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291 lines
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Plaintext
291 lines
10 KiB
Plaintext
[section:igamma Incomplete Gamma Functions]
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[h4 Synopsis]
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``
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#include <boost/math/special_functions/gamma.hpp>
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``
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namespace boost{ namespace math{
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template <class T1, class T2>
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``__sf_result`` gamma_p(T1 a, T2 z);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` gamma_p(T1 a, T2 z, const ``__Policy``&);
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template <class T1, class T2>
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``__sf_result`` gamma_q(T1 a, T2 z);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` gamma_q(T1 a, T2 z, const ``__Policy``&);
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template <class T1, class T2>
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``__sf_result`` tgamma_lower(T1 a, T2 z);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` tgamma_lower(T1 a, T2 z, const ``__Policy``&);
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template <class T1, class T2>
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``__sf_result`` tgamma(T1 a, T2 z);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` tgamma(T1 a, T2 z, const ``__Policy``&);
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}} // namespaces
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[h4 Description]
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There are four [@http://mathworld.wolfram.com/IncompleteGammaFunction.html
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incomplete gamma functions]:
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two are normalised versions (also known as /regularized/ incomplete gamma functions)
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that return values in the range [0, 1], and two are non-normalised and
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return values in the range [0, [Gamma](a)]. Users interested in statistical
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applications should use the
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[@http://mathworld.wolfram.com/RegularizedGammaFunction.html normalised versions (gamma_p and gamma_q)].
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All of these functions require /a > 0/ and /z >= 0/, otherwise they return
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the result of __domain_error.
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[optional_policy]
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The return type of these functions is computed using the __arg_promotion_rules
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when T1 and T2 are different types, otherwise the return type is simply T1.
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template <class T1, class T2>
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``__sf_result`` gamma_p(T1 a, T2 z);
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template <class T1, class T2, class Policy>
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``__sf_result`` gamma_p(T1 a, T2 z, const ``__Policy``&);
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Returns the normalised lower incomplete gamma function of a and z:
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[equation igamma4]
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This function changes rapidly from 0 to 1 around the point z == a:
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[graph gamma_p]
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template <class T1, class T2>
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``__sf_result`` gamma_q(T1 a, T2 z);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` gamma_q(T1 a, T2 z, const ``__Policy``&);
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Returns the normalised upper incomplete gamma function of a and z:
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[equation igamma3]
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This function changes rapidly from 1 to 0 around the point z == a:
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[graph gamma_q]
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template <class T1, class T2>
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``__sf_result`` tgamma_lower(T1 a, T2 z);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` tgamma_lower(T1 a, T2 z, const ``__Policy``&);
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Returns the full (non-normalised) lower incomplete gamma function of a and z:
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[equation igamma2]
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template <class T1, class T2>
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``__sf_result`` tgamma(T1 a, T2 z);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` tgamma(T1 a, T2 z, const ``__Policy``&);
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Returns the full (non-normalised) upper incomplete gamma function of a and z:
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[equation igamma1]
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[h4 Accuracy]
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The following tables give peak and mean relative errors in over various domains of
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a and z, along with comparisons to the __gsl and __cephes libraries.
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Note that only results for the widest floating point type on the system are given as
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narrower types have __zero_error.
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Note that errors grow as /a/ grows larger.
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Note also that the higher error rates for the 80 and 128 bit
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long double results are somewhat misleading: expected results that are
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zero at 64-bit double precision may be non-zero - but exceptionally small -
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with the larger exponent range of a long double. These results therefore
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reflect the more extreme nature of the tests conducted for these types.
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All values are in units of epsilon.
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[table_gamma_p]
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[table_gamma_q]
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[table_tgamma_lower]
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[table_tgamma_incomplete_]
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[h4 Testing]
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There are two sets of tests: spot tests compare values taken from
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[@http://functions.wolfram.com/GammaBetaErf/ Mathworld's online evaluator]
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with this implementation to perform a basic "sanity check".
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Accuracy tests use data generated at very high precision
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(using NTL's RR class set at 1000-bit precision) using this implementation
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with a very high precision 60-term __lanczos, and some but not all of the special
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case handling disabled.
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This is less than satisfactory: an independent method should really be used,
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but apparently a complete lack of such methods are available. We can't even use a deliberately
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naive implementation without special case handling since Legendre's continued fraction
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(see below) is unstable for small a and z.
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[h4 Implementation]
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These four functions share a common implementation since
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they are all related via:
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1) [equation igamma5]
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2) [equation igamma6]
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3) [equation igamma7]
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The lower incomplete gamma is computed from its series representation:
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4) [equation igamma8]
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Or by subtraction of the upper integral from either [Gamma](a) or 1
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when /x - (1/(3x)) > a and x > 1.1/.
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The upper integral is computed from Legendre's continued fraction representation:
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5) [equation igamma9]
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When /(x > 1.1)/ or by subtraction of the lower integral from either [Gamma](a) or 1
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when /x - (1/(3x)) < a/.
