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[mathpart gcd_lcm Integer Utilities (Greatest Common Divisor and Least Common Multiple)]
[section Introduction]
This code has now been moved to Boost.Integer, please see [@http://www.boost.org/doc/libs/release/libs/integer/doc/html/index.html here].
The class and function templates in `<boost/math/common_factor.hpp>`
provide both run-time and compile-time evaluation of the greatest common divisor
(GCD) or least common multiple (LCM) of two integers.
These facilities are useful for many numeric-oriented generic
programming problems.
[endsect] [/section Introduction]
[section Synopsis]
namespace boost
{
namespace math
{
template < typename IntegerType >
class gcd_evaluator;
template < typename IntegerType >
class lcm_evaluator;
template < typename IntegerType >
IntegerType gcd( IntegerType const &a, IntegerType const &b );
template < typename ForwardIterator >
std::pair<typename std::iterator_traits<I>::value_type, I> gcd_range(I first, I last);
template < typename IntegerType >
IntegerType lcm( IntegerType const &a, IntegerType const &b );
typedef ``['see-below]`` static_gcd_type;
template < static_gcd_type Value1, static_gcd_type Value2 >
struct static_gcd;
template < static_gcd_type Value1, static_gcd_type Value2 >
struct static_lcm;
}
}
[endsect] [/section Introduction]
[section GCD Function Object]
[*Header: ] [@../../../../boost/math/common_factor_rt.hpp <boost/math/common_factor_rt.hpp>]
template < typename IntegerType >
class boost::math::gcd_evaluator
{
public:
// Types
typedef IntegerType result_type;
typedef IntegerType first_argument_type;
typedef IntegerType second_argument_type;
// Function object interface
result_type operator ()( first_argument_type const &a,
second_argument_type const &b ) const;
};
The `boost::math::gcd`_evaluator class template defines a function object
class to return the greatest common divisor of two integers.
The template is parameterized by a single type, called `IntegerType` here.
This type should be a numeric type that represents integers.
The result of the function object is always nonnegative, even if either of
the operator arguments is negative.
This function object class template is used in the corresponding version of
the GCD function template. If a numeric type wants to customize evaluations
of its greatest common divisors, then the type should specialize on the
`gcd_evaluator` class template.
[endsect] [/section GCD Function Object]
[section LCM Function Object]
[*Header: ] [@../../../../boost/math/common_factor_rt.hpp <boost/math/common_factor_rt.hpp>]
template < typename IntegerType >
class boost::math::lcm_evaluator
{
public:
// Types
typedef IntegerType result_type;
typedef IntegerType first_argument_type;
typedef IntegerType second_argument_type;
// Function object interface
result_type operator ()( first_argument_type const &a,
second_argument_type const &b ) const;
};
The `boost::math::lcm_evaluator` class template defines a function object
class to return the least common multiple of two integers. The template
is parameterized by a single type, called `IntegerType `here. This type
should be a numeric type that represents integers. The result of the
function object is always nonnegative, even if either of the operator
arguments is negative. If the least common multiple is beyond the range
of the integer type, the results are undefined.
This function object class template is used in the corresponding version
of the LCM function template. If a numeric type wants to customize
evaluations of its least common multiples, then the type should
specialize on the `lcm_evaluator` class template.
[endsect] [/section LCM Function Object]
[section:run_time Run-time GCD & LCM Determination]
[*Header: ] [@../../../../boost/math/common_factor_rt.hpp <boost/math/common_factor_rt.hpp>]
template < typename IntegerType >
IntegerType boost::math::gcd( IntegerType const &a, IntegerType const &b );
template < typename ForwardIterator >
std::pair<typename std::iterator_traits<I>::value_type, I> gcd_range(I first, I last);
template < typename IntegerType >
IntegerType boost::math::lcm( IntegerType const &a, IntegerType const &b );
The `boost::math::gcd` function template returns the greatest common
(nonnegative) divisor of the two integers passed to it.
