm should be printed.

Value

References

Kaneko, Masanobu (2000) The Akiyama-Tanigawa algorithm for Bernoulli numbers; Journal of Integer Sequences 3, article 00.2.9

See Also

Eulerian, Stirling1, etc.

Examples

## The example for the paper
MASS::fractions(Bernoulli.all(8, verbose=TRUE))

B10 <- Bernoulli.all(10)
MASS::fractions(B10)

system.time(B50  <- Bernoulli.all(50))#  {does not cache} -- still "no time"
system.time(B100 <- Bernoulli.all(100))# still less than a milli second

## Using Bernoulli() is not much slower, but hopefully *more* accurate!
## Check first - TODO
system.time(B.1c <- Bernoulli(100))# caches ..
system.time(B1c. <- Bernoulli(100))# ==> now much faster
stopifnot(identical(B.1c, B1c.))

if(FALSE)## reset the cache:
assign("Bern.tab", list(), envir = copula:::.nacopEnv)

## More experiments in the source of the copula package ../tests/Stirling-etc.R