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

\[\frac{x}{e^x-1} = \sum_{n=0}^{\infty} B_n \frac{x^n}{n!}.\]

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.

void bernoulli_rev_init(bernoulli_rev_t iter, ulong n)

Initializes the iterator iter. The first Bernoulli number to be generated by calling bernoulli_rev_next() is \(B_n\). It is assumed that \(n\) is even.

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 \(B_{k-2}\).

void bernoulli_rev_clear(bernoulli_rev_t iter)

Frees all memory allocated internally by iter.

void bernoulli_fmpq_vec_no_cache(fmpq *res, ulong a, slong num)

Writes num consecutive Bernoulli numbers to res starting with \(B_a\). This function is not currently optimized for a small count num. The entries are not read from or written to the Bernoulli number cache; if retrieving a vector of Bernoulli numbers is needed more than once, use bernoulli_cache_compute() followed by bernoulli_fmpq_ui() instead.

This function is a wrapper for the rev iterators. It can use multiple threads internally.

Caching

slong bernoulli_cache_num

fmpq *bernoulli_cache

Cache of Bernoulli numbers. Uses thread-local storage if enabled in FLINT.

void bernoulli_cache_compute(slong n)

Makes sure that the Bernoulli numbers up to at least \(B_{n-1}\) are cached. Calling flint_cleanup() frees the cache.

The cache is extended by calling bernoulli_fmpq_vec_no_cache() internally.

Bounding

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

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.

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 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.

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.