`boost::math::gcd_range` is the iteration of the above gcd algorithm over a
range, returning the greatest common divisor of all the elements. The algorithm
terminates when the gcd reaches unity or the end of the range. Thus it also
returns the iterator after the last element inspected because this may not be
equal to the end of the range.
The boost::math::lcm function template returns the least common
(nonnegative) multiple of the two integers passed to it.
The function templates are parameterized on the function arguments'
IntegerType, which is also the return type. Internally, these function
templates use an object of the corresponding version of the
`gcd_evaluator` and `lcm_evaluator` class templates, respectively.
[endsect] [/section:run_time Run-time GCD & LCM Determination]
[section:compile_time Compile-time GCD and LCM determination]
[*Header: ] [@../../../../boost/math/common_factor_ct.hpp <boost/math/common_factor_ct.hpp>]
typedef ``['unspecified]`` static_gcd_type;
template < static_gcd_type Value1, static_gcd_type Value2 >
struct boost::math::static_gcd : public mpl::integral_c<static_gcd_type, implementation_defined>
{
};
template < static_gcd_type Value1, static_gcd_type Value2 >
struct boost::math::static_lcm : public mpl::integral_c<static_gcd_type, implementation_defined>
{
};
The type `static_gcd_type` is the widest unsigned-integer-type that is supported
for use in integral-constant-expressions by the compiler. Usually this
the same type as `boost::uintmax_t`, but may fall back to being `unsigned long`
for some older compilers.
The boost::math::static_gcd and boost::math::static_lcm class templates
take two value-based template parameters of the ['static_gcd_type] type
and inherit from the type `boost::mpl::integral_c`.
Inherited from the base class, they have a member /value/
that is the greatest common factor or least
common multiple, respectively, of the template arguments.
A compile-time error will occur if the least common multiple
is beyond the range of `static_gcd_type`.
[h3 Example]
#include <boost/math/common_factor.hpp>
#include <algorithm>
#include <iterator>
#include <iostream>
int main()
{
using std::cout;
using std::endl;
cout << "The GCD and LCM of 6 and 15 are "
<< boost::math::gcd(6, 15) << " and "
<< boost::math::lcm(6, 15) << ", respectively."
<< endl;
cout << "The GCD and LCM of 8 and 9 are "
<< boost::math::static_gcd<8, 9>::value
<< " and "
<< boost::math::static_lcm<8, 9>::value
<< ", respectively." << endl;
int a[] = { 4, 5, 6 }, b[] = { 7, 8, 9 }, c[3];
std::transform( a, a + 3, b, c, boost::math::gcd_evaluator<int>() );
std::copy( c, c + 3, std::ostream_iterator<int>(cout, " ") );
}
[endsect] [/section:compile_time Compile time GCD and LCM determination]
[section:gcd_header Header <boost/math/common_factor.hpp>]
This header simply includes the headers
[@../../../../boost/math/common_factor_ct.hpp <boost/math/common_factor_ct.hpp>]
and [@../../../../boost/math/common_factor_rt.hpp <boost/math/common_factor_rt.hpp>].
[note This is a legacy header: it used to contain the actual implementation,
but the compile-time and run-time facilities
were moved to separate headers (since they were independent of each other).]
[endsect] [/section:gcd_header Header <boost/math/common_factor.hpp>]
[section:demo Demonstration Program]
The program [@../../../../libs/math/test/common_factor_test.cpp common_factor_test.cpp]
is a demonstration of the results from
instantiating various examples of the run-time GCD and LCM function
templates and the compile-time GCD and LCM class templates.
(The run-time GCD and LCM class templates are tested indirectly through
the run-time function templates.)
[endsect] [/section:demo Demonstration Program]
[section Rationale]
The greatest common divisor and least common multiple functions are
greatly used in some numeric contexts, including some of the other
Boost libraries. Centralizing these functions to one header improves
code factoring and eases maintainence.
[endsect] [/section Rationale]
[section:gcd_history History]
* 13 May 2013 Moved into main Boost.Math Quickbook documentation.
