std::assoc_legendre, std::assoc_legendref, std::assoc_legendrel
Defined in header <cmath>
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(1) | ||
float assoc_legendre ( unsigned int n, unsigned int m, float x ); double assoc_legendre ( unsigned int n, unsigned int m, double x ); |
(since C++17) (until C++23) |
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/* floating-point-type */ assoc_legendre( unsigned int n, unsigned int m, /* floating-point-type */ x ); |
(since C++23) | |
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) |
Defined in header <cmath>
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template< class Integer > double assoc_legendre ( unsigned int n, unsigned int m, Integer x ); |
(A) | (since C++17) |
std::assoc_legendre
for all cv-unqualified floating-point types as the type of the parameter x.(since C++23)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 Legendre polynomial Pmn of x, that is (1-x2
)m/2
dm |
dxm |
Note that the Condon-Shortley phase term (-1)m
is omitted from this definition.
[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| > 1, a domain error may occur
- If
n
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 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 - x2 )1/2 | ||
assoc_legendre(2, 0, x) |
- 1) | ||
assoc_legendre(2, 1, x) | 3x(1 - x2 )1/2 | ||
assoc_legendre(2, 2, x) | 3(1 - x2 ) |
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_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
(C++17)(C++17)(C++17) |
Legendre polynomials (function) |
[edit] External links
Weisstein, Eric W. "Associated Legendre Polynomial." From MathWorld — A Wolfram Web Resource. |