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Difference between revisions of "cpp/numeric/special functions/beta"

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m (→‎Return value: mathjax-or)
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{{cpp/numeric/special_functions/navbar}}
 
{{cpp/numeric/special_functions/navbar}}
 
{{dcl begin}}
 
{{dcl begin}}
−
{{dcl header | cmath}}
+
{{dcl header|cmath}}
−
{{dcl |num=1|since=c++17|
+
{{dcl rev multi|num=1|since1=c++17|dcl1=
−
double      beta( double x, double y );
+
float      beta ( float x, float y );
 +
double      beta ( double x, double y );
 +
long double beta ( long double x, long double y );
 +
|since2=c++23|dcl2=
 +
/* floating-point-type */ beta( /* floating-point-type */ x,
 +
                                /* floating-point-type */ y );
 +
}}
 +
{{dcl|num=2|since=c++17|
 
float      betaf( float x, float y );
 
float      betaf( float x, float y );
 +
}}
 +
{{dcl|num=3|since=c++17|
 
long double betal( long double x, long double y );
 
long double betal( long double x, long double y );
 
}}
 
}}
−
{{dcl |num=2|since=c++17|
+
{{dcl h|[[#Notes|Additional overloads]]}}
−
Promoted    beta( Arithmetic x, Arithmetic y );
+
{{dcl header|cmath}}
 +
{{dcl|num=A|since=c++17|
 +
template< class Arithmetic1, class Arithmetic2 >
 +
/* common-floating-point-type */ beta( Arithmetic1 x, Arithmetic2 y );
 
}}
 
}}
 
{{dcl end}}
 
{{dcl end}}
  
−
@1@ Computes the [[enwiki:Beta_function|beta function]] of {{tt|x}} and {{tt|y}}.
+
@1-3@ Computes the {{enwiki|Beta function}} of {{c|x}} and {{c|y}}.{{rev inl|since=c++23| The library provides overloads of {{tt|std::beta}} for all cv-unqualified floating-point types as the type of the parameters {{c|x}} and {{c|y}}.}}
−
@2@ A set of overloads or a function template for all combinations of arguments of arithmetic type not covered by {{v|1}}. If any argument has [[cpp/types/is_integral|integral type]], it is cast to {{c|double}}. If any argument is {{c|long double}}, then the return type {{tt|Promoted}} is also {{c|long double}}, otherwise the return type is always {{c|double}}.
+
 
 +
@A@ Additional overloads are provided for all other combinations of arithmetic types.
  
 
===Parameters===
 
===Parameters===
 
{{par begin}}
 
{{par begin}}
−
{{par | x, y | values of a floating-point or integral type}}
+
{{par|x, y|floating-point or integer values}}
 
{{par end}}
 
{{par end}}
  
 
===Return value===
 
===Return value===
−
If no errors occur, value of the beta function of {{tt|x}} and {{tt|y}}, that is {{mathjax-or|\(\int_{0}^{1}{ {t}^{x-1}{(1-t)}^{y-1}{\rm{d} }t}\)|{{minteg|0|1|t{{su|p=x-1}}(1-t){{su|p=(y-1)}}d''t''}}}}, or, equivalently, {{mathjax-or|\(\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\)|{{mfrac|Γ(x)Γ(y)|Γ(x+y)}}}} is returned.
+
If no errors occur, value of the beta function of {{c|x}} and {{c|y}}, that is {{mathjax-or|\(\int_{0}^{1}{ {t}^{x-1}{(1-t)}^{y-1}\mathsf{d}t}\)|{{minteg|0|1|t{{su|p=x-1}}(1-t){{su|p=(y-1)}}d''t''}}}}, or, equivalently, {{mathjax-or|\(\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\)|{{mfrac|Γ(x)Γ(y)|Γ(x+y)}}}} is returned.
  