* 17 Dec 2005: Converted documentation to Quickbook Format.
* 2 Jul 2002: Compile-time and run-time items separated to new headers.
* 7 Nov 2001: Initial version
[endsect] [/section:gcd_history History]
[section:gcd_credits Credits]
The author of the Boost compilation of GCD and LCM computations is
Daryle Walker. The code was prompted by existing code hiding in the
implementations of Paul Moore's rational library and Steve Cleary's
pool library. The code had updates by Helmut Zeisel.
[endsect] [/section:gcd_credits Credits]
Note that for the time being the headers `<boost/math/common_factor.hpp>`, `<boost/math/common_factor_ct.hpp>` and `<boost/math/common_factor_rt.hpp>` will continue to exist
and redirect to the moved headers.
[endmathpart] [/mathpart gcd_lcm Integer Utilities (Greatest Common Divisor and Least Common Multiple)]

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<a name="gcd_lcm"></a>Chapter&#160;11.&#160;Integer Utilities (Greatest Common Divisor and Least Common Multiple)</h1></div></div></div>
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<p><b>Table of Contents</b></p>
<dl>
<dt><span class="section"><a href="math_toolkit/introduction.html">Introduction</a></span></dt>
<dt><span class="section"><a href="math_toolkit/synopsis.html">Synopsis</a></span></dt>
<dt><span class="section"><a href="math_toolkit/gcd_function_object.html">GCD Function Object</a></span></dt>
<dt><span class="section"><a href="math_toolkit/lcm_function_object.html">LCM Function Object</a></span></dt>
<dt><span class="section"><a href="math_toolkit/run_time.html">Run-time GCD &amp; LCM Determination</a></span></dt>
<dt><span class="section"><a href="math_toolkit/compile_time.html">Compile-time GCD and LCM determination</a></span></dt>
<dt><span class="section"><a href="math_toolkit/gcd_header.html">Header &lt;boost/math/common_factor.hpp&gt;</a></span></dt>
<dt><span class="section"><a href="math_toolkit/demo.html">Demonstration Program</a></span></dt>
<dt><span class="section"><a href="math_toolkit/rationale0.html">Rationale</a></span></dt>
<dt><span class="section"><a href="math_toolkit/gcd_history.html">History</a></span></dt>
<dt><span class="section"><a href="math_toolkit/gcd_credits.html">Credits</a></span></dt>
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<p>
This code has now been moved to Boost.Integer, please see <a href="http://www.boost.org/doc/libs/release/libs/integer/doc/html/index.html" target="_top">here</a>.
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<p>
Note that for the time being the headers <code class="computeroutput"><span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">common_factor</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></code>,
<code class="computeroutput"><span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">common_factor_ct</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></code> and
<code class="computeroutput"><span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">common_factor_rt</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></code> will
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@@ -620,7 +620,6 @@
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tr1_ref.html" title="TR1 C Functions Quick Reference"><span class="index-entry-level-1">TR1 C Functions Quick Reference</span></a></p></li>
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</li>
@@ -637,7 +636,6 @@
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_derivatives.html" title="Derivatives of the Bessel Functions"><span class="index-entry-level-1">Derivatives of the Bessel Functions</span></a></p></li>
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">Known Issues, and TODO List</span></a></p></li>
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@@ -694,7 +692,6 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/main_tr1.html" title="C99 and TR1 C Functions Overview"><span class="index-entry-level-1">C99 and TR1 C Functions Overview</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/logs.html" title="Error Logs For Error Rate Tables"><span class="index-entry-level-1">Error Logs For Error Rate Tables</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_first.html#math_toolkit.bessel.bessel_first.table_cyl_neumann_integer_orders_" title="Table&#160;6.42.&#160;Error rates for cyl_neumann (integer orders)"><span class="index-entry-level-1">Error rates for cyl_neumann (integer orders)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">Known Issues, and TODO List</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tr1_ref.html" title="TR1 C Functions Quick Reference"><span class="index-entry-level-1">TR1 C Functions Quick Reference</span></a></p></li>
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</li>
@@ -711,7 +708,6 @@
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_derivatives.html" title="Derivatives of the Bessel Functions"><span class="index-entry-level-1">Derivatives of the Bessel Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/all_table.html#math_toolkit.logs_and_tables.all_table.table_cyl_neumann_prime_integer_orders_" title="Table&#160;17.24.&#160;Error rates for cyl_neumann_prime (integer orders)"><span class="index-entry-level-1">Error rates for cyl_neumann_prime (integer orders)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">Known Issues, and TODO List</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
