std::legendre, std::legendref and std::legendrel functions in C++17

The legendre , legendref and legendrel are built functions in C++ STL that are used to compute the value of unassociated polynomials of degree n and argument x. Value of order-n unassociated Legendre Polynomial of x is given by :

$\ P_{n}(x)= \frac{1}{2^{n}n!}\frac{d^{n}}{dx^{n}}(x^{2}-1)^{n}$

The first few Legendre polynomials are

$Legendre(0,x)= 1 \] \[ Legendre(1,x)= x \] \[ Legendre(2,x)= \frac{1}{2} ( 3x^{2}-1 ) \] \[ Legendre(3,x)= \frac{1}{2} ( 5x^{3}-3x )$

Syntax:

double legendre( unsigned int n, double x )
             or 
double legendre( unsigned int n, float x )
             or 
double legendre( unsigned int n, long double x )
             or
float legendref( unsigned int n, float x )
             or
long double legendrel( unsigned int n, long double x )

Parameters: The function accepts two mandatory parameters which are described below:

n: it specifies the degree of the polynomial.
x: it specifies the argument which denotes a value of a floating-point or integral type

Return Value: The function returns the value of order-n unassociated Legendre Polynomial for argument x. The return type depends on the parameters passed.

Note: The function runs in and above C++ 17(7.1).

Below program illustrates the above mentioned functions:

// C++ program to illustrate the above  
// mentioned three functions 
#include <cmath>  
#include <iostream> 
using namespace std; 
int main() 
{ 
    // int and double as parameter  
    cout << "legendre(2,0.3)= " << legendre(2,0.3); 
      
    //  x as double type parameter 
    cout << "\nlegendre(3,(double)0.4)=" << legendre(3,(double)0.4); 
      
     // x as float  
     cout << "\nlegendre(3,(float)0.4)= " << legendre(3,(float)0.4); 
  
    //  legendref  
    cout << "\nlegendref(3, 0.45)= " << legendref(3, 0.45); 
  
    // legendrel 
    cout << "\nlegendrel(7, 0.50)= " << legendrel(7, 0.50); 
      
    return 0; 
} 

Errors and Exceptions: The function throws an error on three cases which is described below:

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

Below programs illustrate the above errors:

Program 1:

// C++ program to illustrate the above  
// mentioned three functions 
// domain error 
#include <cmath>  
#include <iostream> 
using namespace std; 
int main() 
{ 
    // int and double as parameter  
    cout << "legendre(129, 2)= " << legendre(129, 2); 
          
    return 0; 
} 

Output:

terminate called after throwing an instance of 'std::domain_error'
  what():  Argument out of range in __poly_legendre_p.
legendre(129, 2)=

Program 2:

// C++ program to illustrate the above  
// mentioned three functions 
// when x is NaN 
#include <cmath>  
#include <iostream> 
using namespace std; 
int main() 
{ 
    // int and double as parameter  
    cout << "legendre(129, NaN)= " << legendre(129, sqrt(-2)); 
      
    return 0; 
} 

Output:

legendre(129, NaN)= nan

Program 3:

// C++ program to illustrate the above  
// mentioned three functions 
// domain error 
#include <cmath>  
#include <iostream> 
using namespace std; 
int main() 
{ 
    // int and double as parameter  
    cout << "legendre(129, 2)= " << legendre(129, 2); 
      
    return 0; 
} 

Output:

terminate called after throwing an instance of 'std::domain_error'
  what():  Argument out of range in __poly_legendre_p.

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