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For /x < 1.1/ computation of the upper integral is more complex as the continued
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fraction representation is unstable in this area. However there is another
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series representation for the lower integral:
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6) [equation igamma10]
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That lends itself to calculation of the upper integral via rearrangement
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to:
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7) [equation igamma11]
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Refer to the documentation for __powm1 and __tgamma1pm1 for details
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of their implementation. Note however that the precision of __tgamma1pm1
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is capped to either around 35 digits, or to that of the __lanczos associated with
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type T - if there is one - whichever of the two is the greater.
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That therefore imposes a similar limit on the precision of this
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function in this region.
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For /x < 1.1/ the crossover point where the result is ~0.5 no longer
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occurs for /x ~ y/. Using /x * 0.75 < a/ as the crossover criterion
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for /0.5 < x <= 1.1/ keeps the maximum value computed (whether
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it's the upper or lower interval) to around 0.75. Likewise for
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/x <= 0.5/ then using /-0.4 / log(x) < a/ as the crossover criterion
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keeps the maximum value computed to around 0.7
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(whether it's the upper or lower interval).
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There are two special cases used when a is an integer or half integer,
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and the crossover conditions listed above indicate that we should compute
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the upper integral Q.
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If a is an integer in the range /1 <= a < 30/ then the following
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finite sum is used:
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9) [equation igamma1f]
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While for half integers in the range /0.5 <= a < 30/ then the
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following finite sum is used:
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10) [equation igamma2f]
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These are both more stable and more efficient than the continued fraction
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alternative.
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When the argument /a/ is large, and /x ~ a/ then the series (4) and continued
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fraction (5) above are very slow to converge. In this area an expansion due to
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Temme is used:
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11) [equation igamma16]
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12) [equation igamma17]
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13) [equation igamma18]
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14) [equation igamma19]
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The double sum is truncated to a fixed number of terms - to give a specific
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target precision - and evaluated as a polynomial-of-polynomials. There are
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versions for up to 128-bit long double precision: types requiring
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greater precision than that do not use these expansions. The
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coefficients C[sub k][super n] are computed in advance using the recurrence
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relations given by Temme. The zone where these expansions are used is
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(a > 20) && (a < 200) && fabs(x-a)/a < 0.4
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And:
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(a > 200) && (fabs(x-a)/a < 4.5/sqrt(a))
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The latter range is valid for all types up to 128-bit long doubles, and
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is designed to ensure that the result is larger than 10[super -6], the
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first range is used only for types up to 80-bit long doubles. These
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domains are narrower than the ones recommended by either Temme or Didonato
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and Morris. However, using a wider range results in large and inexact
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(i.e. computed) values being passed to the `exp` and `erfc` functions
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resulting in significantly larger error rates. In other words there is a
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fine trade off here between efficiency and error. The current limits should
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keep the number of terms required by (4) and (5) to no more than ~20
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at double precision.
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For the normalised incomplete gamma functions, calculation of the
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leading power terms is central to the accuracy of the function.
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For smallish a and x combining
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the power terms with the __lanczos gives the greatest accuracy:
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15) [equation igamma12]
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In the event that this causes underflow/overflow then the exponent can
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be reduced by a factor of /a/ and brought inside the power term.
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When a and x are large, we end up with a very large exponent with a base
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near one: this will not be computed accurately via the pow function,
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and taking logs simply leads to cancellation errors. The worst of the
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errors can be avoided by using:
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16) [equation igamma13]
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when /a-x/ is small and a and x are large. There is still a subtraction
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and therefore some cancellation errors - but the terms are small so the absolute
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error will be small - and it is absolute rather than relative error that
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counts in the argument to the /exp/ function. Note that for sufficiently
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large a and x the errors will still get you eventually, although this does
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delay the inevitable much longer than other methods. Use of /log(1+x)-x/ here
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is inspired by Temme (see references below).
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[h4 References]
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* N. M. Temme, A Set of Algorithms for the Incomplete Gamma Functions,
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Probability in the Engineering and Informational Sciences, 8, 1994.
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* N. M. Temme, The Asymptotic Expansion of the Incomplete Gamma Functions,
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Siam J. Math Anal. Vol 10 No 4, July 1979, p757.
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* A. R. Didonato and A. H. Morris, Computation of the Incomplete Gamma
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Function Ratios and their Inverse. ACM TOMS, Vol 12, No 4, Dec 1986, p377.
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* W. Gautschi, The Incomplete Gamma Functions Since Tricomi, In Tricomi's Ideas
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and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147,
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Accademia Nazionale dei Lincei, Roma, 1998, pp. 203--237.
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[@http://citeseer.ist.psu.edu/gautschi98incomplete.html http://citeseer.ist.psu.edu/gautschi98incomplete.html]
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[endsect][/section:igamma The Incomplete Gamma Function]
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[/
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Copyright 2006 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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