@@ -1260,7 +1256,6 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/overview_tr1.html" title="C99 and C++ TR1 C-style Functions"><span class="index-entry-level-1">C99 and C++ TR1 C-style Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/main_tr1.html" title="C99 and TR1 C Functions Overview"><span class="index-entry-level-1">C99 and TR1 C Functions Overview</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/history1.html" title="History and What's New"><span class="index-entry-level-1">History and What's New</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">Known Issues, and TODO List</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/nc_t_dist.html" title="Noncentral T Distribution"><span class="index-entry-level-1">Noncentral T Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sinc/sinc_overview.html" title="Sinus Cardinal and Hyperbolic Sinus Cardinal Functions Overview"><span class="index-entry-level-1">Sinus Cardinal and Hyperbolic Sinus Cardinal Functions Overview</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tr1_ref.html" title="TR1 C Functions Quick Reference"><span class="index-entry-level-1">TR1 C Functions Quick Reference</span></a></p></li>
@@ -1315,10 +1310,6 @@
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses"><span class="index-entry-level-1">Incomplete Gamma Function Inverses</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">gcd</span></p>
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</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">get</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/new_const.html" title="Defining New Constants"><span class="index-entry-level-1">Defining New Constants</span></a></p></li>
@@ -1720,10 +1711,6 @@
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_poly/laguerre.html" title="Laguerre (and Associated) Polynomials"><span class="index-entry-level-1">Laguerre (and Associated) Polynomials</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">lcm</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/synopsis.html" title="Synopsis"><span class="index-entry-level-1">Synopsis</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">ldexp</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/real_concepts.html" title="Conceptual Requirements for Real Number Types"><span class="index-entry-level-1">Conceptual Requirements for Real Number Types</span></a></p></li></ul></div>
</li>
@@ -1768,13 +1755,6 @@
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_poly/legendre.html" title="Legendre (and Associated) Polynomials"><span class="index-entry-level-1">Legendre (and Associated) Polynomials</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">less</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/st_eg/two_sample_students_t.html" title="Comparing the means of two samples with the Students-t test"><span class="index-entry-level-1">Comparing the means of two samples with the Students-t test</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/inverse_chi_squared_eg.html" title="Inverse Chi-Squared Distribution Bayes Example"><span class="index-entry-level-1">Inverse Chi-Squared Distribution Bayes Example</span></a></p></li>
</ul></div>
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<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">lgamma</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/overview_tr1.html" title="C99 and C++ TR1 C-style Functions"><span class="index-entry-level-1">C99 and C++ TR1 C-style Functions</span></a></p></li>
@@ -2197,10 +2177,8 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots/root_finding_examples/5th_root_eg.html" title="Computing the Fifth Root"><span class="index-entry-level-1">Computing the Fifth Root</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/internals/cf.html" title="Continued Fraction Evaluation"><span class="index-entry-level-1">Continued Fraction Evaluation</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots/root_finding_examples/cbrt_eg.html" title="Finding the Cubed Root With and Without Derivatives"><span class="index-entry-level-1">Finding the Cubed Root With and Without Derivatives</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/gcd_function_object.html" title="GCD Function Object"><span class="index-entry-level-1">GCD Function Object</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots/root_finding_examples/nth_root.html" title="Generalizing to Compute the nth root"><span class="index-entry-level-1">Generalizing to Compute the nth root</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/internals/test_data.html" title="Graphing, Profiling, and Generating Test Data for Special Functions"><span class="index-entry-level-1">Graphing, Profiling, and Generating Test Data for Special Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/lcm_function_object.html" title="LCM Function Object"><span class="index-entry-level-1">LCM Function Object</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots/brent_minima.html" title="Locating Function Minima using Brent's algorithm"><span class="index-entry-level-1">Locating Function Minima using Brent's algorithm</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/internals/series_evaluation.html" title="Series Evaluation"><span class="index-entry-level-1">Series Evaluation</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots/roots_noderiv/root_termination.html" title="Termination Condition Functors"><span class="index-entry-level-1">Termination Condition Functors</span></a></p></li>

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@@ -24,7 +24,7 @@
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<title>Typedef Index</title>
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@@ -24,7 +24,7 @@
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@@ -24,7 +24,7 @@
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@@ -4,7 +4,7 @@
