triqs::utility::legendre_generator

#include <triqs/utility/legendre.hpp>

Detailed Description

Recursive generation of Legendre polynomials \( P_l(x) \).

Legendre polynomials are defined on the interval \( [-1, 1] \). They form an orthogonal basis with respect to the inner product

\[ \langle P_k, P_l \rangle = \int_{-1}^1 P_k(x) P_l(x) dx = \frac{2}{2l + 1} \delta_{kl} \; . \]

The recurrence relation is given by

\[ (l + 1) P_{l+1}(x) = (2l + 1) x P_l(x) - l P_{l-1}(x) \; , \]

with \( P_0(x) = 1 \) and \( P_1(x) = x \).

See Wikipedia for more information.

Definition at line 87 of file legendre.hpp.

Public Member Functions
	legendre_generator (double x=0.0)
	Construct a Legendre polynomial generator at a given value \( x \).
double	next ()
	Increase the degree of the polynomial from \( l \) to \( l + 1 \) using \( (l + 1) P_{l+1}(x) = (2l + 1) x P_l(x) - l P_{l-1}(x) \).
void	reset (double x)
	Reset the generator to 0th order and with a new \( x \) value.

Constructor & Destructor Documentation

legendre_generator()

triqs::utility::legendre_generator::legendre_generator ( double x = 0.0 )

inline

Construct a Legendre polynomial generator at a given value \( x \).

Parameters

x	Value \( x \) at which to evaluate the Legendre polynomials.

Definition at line 93 of file legendre.hpp.

Member Function Documentation

next()

double triqs::utility::legendre_generator::next ( )

inline

Increase the degree of the polynomial from \( l \) to \( l + 1 \) using \( (l + 1) P_{l+1}(x) = (2l + 1) x P_l(x) - l P_{l-1}(x) \).

Returns

Value of the l^th order Legendre polynomial evaluated at \( x \).

Definition at line 101 of file legendre.hpp.

reset()

void triqs::utility::legendre_generator::reset ( double x )

inline

Reset the generator to 0^th order and with a new \( x \) value.

Parameters

x	Value \( x \) at which to evaluate the Legendre polynomials.

Definition at line 117 of file legendre.hpp.

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The documentation for this class was generated from the following file:

• triqs/utility/legendre.hpp