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Difference between revisions of "cpp/numeric/math/lgamma"

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< cpp‎ | numeric‎ | math
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{{cpp/title|lgamma}}
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{{cpp/title|lgamma|lgammaf|lgammal}}
 
{{cpp/numeric/math/navbar}}
 
{{cpp/numeric/math/navbar}}
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{{ddcl list begin}}
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{{cpp/numeric/math/declarations
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{{ddcl list header | cmath}}
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|family=lgamma
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{{ddcl list item | notes={{mark since c++11}} |
+
|param1=num
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float      lgamma( float arg );
+
|constexpr_since=26
 +
|desc=Computes the natural logarithm of the absolute value of the {{enwiki|gamma function}} of {{c|num}}.
 
}}
 
}}
−
{{ddcl list item | notes={{mark since c++11}} |
 
−
double      lgamma( double arg );
 
−
}}
 
−
{{ddcl list item | notes={{mark since c++11}} |
 
−
long double lgamma( long double arg );
 
−
}}
 
−
{{ddcl list item | notes={{mark since c++11}} |
 
−
double      lgamma( Integral arg );
 
−
}}
 
−
{{ddcl list end}}
 
−
 
−
Computes the natural logarithm of the absolute value of the [[enwiki:Gamma function|gamma function]] of {{tt|arg}}.
 
  
 
===Parameters===
 
===Parameters===
−
{{param list begin}}
+
{{par begin}}
−
{{param list item | arg | floating point value}}
+
{{par|num|floating-point or integer value}}
−
{{param list end}}
+
{{par end}}
  
 
===Return value===
 
===Return value===
 +
If no errors occur, the value of the logarithm of the gamma function of {{c|num}}, that is {{mathjax-or|1=\(\log_{e}{{!}}{\int_0^\infty t^{num-1} e^{-t} \mathsf{d}t}{{!}}\)|2=log{{su|b=e}}{{!}}{{minteg|0|∞|''t''{{su|p=num-1}} {{mexp|-t}} d''t''}}{{!}}}}, is returned.
  
−
The value of the logarithm of the gamma function of {{tt|arg}}, that is {{math|log{{su|b=e}}{{!}}{{minteg|0|∞|''t''{{su|p=arg-1}} {{mexp|-t}} d''t''}}{{!}}}}.
+
If a pole error occurs, {{lc|HUGE_VAL|+HUGE_VAL}}, {{tt|+HUGE_VALF}}, or {{tt|+HUGE_VALL}} is returned.
  
−
If {{tt|arg}} is a natural number, {{c|std::lgamma(arg)}} is the logarithm of the factorial of {{tt|arg-1}}.
+
If a range error due to overflow occurs, {{lc|HUGE_VAL|±HUGE_VAL}}, {{tt|±HUGE_VALF}}, or {{tt|±HUGE_VALL}} is returned.
  
−
===Exceptions===
+
===Error handling===
−
If {{tt|arg}} is zero or a negative integer, pole error may occur and {{c|FE_INVALID}} or {{c|FE_DIVBYZERO}} may be raised.
+
Errors are reported as specified in {{lc|math_errhandling}}.
  
−
If {{tt|arg}} is too large, range error may occur and {{c|FE_OVERFLOW}} may be raised.
+
If {{c|num}} is zero or is an integer less than zero, a pole error may occur.
 +
 
 +
If the implementation supports IEEE floating-point arithmetic (IEC 60559),
 +
* If the argument is 1, +0 is returned.
 +
* If the argument is 2, +0 is returned.
 +
* If the argument is ±0, +∞ is returned and {{lc|FE_DIVBYZERO}} is raised.
 +
* If the argument is a negative integer, +∞ is returned and {{lc|FE_DIVBYZERO}} is raised.
 +
* If the argument is ±∞, +∞ is returned.
 +
* If the argument is NaN, NaN is returned.
  
 
===Notes===
 
===Notes===
 +
If {{c|num}} is a natural number, {{c|std::lgamma(num)}} is the logarithm of the factorial of {{c|num - 1}}.
  
