Difference between revisions of "cpp/numeric/special functions/laguerre"

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Revision as of 18:46, 22 March 2023

Defined in header `<cmath>`
	(1)
float laguerre ( unsigned int n, float x ); double laguerre ( unsigned int n, double x ); long double laguerre ( unsigned int n, long double x );			(since C++17) (until C++23)
/* floating-point-type / laguerre( unsigned int n, / floating-point-type */ x );			(since C++23)
float laguerref( unsigned int n, float x );		(2)	(since C++17)
long double laguerrel( unsigned int n, long double x );		(3)	(since C++17)
Additional overloads
Defined in header `<cmath>`
template< class Integer > double laguerre ( unsigned int n, Integer x );		(A)	(since C++17)

1-3) Computes the non-associated Laguerre polynomials of the degree n and argument x. The library provides overloads of std::laguerre 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.

1 Parameters
2 Return value
3 Error handling
4 Notes
5 Example
6 See also
7 External links

Parameters

n	-	the degree of the polynomial, an unsigned integer value
x	-	the argument, a floating-point or integer value

Return value

If no errors occur, value of the nonassociated Laguerre polynomial of x, that is

e x n! d n dx n (x n e -x)

, 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
If x is negative, a domain error may occur
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__ 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 Laguerre polynomials are the polynomial solutions of the equation .

The first few are:

laguerre(0, x) = 1
laguerre(1, x) = $-x + 1$
laguerre(2, x) = $12 [x 2 -4x+2]$
laguerre(3, x) = $16 [-x 3 -9x 2 -18x+6]$

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::laguerre(int_num, num) has the same effect as std::laguerre(int_num, static_cast<double>(num)).

Example

Run this code

#include <cmath>
#include <iostream>
 
double L1(double x)
{
    return -x + 1;
}
 
double L2(double x)
{
    return 0.5 * (x * x - 4 * x + 2);
}
 
int main()
{
    // spot-checks
    std::cout << std::laguerre(1, 0.5) << '=' << L1(0.5) << '\n'
              << std::laguerre(2, 0.5) << '=' << L2(0.5) << '\n'
              << std::laguerre(3, 0.0) << '=' << 1.0 << '\n';
}

Output:

0.5=0.5
0.125=0.125
1=1

External links

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

Retrieved from "https://en.cppreference.com/mwiki/index.php?title=cpp/numeric/special_functions/laguerre&oldid=149558"

@@ Line 3: / Line 3: @@
 {{dcl begin}}
 {{dcl header|cmath}}
-{{dcl|num=1|since=c++17|
+{{dcl |num=1|=c++17|
-double      laguerre ( unsigned int n, double x );
 float       laguerre ( unsigned int n, float x );
 long double laguerre ( unsigned int n, long double x );
 float       laguerref( unsigned int n, float x );
 long double laguerrel( unsigned int n, long double x );
 }}
-{{dcl|num=2|since=c++17|
-double      laguerre ( unsigned int n, IntegralType x );
+{{dcl|num=|since=c++17|
+double      laguerre ( unsigned int n, x );
 }}
 {{dcl end}}
-@1@ Computes the non-associated {{enwiki|Laguerre polynomials}} of the degree {{tt|n}} and argument {{tt|x}}
+@1@ Computes the non-associated {{enwiki|Laguerre polynomials}} of the degree {{|n}} and argument {{|x}}overloads of |type {{|}}
-@2@ A set of overloads or a function template accepting an argument of any [[cpp/types/is_integral|integral type]]. Equivalent to {{v|1}} after casting the argument to {{c|double}}.
+{{c|double}}.
 ===Parameters===
 {{par begin}}
-{{par|n|the degree of the polynomial, a value of unsigned integer type}}
+{{par|n|the degree of the polynomial, unsigned integer }}
-{{par|x|the argument, a value of a floating-point or integral type}}
+{{par|x|the argument, a floating-point or }}
 {{par end}}
 ===Return value===
-If no errors occur, value of the nonassociated Laguerre polynomial of {{tt|x}}, that is {{math|{{mfrac|{{mexp|x}}|n!}}{{mfrac|d{{su|p=n}}|dx{{su|p=n}}}}(x{{su|p=n}}{{mexp|-x}})}}, is returned.
+If no errors occur, value of the nonassociated Laguerre polynomial of {{|x}}, that is {{math|{{mfrac|{{mexp|x}}|n!}}{{mfrac|d{{su|p=n}}|dx{{su|p=n}}}}(x{{su|p=n}}{{mexp|-x}})}}, is returned.
 ===Error handling===
 Errors may be reported as specified in {{lc|math_errhandling}}
 * If the argument is NaN, NaN is returned and domain error is not reported
-* If {{tt|x}} is negative, a domain error may occur
+* If {{|x}} is negative, a domain error may occur
-* If {{tt|n}} is greater or equal than 128, the behavior is implementation-defined
+* If {{|n}} is greater or equal than 128, the behavior is implementation-defined
 ===Notes===
-{{cpp/numeric/special_functions/older_impl_note}}
+{{cpp/numeric//}}
 An implementation of this function is also [https://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/sf_poly/laguerre.html available in boost.math]
 The Laguerre polynomials are the polynomial solutions of the equation {{math|xy{{su|p=,,}}+(1-x)y{{su|p=,}}+ny = 0}}
 The first few are:
-* {{tt|laguerre(0, x)}} {{=}} 1
+* {{tt|laguerre(0, x)}} = 1
-* {{tt|laguerre(1, x)}} {{=}} {{math|-x + 1}}
+* {{tt|laguerre(1, x)}} = {{math|-x + 1}}
-* {{tt|laguerre(2, x)}} {{=}} {{math|{{mfrac|1|2}}[x{{su|p=2}}-4x+2]}}
+* {{tt|laguerre(2, x)}} = {{math|{{mfrac|1|2}}[x{{su|p=2}}-4x+2]}}
-* {{tt|laguerre(3, x)}} {{=}} {{math|{{mfrac|1|6}}[-x{{su|p=3}}-9x{{su|p=2}}-18x+6]}}
+* {{tt|laguerre(3, x)}} = {{math|{{mfrac|1|6}}[-x{{su|p=3}}-9x{{su|p=2}}-18x+6]
+}}
 ===Example===
-{{example|code=
+{{example
+|code=
 #include <cmath>
 #include <iostream>
-double L1(double x) { return -x + 1; }
+double L1(double x)
-double L2(double x) { return 0.5 * (x * x - 4 * x + 2); }
+{
+return -x + 1;
+}
+double L2(double x)
+{
+return 0.5 * (x * x - 4 * x + 2);
+}
 int main()

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Difference between revisions of "cpp/numeric/special functions/laguerre"

Revision as of 18:46, 22 March 2023

Contents

Parameters

Return value

Error handling

Notes

Example

See also

External links

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