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@@ -23,7 +23,7 @@
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<div class="variablelist"><dl class="variablelist">
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@@ -1583,10 +1583,6 @@
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/comp_compilers.html" title="Comparing Different Compilers"><span class="index-entry-level-1">performance</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">Comparing the means of two samples with the Students-t test</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/st_eg/two_sample_students_t.html" title="Comparing the means of two samples with the Students-t test"><span class="index-entry-level-1">less</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">Comparison of Cube Root Finding Algorithms</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots/root_comparison/cbrt_comparison.html" title="Comparison of Cube Root Finding Algorithms"><span class="index-entry-level-1">double</span></a></p></li>
@@ -1625,13 +1621,6 @@
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">Compile-time GCD and LCM determination</span></p>
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/compile_time.html" title="Compile-time GCD and LCM determination"><span class="index-entry-level-1">constants</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/compile_time.html" title="Compile-time GCD and LCM determination"><span class="index-entry-level-1">expression</span></a></p></li>
</ul></div>
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<p><span class="index-entry-level-0">Compilers</span></p>
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/compilers_overview.html" title="Compilers"><span class="index-entry-level-1">accuracy</span></a></p></li>
@@ -1814,7 +1803,6 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/binom_eg/binomial_quiz_example.html" title="Binomial Quiz Example"><span class="index-entry-level-1">Binomial Quiz Example</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table&#160;1.11.&#160;Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_tuning" title="Table&#160;1.12.&#160;Boost.Math Tuning"><span class="index-entry-level-1">Boost.Math Tuning</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/compile_time.html" title="Compile-time GCD and LCM determination"><span class="index-entry-level-1">Compile-time GCD and LCM determination</span></a></p></li>
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@@ -2004,7 +1992,6 @@
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">Known Issues, and TODO List</span></a></p></li>
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</li>
@@ -2021,7 +2008,6 @@
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_derivatives.html" title="Derivatives of the Bessel Functions"><span class="index-entry-level-1">Derivatives of the Bessel Functions</span></a></p></li>
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">Known Issues, and TODO List</span></a></p></li>
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</li>
<li class="listitem" style="list-style-type: none">
@@ -2078,7 +2064,6 @@
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/logs.html" title="Error Logs For Error Rate Tables"><span class="index-entry-level-1">Error Logs For Error Rate Tables</span></a></p></li>
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">Known Issues, and TODO List</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tr1_ref.html" title="TR1 C Functions Quick Reference"><span class="index-entry-level-1">TR1 C Functions Quick Reference</span></a></p></li>
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@@ -2095,7 +2080,6 @@
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_derivatives.html" title="Derivatives of the Bessel Functions"><span class="index-entry-level-1">Derivatives of the Bessel Functions</span></a></p></li>
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">Known Issues, and TODO List</span></a></p></li>
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</li>
<li class="listitem" style="list-style-type: none">
@@ -3216,7 +3200,6 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/c99.html" title="C99 C Functions"><span class="index-entry-level-1">C99 C Functions</span></a></p></li>
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@@ -3702,7 +3685,6 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/overview_tr1.html" title="C99 and C++ TR1 C-style Functions"><span class="index-entry-level-1">C99 and C++ TR1 C-style Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/main_tr1.html" title="C99 and TR1 C Functions Overview"><span class="index-entry-level-1">C99 and TR1 C Functions Overview</span></a></p></li>
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">Known Issues, and TODO List</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/nc_t_dist.html" title="Noncentral T Distribution"><span class="index-entry-level-1">Noncentral T Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sinc/sinc_overview.html" title="Sinus Cardinal and Hyperbolic Sinus Cardinal Functions Overview"><span class="index-entry-level-1">Sinus Cardinal and Hyperbolic Sinus Cardinal Functions Overview</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tr1_ref.html" title="TR1 C Functions Quick Reference"><span class="index-entry-level-1">TR1 C Functions Quick Reference</span></a></p></li>
@@ -3795,14 +3777,6 @@
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses"><span class="index-entry-level-1">Incomplete Gamma Function Inverses</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">gcd</span></p>
