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

From cppreference.com
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{{cpp/title|laguerre|laguerref|laguerrel}}
 
{{cpp/title|laguerre|laguerref|laguerrel}}
−
{{cpp/numeric/special_math/navbar}}
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{{cpp/numeric/special_functions/navbar}}
 
{{dcl begin}}
 
{{dcl begin}}
−
{{dcl |num=1|since=c++17|
+
{{dcl header|cmath}}
−
double      laguerre( unsigned int n, double x );
+
{{dcl rev multi|num=1|since1=c++17|dcl1=
−
double      laguerre( unsigned int n, float x );
+
float      laguerre ( unsigned int n, float x );
−
double     laguerre( unsigned int n, long double x );
+
double      laguerre ( unsigned int n, double x );
 +
long double laguerre ( unsigned int n, long double x );
 +
|since2=c++23|dcl2=
 +
/* floating-point-type */ laguerre( unsigned int n,
 +
                                    /* floating-point-type */ x );
 +
}}
 +
{{dcl|num=2|since=c++17|
 
float      laguerref( unsigned int n, float x );
 
float      laguerref( unsigned int n, float x );
 +
}}
 +
{{dcl|num=3|since=c++17|
 
long double laguerrel( unsigned int n, long double x );
 
long double laguerrel( unsigned int n, long double x );
 
}}
 
}}
−
{{dcl |num=2|since=c++17|
+
{{dcl h|[[#Notes|Additional overloads]]}}
−
double      laguerre( unsigned int n, Integral x );
+
{{dcl header|cmath}}
 +
{{dcl|num=A|since=c++17|
 +
template< class Integer >
 +
double      laguerre ( unsigned int n, Integer x );
 
}}
 
}}
 
{{dcl end}}
 
{{dcl end}}
  
−
@1@ Computes the non-associated [[enwiki:Laguerre_polynomials|Laguerre polynomials]] of the degree {{tt|n}} and argument {{tt|x}}
+
@1-3@ Computes the non-associated {{enwiki|Laguerre polynomials}} of the degree {{c|n}} and argument {{c|x}}.{{rev inl|since=c++23| The library provides overloads of {{tt|std::laguerre}} for all cv-unqualified floating-point types as the type of the parameter {{c|x}}.}}
−
@2@ 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 | n | the degree of the polymonial, a value of unsigned integer type}}
+
{{par|n|the degree of the polynomial, an unsigned integer value}}
−
{{par | x | the argument, a value of a floating-point or integral type}}
+
{{par|x|the argument, a floating-point or integer value}}
 
{{par end}}
 
{{par end}}
  
 
===Return value===
 
===Return value===
−
If no errors occur, value of the nonassociated Laguerre polynomial of {{tt|x}}, that is {{math|{{mfrac|{{mexp|x}}|n!}}{{mfrac|d{{su|p=n}}|dx{{su|p=n}}}}(x{{su|p=n}}{{mexp|-x}})}}, is returned.
+
If no errors occur, value of the nonassociated Laguerre polynomial of {{c|x}}, that is {{math|{{mfrac|{{mexp|x}}|n!}}{{mfrac|d{{su|p=n}}|dx{{su|p=n}}}}(x{{su|p=n}}{{mexp|-x}})}}, 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 {{tt|x}} is negative, a domain error may occur
+
* If {{c|x}} is negative, a domain error may occur
−
* If {{tt|n}} is greater or equal than 128, the behavior is implementation-defined
+
* If {{c|n}} is greater or equal than 128, the behavior is implementation-defined
  
 
===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/release/libs/math/doc/html/math_toolkit/sf_poly/laguerre.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_poly/laguerre.html available in boost.math].
  
−
The Laguerre polynomials are the polynomial solutions of the equation {{math|xy{{su|p=,,}}+(1-x)y{{su|p=,}}+ny {{=}} 0}}
+
The Laguerre polynomials are the polynomial solutions of the equation {{math|xy{{su|p=,,}}+(1-x)y{{su|p=,}}+ny = 0}}.
  
 
The first few are:
 
The first few are:
−
* laguerre(0, x) {{=}} 1
+
{| class="wikitable" style="font-size:95%; text-align:center;"
−
* laguerre(1, x) {{=}} {{math|-x + 1}}
+
|-
−
* laguerre(2, x) {{=}} {{math|{{mfrac|1|2}}[x{{su|p=2}}-4x+2]}}
+
! Function
−
* laguerre(3, x) {{=}} {{math|{{mfrac|1|6}}[-x{{su|p=3}}-9x{{su|p=2}}-18x+6]}}
+
! Polynomial
 +
|- style="height:45px;"
 +
| {{nbsp|4}}{{co|laguerre(0, x)}}{{nbsp|4}} || 1
 +
|- style="height:45px;"
 +
| {{co|laguerre(1, x)}} || {{math|-x + 1}}
 +
|- style="height:45px;"
 +
| {{co|laguerre(2, x)}} || {{math|{{mfrac|1|2}}(x{{su|p=2}} - 4x + 2)}}
 +
|- style="height:45px;"
 +
| {{co|laguerre(3, x)}} || {{nbsp|4}}{{math|{{mfrac|1|6}}(-x{{su|p=3}} - 9x{{su|p=2}} - 18x + 6)}}{{nbsp|4}}
 +
|}
 +
 
 +
{{cpp/numeric/special functions/additional integer overload note|laguerre}}
  
 
===Example===
 
===Example===
−
{{example|code=
+
{{example
−
#define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1
+
|code=
 
