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Fix some issues in bernoulli docs.
This commit is contained in:
jzmaddock
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<annotation encoding="StarMath 5.0">matrix {ln(B_n) # "=" # ({1} over {2} + n)ln(n) + ({1} over {2} - n)ln(%pi%) + ({3} over {2} - n)ln(2) - R(n)
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##
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R(n) # "=" # n(1 - {1} over 12 (1 - 1 over 30 (1 - 2 over 7 n^{-2})n^{-2})n^{-2}) }</annotation>
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</semantics>
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</math>
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doc/equations/bernoulli_numbers2.png
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doc/equations/bernoulli_numbers2.svg
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#
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# Paths to tools come first, change these to match your system:
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#
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math2svg='m:\download\open\SVGMath-0.3.1\math2svg.py'
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python=/cygdrive/c/Python26/python.exe
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inkscape=/cygdrive/c/progra~1/Inkscape/inkscape
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math2svg='d:\download\open\SVGMath-0.3.1\math2svg.py'
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python=/cygdrive/c/Python27/python.exe
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inkscape=/cygdrive/c/progra~1/Inkscape/inkscape.exe
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# Image DPI:
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dpi=120
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@@ -18,7 +18,7 @@ for mmlfile in $*; do
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pngfile=$(basename $svgfile .svg).png
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tempfile=temp.mml
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# strip html wrappers put in by MathCast:
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cat $mmlfile | tr -d "\r\n" | sed -e 's/.*\(<math[^>]*>.*<\/math>\).*/\1/' > $tempfile
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cat $mmlfile | tr -d "\r\n" | sed -e 's/.*\(<math[^>]*>.*<\/math>\).*/\1/' -e 's/<semantics>//g' -e 's/<\/semantics>//g' > $tempfile
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echo Generating $svgfile
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$python $math2svg $tempfile > $svgfile
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@@ -31,3 +31,4 @@ done
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@@ -18,16 +18,23 @@ including the __tgamma, __lgamma and polygamma functions.
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namespace boost { namespace math {
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template <class T>
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T bernoulli_b2n(const int i); // Single Bernoulli number (default policy).
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T bernoulli_b2n(const int n); // Single Bernoulli number (default policy).
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template <class T, class Policy>
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T bernoulli_b2n(const int i, const Policy &pol); // User policy for errors etc.
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T bernoulli_b2n(const int n, const Policy &pol); // User policy for errors etc.
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}} // namespaces
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[h4 Description]
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Both return the (2 * i)[super th] Bernoulli number.
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Both return the (2 * n)[super th] Bernoulli number B[sub 2n].
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Note that since all odd numbered Bernoulli numbers are zero (apart from B[sub 1] which is [plusminus][frac12])
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the interface will only return the even numbered Bernoulli numbers.
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This function uses fast table lookup for low-indexed Bernoulli numbers, while larger values are calculated
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as needed and then cached. The caching mechanism requires a certain amount of thread safety code, so
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`unchecked_bernoulli_b2n` may provide a better interface for performance critical code.
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The final __Policy argument is optional and can be used to control the behaviour of the function:
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how it handles errors, what level of precision to use, etc.
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@@ -45,23 +52,32 @@ Refer to __policy_section for more details.
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[h4 Synopsis]
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``
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#include <boost/math/special_functions/detail/unchecked_bernoulli.hpp>
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#include <boost/math/special_functions/bernoulli.hpp>
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``
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template <>
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struct max_bernoulli_b2n<T>;
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template<class T>
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inline T unchecked_bernoulli_b2n(unsigned n);
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`unchecked_bernoulli_b2n` provides access to Bernoulli numbers [*without any checks for overflow or invalid parameters].
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(Not recommended for normal use, but bypassing overflow checks is quicker.)
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It is implemented as a direct (and very fast) table lookup, and while not recomended for general use it can be useful
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inside inner loops where the ultimate performance is required, and error checking is moved outside the loop.
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template <>
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struct max_bernoulli_index<T>::value
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The largest value you can pass to `unchecked_bernoulli_b2n<>` is `max_bernoulli_b2n<>::value`: passing values greater than
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that will result in a buffer overrun error, so it's clearly important to place the error handling in your own code
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when using this direct interface.
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The largest value you can (without overflow) pass to `unchecked_bernoulli_b2n<>`
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is `max_bernoulli_b2n<>::value`.
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The value of `boost::math::max_bernoulli_b2n<T>::value` varies by the type T, for types `float`/`double`/`long double`
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it's the largest value which doesn't overflow the target type: for example, `boost::math::max_bernoulli_b2n<double>::value` is 129.
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However, for multiprecision types, it's the largest value for which the result can be represented as the ratio of two 64-bit
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integers, for example `boost::math::max_bernoulli_b2n<boost::multiprecision::cpp_dec_float_50>::value` is just 17. Of course
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larger indexes can be passed to `bernoulli_b2n<T>(n)`, but then then you loose fast table lookup (i.e. values may need to be calculated).
