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Showing content from https://en.cppreference.com/w/cpp/language/../algorithm/../numeric/special_functions/legendre.html below:

std::legendre, std::legendref, std::legendrel - cppreference.com

(1) float legendre ( unsigned int n, float x ); double legendre ( unsigned int n, double x ); long double legendre ( unsigned int n, long double x ); (since C++17)
(until C++23)

/* floating-point-type */ legendre( unsigned int n, /* floating-point-type */ x );

(since C++23)

float legendref( unsigned int n, float x );

(2) (since C++17)

long double legendrel( unsigned int n, long double x );

(3) (since C++17)

template< class Integer > double legendre ( unsigned int n, Integer x );

(A) (since C++17) 1-3)

Computes the unassociated

Legendre polynomials

of the degree

n

and argument

x

The library provides overloads of std::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 x - the argument, a floating-point or integer value [edit] Return value

If no errors occur, value of the order-

n

unassociated Legendre polynomial of

x

, that is

\(\mathsf{P}_n(x) = \frac{1}{2^n n!} \frac{\mathsf{d}^n}{\mathsf{d}x^n} (x^2-1)^n \)(x2 -1)n

, is returned.

[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
The function is not required to be defined for |x|>1
If n is greater or equal than 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.

The first few Legendre polynomials are:

Function Polynomial 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)

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)).

[edit] 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

[edit] See also [edit] External links

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