#include <cmath>
 
#include <cmath>
 
#include <iostream>
 
#include <iostream>
−
double L1(double x) { return -x + 1; }
+
 
−
double L2(double x) { return 0.5*(x*x-4*x+2); }
+
double L1(double x)
 +
{
 +
    return -x + 1;
 +
}
 +
 
 +
double L2(double x)
 +
{
 +
    return 0.5 * (x * x - 4 * x + 2);
 +
}
 +
 
 
int main()
 
int main()
 
{
 
{
 
     // spot-checks
 
     // spot-checks
 
     std::cout << std::laguerre(1, 0.5) << '=' << L1(0.5) << '\n'
 
     std::cout << std::laguerre(1, 0.5) << '=' << L1(0.5) << '\n'
−
               << std::laguerre(2, 0.5) << '=' << L2(0.5) << '\n';
+
               << std::laguerre(2, 0.5) << '=' << L2(0.5) << '\n'
 +
              << std::laguerre(3, 0.0) << '=' << 1.0 << '\n';
 
}
 
}
 
|output=
 
|output=
 
0.5=0.5
 
0.5=0.5
 
0.125=0.125
 
0.125=0.125
 +
1=1
 
}}
 
}}
  
 
===See also===
 
===See also===
 
{{dsc begin}}
 
{{dsc begin}}
−
{{dsc inc | cpp/numeric/special_math/dsc assoc_laguerre}}
+
{{dsc inc|cpp/numeric/special_functions/dsc assoc_laguerre}}
 
{{dsc end}}
 
{{dsc end}}
  
 
===External links===
 
===External links===
−
[http://mathworld.wolfram.com/LaguerrePolynomial.html Weisstein, Eric W. "Laguerre Polynomial."] From MathWorld--A Wolfram Web Resource.
+
{{eli|[https://mathworld.wolfram.com/LaguerrePolynomial.html Weisstein, Eric W. "Laguerre Polynomial."] From MathWorld — A Wolfram Web Resource.}}
  
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[[de:cpp/numeric/special_math/laguerre]]
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{{langlinks|de|es|fr|it|ja|pt|ru|zh}}
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[[es:cpp/numeric/special_math/laguerre]]
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[[fr:cpp/numeric/special_math/laguerre]]
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[[it:cpp/numeric/special_math/laguerre]]
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[[ja:cpp/numeric/special_math/laguerre]]
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[[pt:cpp/numeric/special_math/laguerre]]
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−
[[ru:cpp/numeric/special_math/laguerre]]
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−
[[zh:cpp/numeric/special_math/laguerre]]
+

Latest revision as of 09:33, 24 March 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
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Pseudo-random number generation
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Saturation arithmetic
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Bit operations
(C++20)    
(C++20)
(C++20)
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(C++20)
(C++20)
(C++20)
(C++23)
(C++20)
 
Mathematical special functions
 
Defined in header <cmath>
(1)
float       laguerre ( unsigned int n, float x );

double      laguerre ( unsigned int n, double x );

long double laguerre ( unsigned int n, long double x );
 (since C++17)
(until C++23)
/* floating-point-type */ laguerre( unsigned int n,
                                    /* floating-point-type */ x );
 (since C++23)
float       laguerref( unsigned int n, float x );
 (2) (since C++17)
long double laguerrel( unsigned int n, long double x );
 (3) (since C++17)
Defined in header <cmath>
template< class Integer >
double      laguerre ( unsigned int n, Integer x );
 (A) (since C++17)
1-3) Computes the non-associated Laguerre polynomials of the degree n and argument x. The library provides overloads of std::laguerre for all cv-unqualified floating-point types as the type of the parameter x.(since C++23)
A) Additional overloads are provided for all integer types, which are treated as double.

Contents

[edit] Parameters

n - the degree of the polynomial, an unsigned integer value
x - the argument, a floating-point or integer value

[edit] Return value

If no errors occur, value of the nonassociated Laguerre polynomial of x, that is
ex
n!
dn
dxn
(xne-x), 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 x is negative, a domain error may occur
  • If n is greater or equal than 128, the behavior is implementation-defined

[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 Laguerre polynomials are the polynomial solutions of the equation .

The first few are:

Function Polynomial
    laguerre(0, x)     1
laguerre(1, x) -x + 1
laguerre(2, x)
1
2
(x2 - 4x + 2)
laguerre(3, x)     
1
6
(-x3 - 9x2 - 18x + 6)    

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::laguerre(int_num, num) has the same effect as std::laguerre(int_num, static_cast<double>(num)).

[edit] Example

Run this code
#include <cmath>
#include <iostream>
 
double L1(double x)
{
    return -x + 1;
}
 
double L2(double x)
{
    return 0.5 * (x * x - 4 * x + 2);
}
 
int main()
{
    // spot-checks
    std::cout << std::laguerre(1, 0.5) << '=' << L1(0.5) << '\n'
              << std::laguerre(2, 0.5) << '=' << L2(0.5) << '\n'
              << std::laguerre(3, 0.0) << '=' << 1.0 << '\n';
}

Output: 

0.5=0.5
0.125=0.125
1=1

[edit] See also

(C++17)(C++17)(C++17)
 associated Laguerre polynomials
(function) [edit]

[edit] External links

Weisstein, Eric W. "Laguerre Polynomial." From MathWorld — A Wolfram Web Resource.
Retrieved from "https://en.cppreference.com/mwiki/index.php?title=cpp/numeric/special_functions/laguerre&oldid=149660"