arb/doc/source/bernoulli.rst

.. _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.

Isolated Bernoulli numbers
-------------------------------------------------------------------------------

.. function:: ulong bernoulli_mod_p_harvey(ulong n, ulong p)

    Returns the `B_n` modulo the prime number *p*, computed using
    Harvey's algorithm [Har2010]_. The running time is linear in *p*.
    If *p* divides the numerator of `B_n`, *UWORD_MAX* is returned
    as an error code.

.. function:: void _bernoulli_fmpq_ui_zeta(fmpz_t num, fmpz_t den, ulong n)
              void _bernoulli_fmpq_ui_multi_mod(fmpz_t num, fmpz_t den, ulong n, double alpha)

    Sets *num* and *den* to the reduced numerator and denominator
    of the Bernoulli number `B_n`.

    The *zeta* version computes the denominator `d` using the von Staudt-Clausen
    theorem, numerically approximates `B_n` using :func:`arb_bernoulli_ui_zeta`,
    and then rounds `d B_n` to the correct numerator.

    The *multi_mod* version reconstructs `B_n` by computing the high bits
    via the Riemann zeta function and the low bits via Harvey's multimodular
    algorithm. The tuning parameter *alpha* should be a fraction between
    0 and 1 controlling the number of bits to compute by the multimodular
    algorithm. If set to a negative number, a default value will be used.

.. function:: void _bernoulli_fmpq_ui(fmpz_t num, fmpz_t den, ulong n)
              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.