std::legendre, std::legendref, std::legendrel - cppreference.com (original) (raw)

Defined in header
(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)
Additional overloads
Defined in header
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.

Contents

[edit] Parameters

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

[edit] Return value

If no errors occur, value of the order-n unassociated Legendre polynomial of x, that is \(\mathsf{P}_n(x) = \frac{1}{2^n n!} \frac{\mathsf{d}^n}{\mathsf{d}x^n} (x^2-1)^n \)(x2
-1)n
, is returned.

[edit] Error handling

Errors may be reported as specified in math_errhandling.

[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 first few Legendre polynomials are:

Function Polynomial
legendre(0, x) 1
legendre(1, x) x
legendre(2, x) 12(3x2 - 1)
legendre(3, x) 12(5x3 - 3x)
legendre(4, x) 18(35x4 - 30x2 + 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)).

[edit] Example

#include #include   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

[edit] See also