double assoc_legendre ( unsigned int n, unsigned int m, double x );
/* floating-point-type */ assoc_legendre( unsigned int n, unsigned int m,
/* floating-point-type */ x );
float assoc_legendref( unsigned int n, unsigned int m, float x );
(2) (since C++17)long double assoc_legendrel( unsigned int n, unsigned int m, long double x );
(3) (since C++17)template< class Integer >
double assoc_legendre ( unsigned int n, unsigned int m, Integer x );
Computes the
Associated Legendre polynomialsof the degree
n, order
m, and argument
x.
The library provides overloads ofstd::assoc_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.
[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 valueIf no errors occur, value of the associated Legendre polynomial
\(\mathsf{P}_n^m\)Pmof
x, that is
\((1 - x^2) ^ {m/2} \: \frac{ \mathsf{d} ^ m}{ \mathsf{d}x ^ m} \, \mathsf{P}_n(x)\)(1-x2, is returned (where
\(\mathsf{P}_n(x)\)Pn(x)is the unassociated Legendre polynomial,
std::legendre(n, x)).
Note that the Condon-Shortley phase term \((-1)^m\)(-1)m
is omitted from this definition.
Errors may be reported as specified in math_errhandling
n
is greater or equal to 128, the behavior is implementation-definedImplementations 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 as boost::math::legendre_p
, except that the boost.math definition includes the Condon-Shortley phase term.
The first few associated Legendre polynomials are:
Function Polynomial assoc_legendre(0, 0, x) 1 assoc_legendre(1, 0, x) x assoc_legendre(1, 1, x) (1 - x2The 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_legendre(int_num1, int_num2, num) has the same effect as std::assoc_legendre(int_num1, int_num2, static_cast<double>(num)).
[edit] Example#include <cmath> #include <iostream> double P20(double x) { return 0.5 * (3 * x * x - 1); } double P21(double x) { return 3.0 * x * std::sqrt(1 - x * x); } double P22(double x) { return 3 * (1 - x * x); } int main() { // spot-checks std::cout << std::assoc_legendre(2, 0, 0.5) << '=' << P20(0.5) << '\n' << std::assoc_legendre(2, 1, 0.5) << '=' << P21(0.5) << '\n' << std::assoc_legendre(2, 2, 0.5) << '=' << P22(0.5) << '\n'; }
Output:
-0.125=-0.125 1.29904=1.29904 2.25=2.25[edit] See also [edit] External links
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