std::legendre, std::legendref, std::legendrel

From cppreference.com
 
 
Technical specifications
Filesystem library (filesystem TS)
Library fundamentals (library fundamentals TS)
Library fundamentals 2 (library fundamentals TS v2)
Library fundamentals 3 (library fundamentals TS v3)
Extensions for parallelism (parallelism TS)
Extensions for parallelism 2 (parallelism TS v2)
Extensions for concurrency (concurrency TS)
Extensions for concurrency 2 (concurrency TS v2)
Concepts (concepts TS)
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Mathematical special functions (special functions TR)
 
 
double      legendre( unsigned int n, double x );

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

long double legendrel( unsigned int n, long double x );
(1)
double      legendre( unsigned int n, IntegralType x );
(2)
1) Computes the unassociated Legendre polynomials of the degree n and argument x.
2) A set of overloads or a function template accepting an argument of any integral type. Equivalent to (1) after casting the argument to double.

As all special functions, legendre is only guaranteed to be available in <cmath> 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.

Parameters

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

Return value

If no errors occur, value of the order-n unassociated Legendre polynomial of x, that is
1
2n
n!
dn
dxn
(x2
- 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 TR 29124 but support TR 19768, 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:

  • legendre(0, x) = 1.
  • legendre(1, x) = x.
  • legendre(2, x) =
    1
    2
    (3x2
    - 1)
    .
  • legendre(3, x) =
    1
    2
    (5x3
    - 3x)
    .
  • legendre(4, x) =
    1
    8
    (35x4
    - 30x2
    + 3)
    .

Example

(works as shown with gcc 6.0)

#define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1
#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

Laguerre polynomials
(function)
Hermite polynomials
(function)

External links

Weisstein, Eric W. "Legendre Polynomial." From MathWorld — A Wolfram Web Resource.