 
===Error handling===
 
===Error handling===
−
Errors may be reported as specified in [[cpp/numeric/math/math_errhandling|math_errhandling]]
+
Errors may be reported as specified in {{lc|math_errhandling}}.
−
 
+
* If any argument is NaN, NaN is returned and domain error is not reported.
−
* If any argument is NaN, NaN is returned and domain error is not reported
+
* The function is only required to be defined where both {{c|x}} and {{c|y}} are greater than zero, and is allowed to report a domain error otherwise.
−
* The function is only required to be defined where both {{tt|x}} and {{tt|y}} are greater than zero, and is allowed to report a domain error otherwise.
+
  
 
===Notes===
 
===Notes===
−
{{cpp/numeric/special_functions/older_impl_note}}
+
{{cpp/numeric/special functions/older impl note}}
  
−
An implementation of this function is also [http://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/sf_beta/beta_function.html available in boost.math]
+
An implementation of this function is also [https://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/sf_beta/beta_function.html available in boost.math].
  
−
{{c|beta(x, y)}} equals {{c|beta(y, x)}}
+
{{c|std::beta(x, y)}} equals {{c|std::beta(y, x)}}.
  
−
When {{tt|x}} and {{tt|y}} are positive integers, beta(x,y) equals {{mathjax-or|1=\(\frac{(x-1)!(y-1)!}{(x+y-1)!}\)|2={{mfrac|(x-1)!(y-1)!|(x+y-1)!}}}}.
+
When {{c|x}} and {{c|y}} are positive integers, {{c|std::beta(x, y)}} equals {{mathjax-or|1=\(\frac{(x-1)!(y-1)!}{(x+y-1)!}\)|2={{mfrac|(x-1)!(y-1)!|(x+y-1)!}}}}.
  
 +
Binomial coefficients can be expressed in terms of the beta function: {{mathjax-or|1=\(\binom{n}{k} = \frac{1}{(n+1)B(n-k+1,k+1)}\)|2={{mparen|(|)|n|k}}={{mfrac|1|(n+1)Β(n-k+1,k+1)}}}}.
  
−
Binomial coefficients can be expressed in terms of the beta function: {{mathjax-or|1=\(\binom{n}{k} = \frac{1}{(n+1)B(n-k+1,k+1)}\)|2={{mparen|(|)|n|k}}{{=}}{{mfrac|1|(n+1)Β(n-k+1,k+1)}}}}
+
{{cpp/numeric/special functions/additional overload note|beta}}
  
 
===Example===
 
===Example===
−
{{example|code=
+
{{example
 +
|code=
 +
#include <cassert>
 
#include <cmath>
 
#include <cmath>
−
#include <string>
 
−
#include <iostream>
 
 
#include <iomanip>
 
#include <iomanip>
−
double binom(int n, int k) { return 1/((n+1)*std::beta(n-k+1,k+1)); }
+
#include <iostream>
 +
#include <numbers>
 +
#include <string>
 +
 
 +
long binom_via_beta(int n, int k)
 +
{
 +
    return std::lround(1 / ((n + 1) * std::beta(n - k + 1, k + 1)));
 +
}
 +
 
 +
long binom_via_gamma(int n, int k)
 +
{
 +
    return std::lround(std::tgamma(n + 1) /
 +
                      (std::tgamma(n - k + 1) *
 +
                      std::tgamma(k + 1)));
 +
}
 +
 
 
int main()
 
int main()
 
{
 
{
 
     std::cout << "Pascal's triangle:\n";
 
     std::cout << "Pascal's triangle:\n";
−
     for(int n = 1; n < 10; ++n) {
+
     for (int n = 1; n < 10; ++n)
−
         std::cout << std::string(20-n*2, ' ');
+
    {
−
         for(int k = 1; k < n; ++k)
+
         std::cout << std::string(20 - n * 2, ' ');
−
             std::cout << std::setw(3) << binom(n,k) << ' ';
+
         for (int k = 1; k < n; ++k)
 +
        {
 +
             std::cout << std::setw(3) << binom_via_beta(n, k) << ' ';
 +
            assert(binom_via_beta(n, k) == binom_via_gamma(n, k));
 +
        }
 
         std::cout << '\n';
 
         std::cout << '\n';
 
