std::legendre, std::legendref, std::
Defined in header <cmath>
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| (1) | ||
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float legendre (
unsigned
int n, float x )
;
double legendre ( unsigned int n, double x ); |
(since C++17) (until C++23) |
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/* floating-point-type */ legendre(
unsigned
int n,
/* floating-point-type */ x ) ; |
(since C++23) | |
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float legendref(
unsigned
int n, float x )
;
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(2) | (since C++17) |
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long
double legendrel(
unsigned
int n, long
double x )
;
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(3) | (since C++17) |
Defined in header <cmath>
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template
<
class Integer >
double legendre ( unsigned int n, Integer x ) ; |
(A) | (since C++17) |
std::legendre for all cv-unqualified floating-point types as the type of the parameter x.
(since C++23)
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 |
| 2n n! |
| dn |
| dxn |
-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__
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) |
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| legendre(3, x) |
- 3x) |
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| legendre(4, x) |
- 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) )
Example
#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
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(C++17)(C++17)(C++17)
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Laguerre polynomials (function) |
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(C++17)(C++17)(C++17)
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Hermite polynomials (function) |
External links
| Weisstein, Eric W. "Legendre Polynomial." From MathWorld — A Wolfram Web Resource. |