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For example, `boost::math::max_bernoulli_b2n<double>::value` is 129.
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[bernoulli_example_4]
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[bernoulli_output_4]
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[h4 Multiple Bernoulli Numbers]
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@@ -96,20 +112,15 @@ with one call (one with default policy and the other allowing a user-defined pol
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These return a series of Bernoulli numbers:
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Bernoulli(2*start_index),
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Bernoulli(2*(start_index+1))
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...
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Bernoulli(2*(number_of_bernoullis_b2n-1))
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[:B[sub 2*start_index],B[sub 2*(start_index+1)],...,B[sub 2*(start_index+number_of_bernoullis_b2n-1)]]
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[h4 Examples]
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[bernoulli_example_2]
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[bernoulli_output_2]
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[bernoulli_example_3]
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[bernoulli_output_3]
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[bernoulli_example_4]
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[bernoulli_output_4]
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The source of this example is at [../../example/bernoulli_example.cpp bernoulli_example.cpp]
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The source of this example is at [@../../example/bernoulli_example.cpp bernoulli_example.cpp]
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[h4 Accuracy]
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@@ -117,9 +128,10 @@ All the functions usually return values within one ULP (unit in the last place)
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[h4 Implementation]
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The implementation details are in [@../../include/boost/math/special_functions/detail/bernoulli_details.hpp bernoulli_details.hpp].
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The implementation details are in [@../../include/boost/math/special_functions/detail/bernoulli_details.hpp bernoulli_details.hpp]
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and [@../../include/boost/math/special_functions/detail/unchecked_bernoulli.hpp unchecked_bernoulli.hpp].
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For `i <= max_bernoulli_index<T>::value` this is implemented by table lookup;
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For `i <= max_bernoulli_index<T>::value` this is implemented by simple table lookup from a statically initialized table;
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for larger values of `i`, this is implemented by the Tangent Numbers algorithm as described in the paper:
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Fast Computation of Bernoulli, Tangent and Secant Numbers, Richard P. Brent and David Harvey,
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[@http://arxiv.org/pdf/1108.0286v3.pdf] (2011).
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@@ -128,8 +140,6 @@ Fast Computation of Bernoulli, Tangent and Secant Numbers, Richard P. Brent and
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(an even alternating permutation number) are defined
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and their generating function is also given therein.
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[/@http://mathworld.wolfram.com/images/equations/TangentNumber/Inline15.gif]
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The relation of Tangent numbers with Bernoulli numbers ['B[sub i]]
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is given by Brent and Harvey's equation 14:
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@@ -143,7 +153,13 @@ elseif i == 0 then ['B[sub i]] = 1 [br]
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elseif i == 1 then ['B[sub i]] = -1/2 [br]
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elseif i < 0 or i is odd then ['B[sub i]] = 0
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[/@http://s7.postimg.org/mygi2kror/bernoulli_1.png]
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Note that computed values are stored in a fixed-size table, access is thread safe via atomic operations (i.e. lock
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free programming), this imparts a much lower overhead on access to cached values than might overwise be expected -
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typically for multiprecision types the cost of thread synchronisation is negligable, while for built in types
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this code is not normally executed anyway. For very large arguments which cannot be reasonably computed or
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stored in our cache, an asymptotic expansion [@http://www.luschny.de/math/primes/bernincl.html due to Luschny] is used:
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[equation bernoulli_numbers2]
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[endsect] [/section:bernoulli_numbers Bernoulli Numbers]
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@@ -34,7 +34,8 @@ int main()
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{
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//[bernoulli_example_1
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/*`A simple example computes the value of `Bernoulli(2)` where the return type is `double`.
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/*`A simple example computes the value of B[sub 4] where the return type is `double`,
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note that the argument to bernoulli_b2n is ['2] not ['4] since it computes B[sub 2N].
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*/
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@@ -48,7 +49,7 @@ int main()
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/*`So B[sub 4] == -1/30 == -0.0333333333333333
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If we use Boost.Multiprecision and its 50 decimal digit floating-point type `cpp_dec_float_50`,
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we can calculate the value of much larger numbers like `Bernoulli(100)`
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we can calculate the value of much larger numbers like B[sub 200]
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and also obtain much higher precision.
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*/
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@@ -107,11 +108,7 @@ and we will get a helpful error message (provided try'n'catch blocks are used).
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//] //[/bernoulli_example_3]
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//[bernoulli_example_4
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/*`Users can determine from `max_bernoulli_b2n<>::value`
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the largest Bernoulli number you can evaluate for any type
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(or safely pass to `unchecked_bernoulli_b2n<>`).
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For example:
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/*For example:
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*/
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std::cout << "boost::math::max_bernoulli_b2n<float>::value = " << boost::math::max_bernoulli_b2n<float>::value << std::endl;
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std::cout << "Maximum Bernoulli number using float is " << boost::math::bernoulli_b2n<float>( boost::math::max_bernoulli_b2n<float>::value) << std::endl;
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