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</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">GCD Function Object</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/gcd_function_object.html" title="GCD Function Object"><span class="index-entry-level-1">operator</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">Generalizing to Compute the nth root</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots/root_finding_examples/nth_root.html" title="Generalizing to Compute the nth root"><span class="index-entry-level-1">accuracy</span></a></p></li>
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/inverse_chi_squared_eg.html" title="Inverse Chi-Squared Distribution Bayes Example"><span class="index-entry-level-1">data</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/inverse_chi_squared_eg.html" title="Inverse Chi-Squared Distribution Bayes Example"><span class="index-entry-level-1">expression</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/inverse_chi_squared_eg.html" title="Inverse Chi-Squared Distribution Bayes Example"><span class="index-entry-level-1">less</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/inverse_chi_squared_eg.html" title="Inverse Chi-Squared Distribution Bayes Example"><span class="index-entry-level-1">performance</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/inverse_chi_squared_eg.html" title="Inverse Chi-Squared Distribution Bayes Example"><span class="index-entry-level-1">scale</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/inverse_chi_squared_eg.html" title="Inverse Chi-Squared Distribution Bayes Example"><span class="index-entry-level-1">variance</span></a></p></li>
@@ -4618,11 +4591,6 @@
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">accuracy</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">by</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">cyl_bessel_j</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">cyl_bessel_j_prime</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">cyl_neumann</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">cyl_neumann_prime</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">J</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">K</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">Lanczos approximation</span></a></p></li>
@@ -4739,14 +4707,6 @@
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/laplace_dist.html" title="Laplace Distribution"><span class="index-entry-level-0">laplace_distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">lcm</span></p>
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</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">LCM Function Object</span></p>
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</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">ldexp</span></p>
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</li>
@@ -4801,13 +4761,6 @@
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</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">less</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/st_eg/two_sample_students_t.html" title="Comparing the means of two samples with the Students-t test"><span class="index-entry-level-1">Comparing the means of two samples with the Students-t test</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/inverse_chi_squared_eg.html" title="Inverse Chi-Squared Distribution Bayes Example"><span class="index-entry-level-1">Inverse Chi-Squared Distribution Bayes Example</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">lgamma</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/overview_tr1.html" title="C99 and C++ TR1 C-style Functions"><span class="index-entry-level-1">C99 and C++ TR1 C-style Functions</span></a></p></li>
@@ -5576,10 +5529,8 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots/root_finding_examples/5th_root_eg.html" title="Computing the Fifth Root"><span class="index-entry-level-1">Computing the Fifth Root</span></a></p></li>
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots/root_finding_examples/cbrt_eg.html" title="Finding the Cubed Root With and Without Derivatives"><span class="index-entry-level-1">Finding the Cubed Root With and Without Derivatives</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/gcd_function_object.html" title="GCD Function Object"><span class="index-entry-level-1">GCD Function Object</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots/root_finding_examples/nth_root.html" title="Generalizing to Compute the nth root"><span class="index-entry-level-1">Generalizing to Compute the nth root</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/internals/test_data.html" title="Graphing, Profiling, and Generating Test Data for Special Functions"><span class="index-entry-level-1">Graphing, Profiling, and Generating Test Data for Special Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/lcm_function_object.html" title="LCM Function Object"><span class="index-entry-level-1">LCM Function Object</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots/brent_minima.html" title="Locating Function Minima using Brent's algorithm"><span class="index-entry-level-1">Locating Function Minima using Brent's algorithm</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/internals/series_evaluation.html" title="Series Evaluation"><span class="index-entry-level-1">Series Evaluation</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots/roots_noderiv/root_termination.html" title="Termination Condition Functors"><span class="index-entry-level-1">Termination Condition Functors</span></a></p></li>