     }
 
     }
 +
 +
    // A spot-check
 +
    const long double p = 0.123; // a random value in [0, 1]
 +
    const long double q = 1 - p;
 +
    const long double π = std::numbers::pi_v<long double>;
 +
    std::cout << "\n\n" << std::setprecision(19)
 +
              << "β(p,1-p)  = " << std::beta(p, q) << '\n'
 +
              << "π/sin(π*p) = " << π / std::sin(π * p) << '\n';
 
}
 
}
 
|output=
 
|output=
 
Pascal's triangle:
 
Pascal's triangle:
 
                  
 
                  
−
                   2  
+
                   2
−
                 3  3  
+
                 3  3
−
               4  6  4  
+
               4  6  4
−
             5  10  10  5  
+
             5  10  10  5
−
           6  15  20  15  6  
+
           6  15  20  15  6
−
         7  21  35  35  21  7  
+
         7  21  35  35  21  7
−
       8  28  56  70  56  28  8  
+
       8  28  56  70  56  28  8
−
     9  36  84 126 126  84  36  9  
+
     9  36  84 126 126  84  36  9
 +
 
 +
β(p,1-p)  = 8.335989149587307836
 +
π/sin(π*p) = 8.335989149587307834
 
}}
 
}}
  
 
===See also===
 
===See also===
 
{{dsc begin}}
 
{{dsc begin}}
−
{{dsc inc | cpp/numeric/math/dsc tgamma}}
+
{{dsc inc|cpp/numeric/math/dsc tgamma}}
 
{{dsc end}}
 
{{dsc end}}
  
 
===External links===
 
===External links===
−
[http://mathworld.wolfram.com/BetaFunction.html Weisstein, Eric W. "Beta Function."] From MathWorld--A Wolfram Web Resource.
+
{{eli|[https://mathworld.wolfram.com/BetaFunction.html Weisstein, Eric W. "Beta Function."] From MathWorld &mdash; A Wolfram Web Resource.}}
  
 
{{langlinks|de|es|fr|it|ja|pt|ru|zh}}
 
{{langlinks|de|es|fr|it|ja|pt|ru|zh}}

Latest revision as of 23:35, 16 October 2023

 
C++
 
Numerics library
Common mathematical functions
Mathematical special functions (C++17)
Mathematical constants (C++20)
Basic linear algebra algorithms (C++26)
Floating-point environment (C++11)
Complex numbers
Numeric array (valarray)
Pseudo-random number generation
Factor operations
(C++17)
(C++17)
Interpolations
(C++20)
(C++20)
Saturation arithmetic
(C++26)
(C++26)
(C++26)
(C++26)
(C++26)

Generic numeric operations
Bit operations
(C++20)    
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++23)
(C++20)
 
Mathematical special functions
 
Defined in header <cmath>
(1)
float       beta ( float x, float y );

double      beta ( double x, double y );

long double beta ( long double x, long double y );
 (since C++17)
(until C++23)
/* floating-point-type */ beta( /* floating-point-type */ x,
                                /* floating-point-type */ y );
 (since C++23)
float       betaf( float x, float y );
 (2) (since C++17)
long double betal( long double x, long double y );
 (3) (since C++17)
Defined in header <cmath>
template< class Arithmetic1, class Arithmetic2 >
/* common-floating-point-type */ beta( Arithmetic1 x, Arithmetic2 y );
 (A) (since C++17)
1-3) Computes the Beta function of x and y. The library provides overloads of std::beta for all cv-unqualified floating-point types as the type of the parameters x and y.(since C++23)
A) Additional overloads are provided for all other combinations of arithmetic types.

Contents

[edit] Parameters

x, y - floating-point or integer values

[edit] Return value

If no errors occur, value of the beta function of x and y, that is ∫10tx-1(1-t)(y-1)dt, or, equivalently,
Γ(x)Γ(y)
Γ(x+y)
 is returned.

[edit] Error handling

Errors may be reported as specified in math_errhandling.

  • If any argument is NaN, NaN is returned and domain error is not reported.
  • The function is only required to be defined where both x and y are greater than zero, and is allowed to report a domain error otherwise.

