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Difference between revisions of "cpp/numeric/special functions/comp ellint 1"

From cppreference.com
m (proper parameter name)
(Applied P1467R9.)
 
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{{cpp/title|comp_ellint_1|comp_ellint_1f|comp_ellint_1l}}
 
{{cpp/title|comp_ellint_1|comp_ellint_1f|comp_ellint_1l}}
−
{{cpp/numeric/special_math/navbar}}
+
{{cpp/numeric/special_functions/navbar}}
 
{{dcl begin}}
 
{{dcl begin}}
−
{{dcl |num=1|since=c++17|
+
{{dcl header|cmath}}
−
double      comp_ellint_1( double k );
+
{{dcl rev multi|num=1|since1=c++17|dcl1=
−
double      comp_ellint_1( float k );
+
double      comp_ellint_1 ( double k );
−
double     comp_ellint_1( long double k );
+
float      comp_ellint_1 ( float k );
 +
long double comp_ellint_1 ( long double k );
 +
|since2=c++23|dcl2=
 +
/* floating-point-type */ comp_ellint_1( /* floating-point-type */ k );
 +
}}
 +
{{dcl|num=2|since=c++17|
 
float      comp_ellint_1f( float k );
 
float      comp_ellint_1f( float k );
 +
}}
 +
{{dcl|num=3|since=c++17|
 
long double comp_ellint_1l( long double k );
 
long double comp_ellint_1l( long double k );
 
}}
 
}}
−
{{dcl |num=2|since=c++17|
+
{{dcl h|[[#Notes|Additional overloads]]}}
−
double      comp_ellint_1( Integral k );
+
{{dcl header|cmath}}
 +
{{dcl|num=A|since=c++17|
 +
template< class Integer >
 +
double      comp_ellint_1 ( Integer k );
 
}}
 
}}
 
{{dcl end}}
 
{{dcl end}}
  
−
@1@ Computes the [[enwiki:Elliptic_integral#Complete_elliptic_integral_of_the_first_kind|complete elliptic integral of the first kind]] of {{tt|k}}.
+
@1-3@ Computes the {{enwiki|Elliptic integral#Complete elliptic integral of the first kind|complete elliptic integral of the first kind}} of {{c|k}}.{{rev inl|since=c++23| The library provides overloads of {{tt|std::comp_ellint_1}} for all cv-unqualified floating-point types as the type of the parameter {{c|k}}.}}
−
@4@ A set of overloads or a function template accepting an argument of any [[cpp/types/is_integral|integral type]]. Equivalent to {{v|1}} after casting the argument to {{c|double}}.
+
@A@ Additional overloads are provided for all integer types, which are treated as {{c/core|double}}.
  
 
===Parameters===
 
===Parameters===
 
{{par begin}}
 
{{par begin}}
−
{{par | k | elliptic modulus or eccentricity (a value of a floating-point or integral type)}}
+
{{par|k|elliptic modulus or eccentricity (a floating-point or integer value)}}
 
{{par end}}
 
{{par end}}
  
 
===Return value===
 
===Return value===
−
If no errors occur, value of the complete elliptic integral of the first kind of {{tt|k}}, that is {{lc|ellint_1(k,π/2)}}, is returned.
+
If no errors occur, value of the complete elliptic integral of the first kind of {{c|k}}, that is {{c|std::ellint_1(k, π/2)}}, 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 the argument is NaN, NaN is returned and domain error is not reported.
−
* If the argument is NaN, NaN is returned and domain error is not reported
+
* If {{math|{{!}}k{{!}}>1}}, a domain error may occur.
−
* If {{math|{{!}}k{{!}}>1}}, a domain error may occur
+
  
 
===Notes===
 
===Notes===
−
{{cpp/numeric/special_math/older_impl_note}}
+
{{cpp/numeric/special functions/older impl note}}
  
−
An implementation of this function is also [http://www.boost.org/doc/libs/1_60_0/libs/math/doc/html/math_toolkit/ellint/ellint_1.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/ellint/ellint_1.html available in boost.math].
 +
 
 +
{{cpp/numeric/special functions/additional integer overload note|comp_ellint_1}}
  
 
===Example===
 
===Example===
−
{{example|code=
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{{example
 +
|The {{enwiki|Pendulum (mechanics)#Arbitrary-amplitude period|period of a pendulum}} of length {{math|l}}, given acceleration due to gravity {{math|g}}, and initial angle θ equals {{math|4&sdot;{{mrad|l/g}}&sdot;K(sin(θ/2))}}, where {{math|K}} is {{tt|std::comp_ellint_1}}.
 +
|code=
 
#include <cmath>
 
#include <cmath>
 
#include <iostream>
 
#include <iostream>
 +
#include <numbers>
 +
 
int main()
 
int main()
 
{
 
{
−
     double hpi = std::acos(-1)/2;
+
     constexpr double π{std::numbers::pi};
−
     std::cout << "K(0) = " << std::comp_ellint_1(0) << '\n'
+
−
               << "π/2 = " << hpi << '\n'
+
     std::cout << "K(0) ≈ " << std::comp_ellint_1(0) << '\n'
−
               << "K(0.5) = " << std::comp_ellint_1(0.5) << '\n'
+
               << "π/2 ≈ " << π / 2 << '\n'
−
               << "F(0.5, π/2) = " << std::ellint_1(0.5, hpi) << '\n';
+
               << "K(0.5) ≈ " << std::comp_ellint_1(0.5) << '\n'
 +
               << "F(0.5, π/2) ≈ " << std::ellint_1(0.5, π / 2) << '\n'
 +
              << "The period of a pendulum length 1m at 10° initial angle ≈ "
 +
              << 4 * std::sqrt(1 / 9.80665) * std::comp_ellint_1(std::sin(π / 18 / 2))
 +
              << "s,\n" "whereas the linear approximation gives ≈ "
 +
              << 2 * π * std::sqrt(1 / 9.80665) << '\n';
 