@@ -6701,9 +6652,7 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/quat_synopsis.html" title="Synopsis"><span class="index-entry-level-1">conj</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/quat_synopsis.html" title="Synopsis"><span class="index-entry-level-1">cylindrical</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/quat_synopsis.html" title="Synopsis"><span class="index-entry-level-1">cylindrospherical</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/synopsis.html" title="Synopsis"><span class="index-entry-level-1">gcd</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/quat_synopsis.html" title="Synopsis"><span class="index-entry-level-1">l1</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/synopsis.html" title="Synopsis"><span class="index-entry-level-1">lcm</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/quat_synopsis.html" title="Synopsis"><span class="index-entry-level-1">multipolar</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/quat_synopsis.html" title="Synopsis"><span class="index-entry-level-1">norm</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/quat_synopsis.html" title="Synopsis"><span class="index-entry-level-1">semipolar</span></a></p></li>

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@@ -4,8 +4,8 @@
<title>Chapter&#160;13.&#160;Internal Details: Series, Rationals and Continued Fractions, Testing, and Development Tools</title>
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@@ -4,8 +4,8 @@
<title>Chapter&#160;8.&#160;Complex Number Functions</title>
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<meta name="generator" content="DocBook XSL Stylesheets V1.77.1">
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<title>Compile-time GCD and LCM determination</title>
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<title>The Mathematical Constants</title>
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<p>
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<a class="indexterm" name="id897378"></a>
</p>
<p>
This documentation aims to use of the following naming and formatting conventions.

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<title>Quaternion Creation Functions</title>
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<title>Distribution Algorithms</title>
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<title>Bernoulli Distribution</title>
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<title>Beta Distribution</title>
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<title>Binomial Distribution</title>
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<title>Cauchy-Lorentz Distribution</title>
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<title>Chi Squared Distribution</title>
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<title>Exponential Distribution</title>
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<title>Extreme Value Distribution</title>
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<title>F Distribution</title>
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<title>Gamma (and Erlang) Distribution</title>
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<title>Geometric Distribution</title>
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<title>Hyperexponential Distribution</title>
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<title>Hypergeometric Distribution</title>
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<title>Inverse Chi Squared Distribution</title>
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<title>Inverse Gamma Distribution</title>
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<title>Inverse Gaussian (or Inverse Normal) Distribution</title>
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<title>Laplace Distribution</title>
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<title>Logistic Distribution</title>
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<title>Log Normal Distribution</title>
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<title>Noncentral Beta Distribution</title>
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<title>Noncentral Chi-Squared Distribution</title>
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<title>Noncentral F Distribution</title>
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<title>Noncentral T Distribution</title>
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<title>Negative Binomial Distribution</title>
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<title>Normal (Gaussian) Distribution</title>
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<title>Pareto Distribution</title>
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<title>Poisson Distribution</title>
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<title>Rayleigh Distribution</title>
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<title>Skew Normal Distribution</title>
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<title>Students t Distribution</title>
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<title>Triangular Distribution</title>
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<title>Uniform Distribution</title>
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<title>Weibull Distribution</title>
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<title>Elliptic Integrals</title>
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<title>Elliptic Integrals of the First Kind - Legendre Form</title>
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<title>Elliptic Integrals of the Second Kind - Legendre Form</title>
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