−
[http://pubs.opengroup.org/onlinepubs/9699919799/functions/lgamma.html POSIX specification] additionally requires that each execution of {{c|lgamma}} stores the sign of the gamma function of {{tt|arg}} in the external variable {{tt|signgam}}.
+
The [https://pubs.opengroup.org/onlinepubs/9699919799/functions/lgamma.html POSIX version of {{tt|lgamma}}] is not thread-safe: each execution of the function stores the sign of the gamma function of {{c|num}} in the static external variable {{tt|signgam}}. Some implementations provide {{tt|lgamma_r}}, which takes a pointer to user-provided storage for {{tt|singgam}} as the second parameter, and is thread-safe.
 +
 
 +
There is a non-standard function named {{tt|gamma}} in various implementations, but its definition is inconsistent. For example, glibc and 4.2BSD version of {{tt|gamma}} executes {{tt|lgamma}}, but 4.4BSD version of {{tt|gamma}} executes {{tt|tgamma}}.
 +
 
 +
{{cpp/numeric/math/additional integer overload note|lgamma}}
 +
 
 +
===Example===
 +
{{example
 +
|code=
 +
#include <cerrno>
 +
#include <cfenv>
 +
#include <cmath>
 +
#include <cstring>
 +
#include <iostream>
 +
 
 +
// #pragma STDC FENV_ACCESS ON
 +
 
 +
const double pi = std::acos(-1); // or std::numbers::pi since C++20
 +
 
 +
int main()
 +
{
 +
    std::cout << "lgamma(10) = " << std::lgamma(10)
 +
              << ", log(9!) = " << std::log(std::tgamma(10))
 +
              << ", exp(lgamma(10)) = " << std::exp(std::lgamma(10)) << '\n'
 +
              << "lgamma(0.5) = " << std::lgamma(0.5)
 +
              << ", log(sqrt(pi)) = " << std::log(std::sqrt(pi)) << '\n';
 +
   
 +
    // special values
 +
    std::cout << "lgamma(1) = " << std::lgamma(1) << '\n'
 +
              << "lgamma(+Inf) = " << std::lgamma(INFINITY) << '\n';
 +
   
 +
    // error handling
 +
    errno = 0;
 +
    std::feclearexcept(FE_ALL_EXCEPT);
 +
   
 +
    std::cout << "lgamma(0) = " << std::lgamma(0) << '\n';
 +
   
 +
    if (errno == ERANGE)
 +
        std::cout << "    errno == ERANGE: " << std::strerror(errno) << '\n';
 +
    if (std::fetestexcept(FE_DIVBYZERO))
 +
        std::cout << "    FE_DIVBYZERO raised\n";
 +
}
 +
|output=
 +
lgamma(10) = 12.8018, log(9!) = 12.8018, exp(lgamma(10)) = 362880
 +
lgamma(0.5) = 0.572365, log(sqrt(pi)) = 0.572365
 +
lgamma(1) = 0
 +
lgamma(+Inf) = inf
 +
lgamma(0) = inf
 +
    errno == ERANGE: Numerical result out of range
 +
    FE_DIVBYZERO raised
 +
}}
  
 
===See also===
 
===See also===
−
{{dcl list begin}}
+
{{dsc begin}}
−
{{dcl list template | cpp/numeric/math/dcl list tgamma}}
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{{dsc inc|cpp/numeric/math/dsc tgamma}}
−
{{dcl list see c | c/numeric/math/lgamma}}
+
{{dsc see c|c/numeric/math/lgamma}}
−
{{dcl list end}}
+
{{dsc end}}
  
 
===External links===
 
===External links===
−
[http://mathworld.wolfram.com/LogGammaFunction.html Weisstein, Eric W. "Log Gamma Function."] From MathWorld--A Wolfram Web Resource.
+
{{eli|[https://mathworld.wolfram.com/LogGammaFunction.html Weisstein, Eric W. "Log Gamma Function."] From MathWorld &mdash; A Wolfram Web Resource.}}
  
−
[[de:cpp/numeric/math/lgamma]]
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{{langlinks|de|es|fr|it|ja|pt|ru|zh}}
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[[es:cpp/numeric/math/lgamma]]
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−
[[fr:cpp/numeric/math/lgamma]]
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−
[[it:cpp/numeric/math/lgamma]]
+
−
[[ja:cpp/numeric/math/lgamma]]
+
−
[[pt:cpp/numeric/math/lgamma]]
+
−
[[ru:cpp/numeric/math/lgamma]]
+
−
[[zh:cpp/numeric/math/lgamma]]
+

