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

From cppreference.com
m (Wording fix.)
m (→‎Notes: list → table)
 
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The first few are:
 
The first few are:
−
* {{tt|1=assoc_laguerre(0, m, x)}} = 1
+
{| class="wikitable" style="font-size:95%; text-align:center;"
−
* {{tt|1=assoc_laguerre(1, m, x)}} = {{math|-x + m + 1}}
+
|-
−
* {{tt|1=assoc_laguerre(2, m, x)}} = {{math|{{mfrac|1|2}}[x{{su|p=2}}-2(m+2)x+(m+1)(m+2)]}}
+
! Function
−
* {{tt|1=assoc_laguerre(3, m, x)}} = {{math|{{mfrac|1|6}}[-x{{su|p=3}}-3(m+3)x{{su|p=2}}-3(m+2)(m+3)x+(m+1)(m+2)(m+3)]}}
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! Polynomial
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|- style="height:45px;"
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| {{nbsp|4}}{{co|1=assoc_laguerre(0, m, x)}}{{nbsp|4}} || 1
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|- style="height:45px;"
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| {{co|1=assoc_laguerre(1, m, x)}} || {{math|-x + m + 1}}
 +
|- style="height:45px;"
 +
| {{co|1=assoc_laguerre(2, m, x)}} || {{math|{{mfrac|1|2}}[x{{su|p=2}} - 2(m + 2)x + (m + 1)(m + 2)]}}
 +
|- style="height:45px;"
 +
| {{co|1=assoc_laguerre(3, m, x)}} || {{nbsp|4}}{{math|{{mfrac|1|6}}[-x{{su|p=3}} - 3(m + 3)x{{su|p=2}} - 3(m + 2)(m + 3)x + (m + 1)(m + 2)(m + 3)]}}{{nbsp|4}}
 +
|}
  
 
{{cpp/numeric/special functions/additional integer overload note|assoc_laguerre}}
 
{{cpp/numeric/special functions/additional integer overload note|assoc_laguerre}}

Latest revision as of 09:38, 24 March 2023

 
C++
 
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Mathematical special functions
 
Defined in header <cmath>
(1)
float       assoc_laguerre ( unsigned int n, unsigned int m, float x );

double      assoc_laguerre ( unsigned int n, unsigned int m, double x );

long double assoc_laguerre ( unsigned int n, unsigned int m, long double x );
 (since C++17)
(until C++23)
/* floating-point-type */ assoc_laguerre( unsigned int n, unsigned int m,
                                          /* floating-point-type */ x );
 (since C++23)
float       assoc_laguerref( unsigned int n, unsigned int m, float x );
 (2) (since C++17)
long double assoc_laguerrel( unsigned int n, unsigned int m, long double x );
 (3) (since C++17)
Defined in header <cmath>
template< class Integer >
double      assoc_laguerre ( unsigned int n, unsigned int m, Integer x );
 (A) (since C++17)
1-3) Computes the associated Laguerre polynomials of the degree n, order m, and argument x. The library provides overloads of std::assoc_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
m - the order 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 associated Laguerre polynomial of x, that is (-1)m
dm
dxm
Ln+m(x), is returned (where Ln+m(x) is the unassociated Laguerre polynomial, std::laguerre(n + m, x)).

[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 or m is greater or equal to 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 associated Laguerre polynomials are the polynomial solutions of the equation xy,,+(m+1-x)y,+ny = 0.

The first few are:

Function Polynomial
    assoc_laguerre(0, m, x)     1
assoc_laguerre(1, m, x) -x + m + 1
assoc_laguerre(2, m, x)
1
2
[x2 - 2(m + 2)x + (m + 1)(m + 2)]
assoc_laguerre(3, m, x)     
1
6
[-x3 - 3(m + 3)x2 - 3(m + 2)(m + 3)x + (m + 1)(m + 2)(m + 3)]    

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::assoc_laguerre(int_num1, int_num2, num) has the same effect as std::assoc_laguerre(int_num1, int_num2, static_cast<double>(num)).

[edit] Example

Run this code
#include <cmath>
#include <iostream>
 
double L1(unsigned m, double x)
{
    return -x + m + 1;
}
 
double L2(unsigned m, double x)
{
    return 0.5 * (x * x - 2 * (m + 2) * x + (m + 1) * (m + 2));
}
 
int main()
{
    // spot-checks
    std::cout << std::assoc_laguerre(1, 10, 0.5) << '=' << L1(10, 0.5) << '\n'
              << std::assoc_laguerre(2, 10, 0.5) << '=' << L2(10, 0.5) << '\n';
}

Output: 

10.5=10.5
60.125=60.125

[edit] See also

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

[edit] External links

Weisstein, Eric W. "Associated Laguerre Polynomial." From MathWorld — A Wolfram Web Resource.
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