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46 lines
1.8 KiB
ReStructuredText
46 lines
1.8 KiB
ReStructuredText
**bernoulli.h** -- support for Bernoulli numbers
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===============================================================================
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Generation of Bernoulli numbers
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--------------------------------------------------------------------------------
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.. type:: bernoulli_rev_t
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.. function:: bernoulli_rev_init(bernrev_iter_t iter, ulong n)
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.. function:: bernoulli_rev_next(fmpz_t numer, fmpz_t denom, bernrev_iter_t iter)
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.. function:: bernoulli_rev_clear(bernrev_iter_t iter)
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Generates a range of even-indexed Bernoulli numbers in reverse order,
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i.e. generates `B_n, B_{n-2}, B_{n-4}, \ldots, B_0`.
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The exact, minimal numerators and denominators are produced.
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The Bernoulli numbers are computed by direct summation of the zeta series.
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This is made fast by storing a table of powers (as done by Bloemen et al.
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http://remcobloemen.nl/2009/11/even-faster-zeta-calculation.html).
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As an optimization, we only include the odd powers, and use
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fixed-point arithmetic.
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The reverse iteration order is preferred for performance reasons,
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as the powers can be updated using multiplications instead of divisions,
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and we avoid having to periodically recompute terms to higher precision.
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To generate Bernoulli numbers in the forward direction without having
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to store all of them, one can split the desired range into smaller
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blocks and compute each block with a single reverse pass.
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Caching
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-------------------------------------------------------------------------------
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.. var:: long bernoulli_cache_num;
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.. var:: fmpq * bernoulli_cache;
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Global cache of Bernoulli numbers.
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.. function:: void bernoulli_cache_compute(long n)
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Makes sure that the Bernoulli numbers up to at least `B_{n-1}` are cached
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globally. Warning: this function is not currently threadsafe.
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