std:: legendre, std:: legendref, std:: legendrel

From cppreference.com
C++
Numerics library
Common mathematical functions
Mathematical special functions (C++17)
Mathematical constants (C++20)
Basic linear algebra algorithms (C++26)
Data-parallel types ( simd ) (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)
Mathematical special functions
Defined in header <cmath>
(1)
float legendre ( unsigned int n, float x ) ;

double legendre ( unsigned int n, double x ) ;

long double legendre ( unsigned int n, long double x ) ;
(since C++17)
(until C++23)
/* floating-point-type */ legendre ( unsigned int n,
/* floating-point-type */ x ) ;
(since C++23)
float legendref ( unsigned int n, float x ) ;
(2) (since C++17)
long double legendrel ( unsigned int n, long double x ) ;
(3) (since C++17)
Defined in header <cmath>
template < class Integer >
double legendre ( unsigned int n, Integer x ) ;
(A) (since C++17)
1-3) Computes the unassociated Legendre polynomials of the degree n and argument x . The library provides overloads of std::legendre 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 .

Parameters

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

Return value

If no errors occur, value of the order- n unassociated Legendre polynomial of x , that is
1
2 n n!
d n
dx n
(x 2 -1) n , is returned.

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
  • The function is not required to be defined for |x|>1
  • If n is greater or equal than 128, the behavior is implementation-defined

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 first few Legendre polynomials are:

Function Polynomial
legendre ( 0 , x ) 1
legendre ( 1 , x ) x
legendre ( 2 , x )
1
2
(3x 2 - 1)
legendre ( 3 , x )
1
2
(5x 3 - 3x)
legendre ( 4 , x )
1
8
(35x 4 - 30x 2 + 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 :: legendre ( int_num, num ) has the same effect as std :: legendre ( int_num, static_cast < double > ( num ) ) .

Example

Run this code 
#include <cmath>
#include <iostream>
 
double P3(double x)
{
    return 0.5 * (5 * std::pow(x, 3) - 3 * x);
}
 
double P4(double x)
{
    return 0.125 * (35 * std::pow(x, 4) - 30 * x * x + 3);
}
 
int main()
{
    // spot-checks
    std::cout << std::legendre(3, 0.25) << '=' << P3(0.25) << '\n'
              << std::legendre(4, 0.25) << '=' << P4(0.25) << '\n';
}

Output: 

-0.335938=-0.335938
0.157715=0.157715

See also

(C++17) (C++17) (C++17)
Laguerre polynomials
(function)
(C++17) (C++17) (C++17)
Hermite polynomials
(function)

External links

Weisstein, Eric W. "Legendre Polynomial." From MathWorld — A Wolfram Web Resource.
Retrieved from " https://en.cppreference.com/mwiki/index.php?title=cpp/numeric/special_functions/legendre&oldid=149619 "