}
 
}
 
|output=
 
|output=
−
K(0) = 1.5708
+
K(0) ≈ 1.5708
−
π/2 = 1.5708
+
π/2 ≈ 1.5708
−
K(0.5) = 1.68575
+
K(0.5) ≈ 1.68575
−
F(0.5, π/2) = 1.68575
+
F(0.5, π/2) ≈ 1.68575
 +
The period of a pendulum length 1 m at 10° initial angle ≈ 2.01024s,
 +
whereas the linear approximation gives ≈ 2.00641
 
}}
 
}}
−
 
−
===External links===
 
−
[http://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html Weisstein, Eric W. "Complete Elliptic Integral of the First Kind."] From MathWorld--A Wolfram Web Resource.
 
  
 
===See also===
 
===See also===
 
{{dsc begin}}
 
{{dsc begin}}
−
{{dsc inc | cpp/numeric/special_math/dsc ellint_1}}
+
{{dsc inc|cpp/numeric/special_functions/dsc ellint_1}}
 
{{dsc end}}
 
{{dsc end}}
  
−
[[de:cpp/numeric/special_math/comp_ellint_1]]
+
===External links===
−
[[es:cpp/numeric/special_math/comp_ellint_1]]
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{{eli|[https://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html Weisstein, Eric W. "Complete Elliptic Integral of the First Kind."] From MathWorld — A Wolfram Web Resource.}}
−
[[fr:cpp/numeric/special_math/comp_ellint_1]]
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[[it:cpp/numeric/special_math/comp_ellint_1]]
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{{langlinks|de|es|fr|it|ja|pt|ru|zh}}
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[[ja:cpp/numeric/special_math/comp_ellint_1]]
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−
[[pt:cpp/numeric/special_math/comp_ellint_1]]
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[[ru:cpp/numeric/special_math/comp_ellint_1]]
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−
[[zh:cpp/numeric/special_math/comp_ellint_1]]
+

Latest revision as of 18:17, 21 March 2023

 
C++
 
Numerics library
Common mathematical functions
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(C++23)
(C++20)
 
Mathematical special functions
 
Defined in header <cmath>
(1)
double      comp_ellint_1 ( double k );

float       comp_ellint_1 ( float k );

long double comp_ellint_1 ( long double k );
 (since C++17)
(until C++23)
/* floating-point-type */ comp_ellint_1( /* floating-point-type */ k );
 (since C++23)
float       comp_ellint_1f( float k );
 (2) (since C++17)
long double comp_ellint_1l( long double k );
 (3) (since C++17)
Defined in header <cmath>
template< class Integer >
double      comp_ellint_1 ( Integer k );
 (A) (since C++17)
1-3) Computes the complete elliptic integral of the first kind of k. The library provides overloads of std::comp_ellint_1 for all cv-unqualified floating-point types as the type of the parameter k.(since C++23)
A) Additional overloads are provided for all integer types, which are treated as double.

Contents

[edit] Parameters

k - elliptic modulus or eccentricity (a floating-point or integer value)

[edit] Return value

If no errors occur, value of the complete elliptic integral of the first kind of k, that is std::ellint_1(k, π/2), is returned.

[edit] Error handling

Errors may be reported as specified in math_errhandling.

  • If the argument is NaN, NaN is returned and domain error is not reported.
  • If |k|>1, a domain error may occur.

[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.

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::comp_ellint_1(num) has the same effect as std::comp_ellint_1(static_cast<double>(num)).

[edit] Example

The period of a pendulum of length l, given acceleration due to gravity g, and initial angle θ equals 4⋅√l/g⋅K(sin(θ/2)), where K is std::comp_ellint_1.

Run this code
#include <cmath>
#include <iostream>
#include <numbers>
 
int main()
{
    constexpr double π{std::numbers::pi};
 
    std::cout << "K(0) ≈ " << std::comp_ellint_1(0) << '\n'
              << "π/2 ≈ " << π / 2 << '\n'
              << "K(0.5) ≈ " << std::comp_ellint_1(0.5) << '\n'
              << "F(0.5, π/2) ≈ " << std::ellint_1(0.5, π / 2) << '\n'
              << "The period of a pendulum length 1m at 10° initial angle ≈ "
              << 4 * std::sqrt(1 / 9.80665) * std::comp_ellint_1(std::sin(π / 18 / 2))
              << "s,\n" "whereas the linear approximation gives ≈ "
              << 2 * π * std::sqrt(1 / 9.80665) << '\n';
}

Output: 

K(0) ≈ 1.5708
π/2 ≈ 1.5708
K(0.5) ≈ 1.68575
F(0.5, π/2) ≈ 1.68575
The period of a pendulum length 1 m at 10° initial angle ≈ 2.01024s,
whereas the linear approximation gives ≈ 2.00641

[edit] See also

(C++17)(C++17)(C++17)
 (incomplete) elliptic integral of the first kind
(function) [edit]

[edit] External links

Weisstein, Eric W. "Complete Elliptic Integral of the First Kind." From MathWorld — A Wolfram Web Resource.
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