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37 lines
1.4 KiB
ReStructuredText
37 lines
1.4 KiB
ReStructuredText
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**gamma.h** -- support for the gamma function
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===============================================================================
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Evaluation using Taylor series
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--------------------------------------------------------------------------------
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.. function :: long gamma_taylor_bound_mag(long n)
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Letting `1/\Gamma(x) = \sum_{n=1}^{\infty} c_n x^n`, returns an integer
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`p` such that `|c_n| \le 2^{-p}`.
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Uses the bound `\log |c_n| \le b(n) = ((\pi-1)n + (3-5n) \log(n/6))/6`
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(to be published).
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.. function :: void gamma_taylor_bound_ratio(fmpr_t r, long n)
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Sets `r` to a bound for `B(n+1) / B(n)` where `B(n) = \exp b(n)`.
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We use `r(n) = 3n^{-5/6}`.
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.. function :: void gamma_taylor_bound_remainder(fmpr_t err, const fmpr_t z, long n)
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Given a nonnegative `z` and `n \ge 1`, computes a bound for
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`\sum_{k=n}^{\infty} |c_{k+1}| z^k`. This is given by
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`B(n+1) z^n / (1 - r(n+1) z)`, provided that the denominator is
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positive (otherwise the bound is infinity).
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.. function :: void gamma_taylor_fmprb(fmprb_t y, const fmprb_t x, long prec)
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Sets `y = \Gamma(x)`, computed by translating `x` to the interval
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`[-1/2,1/2]` and then evaluating the Taylor series for the
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reciprocal gamma function.
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Assumes that the absolute value of the midpoint of `x` is not too large
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in order for the translation to be reasonably efficient (too large
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values will result in extreme slowness and eventually an exception).
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