table of contents
std::assoc_laguerre,std::assoc_laguerref,std::assoc_laguerrel(3) | C++ Standard Libary | std::assoc_laguerre,std::assoc_laguerref,std::assoc_laguerrel(3) |
NAME¶
std::assoc_laguerre,std::assoc_laguerref,std::assoc_laguerrel - std::assoc_laguerre,std::assoc_laguerref,std::assoc_laguerrel
Synopsis¶
Defined in header <cmath>
double assoc_laguerre( unsigned int n, unsigned int m, double x );
float assoc_laguerre( unsigned int n, unsigned int m, float x );
long double assoc_laguerre( unsigned int n, unsigned int m, long
double x ); (1) (since C++17)
float assoc_laguerref( unsigned int n, unsigned int m, float x );
long double assoc_laguerrel( unsigned int n, unsigned int m, long
double x );
double assoc_laguerre( unsigned int n, unsigned int m, (2) (since
C++17)
IntegralType x );
1) Computes the associated Laguerre polynomials of the degree n, order m, 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.
Parameters¶
n - the degree of the polymonial, a value of unsigned integer
type
m - the order of the polynomial, a value of unsigned integer type
x - the argument, a value of a floating-point or integral type
Return value¶
If no errors occur, value of the associated Laguerre polynomial
of x, that is
\((-1)^m \: \frac{ \mathsf{d} ^ m}{ \mathsf{d}x ^ m} \,
\mathsf{L}_{n+m}(x)\)(-1)m
dm
dxm
L
n+m(x), is returned (where \(\mathsf{L}_{n+m}(x)\)L
n+m(x) is the unassociated Laguerre polynomial, std::laguerre(n+m, x)).
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 is negative, a domain error may occur
* If n or m is greater or equal to 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__
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 associated Laguerre polynomials are the polynomial solutions of the
equation
\(x\ddot{y} + (m+1-x)\dot{y} + ny = 0\)xy,,
+(m+1-x)y,
+ny = 0
The first few are:
* assoc_laguerre(0, m, x) = 1
* assoc_laguerre(1, m, x) = -x + m + 1
* assoc_laguerre(2, m, x) =
1
2
[x2
-2(m+2)x+(m+1)(m+2)]
* assoc_laguerre(3, m, x) =
1
6
[-x3
-3(m+3)x2
-3(m+2)(m+3)x+(m+1)(m+2)(m+3)]
Example¶
// Run this code
#include <cmath>
#include <iostream>
double L1(unsigned m, double x) { return -x + m + 1; }
double L2(unsigned m, double x) { return 0.5*(x*x-2*(m+2)*x+(m+1)*(m+2)); }
int main()
{
// spot-checks
std::cout << std::assoc_laguerre(1, 10, 0.5) << '=' <<
L1(10, 0.5) << '\n'
<< std::assoc_laguerre(2, 10, 0.5) << '=' << L2(10, 0.5)
<< '\n';
}
Output:¶
10.5=10.5
60.125=60.125
See also¶
laguerre
laguerref
laguerrel Laguerre polynomials
(C++17) (function)
(C++17)
(C++17)
External links¶
Weisstein, Eric W. "Associated Laguerre Polynomial."
From MathWorld — A Wolfram
Web Resource.
2022.07.31 | http://cppreference.com |