.. _bernoulli:
**bernoulli.h** -- support for Bernoulli numbers
===============================================================================
This module provides helper functions for exact or approximate calculation
of the Bernoulli numbers, which are defined by the exponential
generating function
.. math ::
\frac{x}{e^x-1} = \sum_{n=0}^{\infty} B_n \frac{x^n}{n!}.
Efficient algorithms are implemented for both multi-evaluation
and calculation of isolated Bernoulli numbers.
A global (or thread-local) cache is also provided,
to support fast repeated evaluation of various special functions
that depend on the Bernoulli numbers (including the gamma function
and the Riemann zeta function).
Generation of Bernoulli numbers
--------------------------------------------------------------------------------
.. type:: bernoulli_rev_t
An iterator object for generating a range of even-indexed Bernoulli numbers
exactly in reverse order, i.e. computing the exact
fractions `B_n, B_{n-2}, B_{n-4}, \ldots, B_0`.
The Bernoulli numbers are generated from scratch, i.e.
no caching is performed.
The Bernoulli numbers are computed by direct summation of the zeta series.
This is made fast by storing a table of powers (as done by [Blo2009]_).
As an optimization, we only include the odd powers, and use
fixed-point arithmetic.
The reverse iteration order is preferred for performance reasons,
as the powers can be updated using multiplications instead of divisions,
and we avoid having to periodically recompute terms to higher precision.
To generate Bernoulli numbers in the forward direction without having
to store all of them, one can split the desired range into smaller
blocks and compute each block with a single reverse pass.
.. function:: void bernoulli_rev_init(bernoulli_rev_t iter, ulong n)
Initializes the iterator *iter*. The first Bernoulli number to
be generated by calling :func:`bernoulli_rev_next` is `B_n`.
It is assumed that `n` is even.
.. function:: void bernoulli_rev_next(fmpz_t numer, fmpz_t denom, bernoulli_rev_t iter)
Sets *numer* and *denom* to the exact, reduced numerator and denominator
of the Bernoulli number `B_k` and advances the state of *iter*
so that the next invocation generates `B_{k-2}`.
.. function:: void bernoulli_rev_clear(bernoulli_rev_t iter)
Frees all memory allocated internally by *iter*.
Caching
-------------------------------------------------------------------------------
.. var:: slong bernoulli_cache_num
.. var:: fmpq * bernoulli_cache
Cache of Bernoulli numbers. Uses thread-local storage if enabled
in FLINT.
.. function:: void bernoulli_cache_compute(slong n)
Makes sure that the Bernoulli numbers up to at least `B_{n-1}` are cached.
Calling :func:`flint_cleanup()` frees the cache.
Bounding
-------------------------------------------------------------------------------
.. function:: slong bernoulli_bound_2exp_si(ulong n)
Returns an integer `b` such that `|B_n| \le 2^b`. Uses a lookup table
for small `n`, and for larger `n` uses the inequality
`|B_n| < 4 n! / (2 \pi)^n < 4 (n+1)^{n+1} e^{-n} / (2 \pi)^n`.
Uses integer arithmetic throughout, with the bound for the logarithm
being looked up from a table. If `|B_n| = 0`, returns *LONG_MIN*.
Otherwise, the returned exponent `b` is never more than one percent
larger than the true magnitude.
This function is intended for use when `n` small enough that one might
comfortably compute `B_n` exactly. It aborts if `n` is so large that
internal overflow occurs.
.. function:: void _bernoulli_fmpq_ui_zeta(fmpz_t num, fmpz_t den, ulong n)
Sets *num* and *den* to the reduced numerator and denominator
of the Bernoulli number `B_n`.
This function computes the denominator `d` using von Staudt-Clausen
theorem, numerically approximates `B_n` using :func:`arb_bernoulli_ui_zeta`,
and then rounds `d B_n` to the correct numerator.
If the working precision is insufficient to determine the numerator,
the function prints a warning message and retries with increased
precision (this should not be expected to happen).
.. function:: void _bernoulli_fmpq_ui(fmpz_t num, fmpz_t den, ulong n)
.. function:: void bernoulli_fmpq_ui(fmpq_t b, ulong n)
Computes the Bernoulli number `B_n` as an exact fraction, for an
isolated integer `n`. This function reads `B_n` from the global cache
if the number is already cached, but does not automatically extend
the cache by itself.