Latest revision as of 09:10, 28 June 2023

 
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Defined in header <cmath>
(1)
float       lgamma ( float num );

double      lgamma ( double num );

long double lgamma ( long double num );
 (until C++23)
/* floating-point-type */
            lgamma ( /* floating-point-type */ num );
 (since C++23)
(constexpr since C++26)
float       lgammaf( float num );
 (2) (since C++11)
(constexpr since C++26)
long double lgammal( long double num );
 (3) (since C++11)
(constexpr since C++26)
Additional overloads (since C++11)
Defined in header <cmath>
template< class Integer >
double      lgamma ( Integer num );
 (A) (constexpr since C++26)
1-3) Computes the natural logarithm of the absolute value of the gamma function of num. The library provides overloads of std::lgamma for all cv-unqualified floating-point types as the type of the parameter.(since C++23)
A) Additional overloads are provided for all integer types, which are treated as double.
 (since C++11)

Contents

[edit] Parameters

num - floating-point or integer value

[edit] Return value

If no errors occur, the value of the logarithm of the gamma function of num, that is loge|∫∞0tnum-1 e-t dt|, is returned.

If a pole error occurs, +HUGE_VAL, +HUGE_VALF, or +HUGE_VALL is returned.

If a range error due to overflow occurs, ±HUGE_VAL, ±HUGE_VALF, or ±HUGE_VALL is returned.

[edit] Error handling

Errors are reported as specified in math_errhandling.

If num is zero or is an integer less than zero, a pole error may occur.

If the implementation supports IEEE floating-point arithmetic (IEC 60559),

  • If the argument is 1, +0 is returned.
  • If the argument is 2, +0 is returned.
  • If the argument is ±0, +∞ is returned and FE_DIVBYZERO is raised.
  • If the argument is a negative integer, +∞ is returned and FE_DIVBYZERO is raised.
  • If the argument is ±∞, +∞ is returned.
  • If the argument is NaN, NaN is returned.

[edit] Notes

If num is a natural number, std::lgamma(num) is the logarithm of the factorial of num - 1.

The POSIX version of lgamma is not thread-safe: each execution of the function stores the sign of the gamma function of num in the static external variable signgam. Some implementations provide lgamma_r, which takes a pointer to user-provided storage for singgam as the second parameter, and is thread-safe.

There is a non-standard function named gamma in various implementations, but its definition is inconsistent. For example, glibc and 4.2BSD version of gamma executes lgamma, but 4.4BSD version of gamma executes tgamma.

The additional overloads are not required to be provided exactly as (A). They only need to be sufficient to ensure that for their argument num of integer type, std::lgamma(num) has the same effect as std::lgamma(static_cast<double>(num)).

[edit] Example

Run this code
#include <cerrno>
#include <cfenv>
#include <cmath>
#include <cstring>
#include <iostream>
 
// #pragma STDC FENV_ACCESS ON
 
const double pi = std::acos(-1); // or std::numbers::pi since C++20
 
int main()
{
    std::cout << "lgamma(10) = " << std::lgamma(10)
              << ", log(9!) = " << std::log(std::tgamma(10))
              << ", exp(lgamma(10)) = " << std::exp(std::lgamma(10)) << '\n'
              << "lgamma(0.5) = " << std::lgamma(0.5)
              << ", log(sqrt(pi)) = " << std::log(std::sqrt(pi)) << '\n';
 
    // special values
    std::cout << "lgamma(1) = " << std::lgamma(1) << '\n'
              << "lgamma(+Inf) = " << std::lgamma(INFINITY) << '\n';
 
    // error handling
    errno = 0;
    std::feclearexcept(FE_ALL_EXCEPT);
 
    std::cout << "lgamma(0) = " << std::lgamma(0) << '\n';
 
    if (errno == ERANGE)
        std::cout << "    errno == ERANGE: " << std::strerror(errno) << '\n';
    if (std::fetestexcept(FE_DIVBYZERO))
        std::cout << "    FE_DIVBYZERO raised\n";
}

Output: 

lgamma(10) = 12.8018, log(9!) = 12.8018, exp(lgamma(10)) = 362880
lgamma(0.5) = 0.572365, log(sqrt(pi)) = 0.572365
lgamma(1) = 0
lgamma(+Inf) = inf
lgamma(0) = inf
    errno == ERANGE: Numerical result out of range
    FE_DIVBYZERO raised

[edit] See also

(C++11)(C++11)(C++11)
 gamma function
(function) [edit]
C documentation for lgamma

[edit] External links

Weisstein, Eric W. "Log Gamma Function." From MathWorld — A Wolfram Web Resource.
Retrieved from "https://en.cppreference.com/mwiki/index.php?title=cpp/numeric/math/lgamma&oldid=154203"