std::legend...d::legendrel(3)

🎀 NAME 🎀

okay so this is the manual page for std::legendre, std::legendref, and std::legendrel — the whole little trio, together. The header title is std::legend...d::legendrel(3), and the manual name folded in near the top is C++ Standard Libary (that spelling is straight from the source, bestie, I’m not fixing it 💅).

The NAME line itself is exactly:

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

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🌸 SYNOPSIS 🌸

Here’s the whole cast, no cap:

double      legendre( unsigned int n, double x );

double      legendre( unsigned int n, float x );
double      legendre( unsigned int n, long double x );  (1)
float       legendref( unsigned int n, float x );

long double legendrel( unsigned int n, long double x );
double      legendre( unsigned int n, IntegralType x ); (2)

1. (1) computes the unassociated Legendre polynomials of the degree n and argument x.
2. (2) is a set of overloads or a function template accepting an argument of any integral type. It’s equivalent to (1) after casting the argument to double.

Availability tea ☕: 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.

bestie, those macro names are long and dramatic, but they are not optional if you want the guarantee. copy them exactly. periodt.

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💖 PARAMETERS 💖

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

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✨ RETURN VALUE ✨

If no errors occur, you get the value of the order-n unassociated Legendre polynomial of x, that is:

1
2n
n!

dn
dxn

(x2
- 1)n

…and that is what gets returned. yes, the formula is giving very “stacked symbols on purpose.” I preserved it as-is so nobody comes for me.

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💀 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.

warning bestie: |x| > 1 is not required to be defined, and n >= 128 goes straight into implementation-defined mystery land. do not act surprised when it gets weird 💀

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🦋 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).

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🍓 EXAMPLE 🍓

(works as shown with gcc 6.0)

// Run this code

#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';
}

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🌈 OUTPUT 🌈

-0.335938=-0.335938
0.157715=0.157715

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👑 SEE ALSO 👑

• laguerre — Laguerre polynomials
• laguerref — (function)
• laguerrel
• hermite — Hermite polynomials
• hermitef — (function)
• hermitel

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💜 EXTERNAL LINKS 💜

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

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and that’s the whole page, bestie ✨ source/version and date from the footer: http://cppreference.com, dated 2024.06.10, with std::legend...d::legendrel(3) doing its little footer cameo. bye 💖🎀