[edit] Notes

Implementations that do not support C++17, but support ISO 29124:2010, provide this function if __STDCPP_MATH_SPEC_FUNCS__ is defined by the implementation to a value at least 201003L and if the user defines __STDCPP_WANT_MATH_SPEC_FUNCS__ before including any standard library headers.

Implementations that do not support ISO 29124:2010 but support TR 19768:2007 (TR1), provide this function in the header tr1/cmath and namespace std::tr1.

An implementation of this function is also available in boost.math.

std::beta(x, y) equals std::beta(y, x).

When x and y are positive integers, std::beta(x, y) equals
(x-1)!(y-1)!
(x+y-1)!
. Binomial coefficients can be expressed in terms of the beta function: ⎛
⎜
⎝n
k⎞
⎟
⎠=
1
(n+1)Β(n-k+1,k+1)
.

The additional overloads are not required to be provided exactly as (A). They only need to be sufficient to ensure that for their first argument num1 and second argument num2:

  • If num1 or num2 has type long double, then std::beta(num1, num2) has the same effect as std::beta(static_cast<long double>(num1),
              static_cast<long double>(num2))    .
  • Otherwise, if num1 and/or num2 has type double or an integer type, then std::beta(num1, num2) has the same effect as std::beta(static_cast<double>(num1),
              static_cast<double>(num2))    .
  • Otherwise, if num1 or num2 has type float, then std::beta(num1, num2) has the same effect as std::beta(static_cast<float>(num1),
              static_cast<float>(num2))    .
(until C++23)

If num1 and num2 have arithmetic types, then std::beta(num1, num2) has the same effect as std::beta(static_cast</* common-floating-point-type */>(num1),
          static_cast</* common-floating-point-type */>(num2)), where /* common-floating-point-type */ is the floating-point type with the greatest floating-point conversion rank and greatest floating-point conversion subrank between the types of num1 and num2, arguments of integer type are considered to have the same floating-point conversion rank as double.

If no such floating-point type with the greatest rank and subrank exists, then overload resolution does not result in a usable candidate from the overloads provided.

(since C++23)

[edit] Example

Run this code
#include <cassert>
#include <cmath>
#include <iomanip>
#include <iostream>
#include <numbers>
#include <string>
 
long binom_via_beta(int n, int k)
{
    return std::lround(1 / ((n + 1) * std::beta(n - k + 1, k + 1)));
}
 
long binom_via_gamma(int n, int k)
{
    return std::lround(std::tgamma(n + 1) /
                      (std::tgamma(n - k + 1) * 
                       std::tgamma(k + 1)));
}
 
int main()
{
    std::cout << "Pascal's triangle:\n";
    for (int n = 1; n < 10; ++n)
    {
        std::cout << std::string(20 - n * 2, ' ');
        for (int k = 1; k < n; ++k)
        {
            std::cout << std::setw(3) << binom_via_beta(n, k) << ' ';
            assert(binom_via_beta(n, k) == binom_via_gamma(n, k));
        }
        std::cout << '\n';
    }
 
    // A spot-check
    const long double p = 0.123; // a random value in [0, 1]
    const long double q = 1 - p;
    const long double π = std::numbers::pi_v<long double>;
    std::cout << "\n\n" << std::setprecision(19)
              << "β(p,1-p)   = " << std::beta(p, q) << '\n'
              << "π/sin(π*p) = " << π / std::sin(π * p) << '\n';
}

Output: 

Pascal's triangle:
 
                  2
                3   3
              4   6   4
            5  10  10   5
          6  15  20  15   6
        7  21  35  35  21   7
      8  28  56  70  56  28   8
    9  36  84 126 126  84  36   9
 
β(p,1-p)   = 8.335989149587307836
π/sin(π*p) = 8.335989149587307834

[edit] See also

(C++11)(C++11)(C++11)
 gamma function
(function) [edit]

[edit] External links

Weisstein, Eric W. "Beta Function." From MathWorld — A Wolfram Web Resource.
Retrieved from "https://en.cppreference.com/mwiki/index.php?title=cpp/numeric/special_functions/beta&oldid=160789"