Plm Class Reference

Class that computes and stores the Associated Legendre Functions (ALFs) and its derivatives at a given co-latitude. More...

#include <Plm.hpp>

Public Member Functions
	Plm ()
	Plm (int l_max, double theta, bool derivatives=false, bool second_derivatives=false)
	Plm (const Plm &other)
Plm &	operator= (const Plm &other)
double	get_Plm_bar (int l, int m)
	Getter for fully-normalized ALF.
double	get_Plm (int l, int m)
	Getter for unnormalized ALF.
double	get_dPlm_bar (int l, int m)
	Getter for fully-normalized ALF derivative.
double	get_dPlm (int l, int m)
	Getter for unnormalized ALF derivative.
double	get_ddPlm_bar (int l, int m)
	Getter for fully-normalized ALF 2nd order derivative.
double	get_ddPlm (int l, int m)
	Getter for unnormalized ALF derivative.
double	get_theta () const
	Getter for associated colatitude.

Detailed Description

Class that computes and stores the Associated Legendre Functions (ALFs) and its derivatives at a given co-latitude.

This class computes the Associated Legendre Functions (ALFs) up to a certain degree and order employing a recursive standard forward column method, so-called Fixed-Order-Increase-Degree (FOID), as described by Holmes and Featherstone, 2002 (sec. 2.1).

First the following constants are defined:

\[ a_{lm} = \sqrt{\frac{(2l-1)(2l+1)}{(l-m)(l+m)}} \quad\quad\quad b_{lm} = \sqrt{\frac{(2l+1)(l+m-1)(l-m+1)}{(l-m)(l+m)(2l-3)}} \]

We define that \( t=\cos{\theta}, u=\sin{\theta}\) and the algorithm is initialised with \( \bar{P}_{0,0}(\theta)=1, \bar{P}_{1,1}(\theta)=\sqrt{3}u\). Then, sectorial recursive relationships are applied:

\[ \bar{P}_{mm} = u \sqrt{\frac{2m+1}{2m}} \bar{P}_{m-1,m-1}(\theta) \]

Next, we sequentially fix the order \( m\) and increase the degree \( l\) applying the following recursive relationships:

\[ \bar{P}_{lm}(\theta) = a_{lm} t \bar{P}_{lm}(\theta) - b_{lm} \bar{P}_{l-2,m} \]

Derivatives with respect to the co-latitude are of interest for multiple applications and can be computed from the ALFs with the following expressions:

\[ f_{lm} = \sqrt{\frac{(l^2-m^2)(2l+1)}{2l-1}} \]

\[ \frac{d \bar{P}_{lm}(\theta)}{d\theta} = \frac{1}{u} (lt \bar{P}_{lm}(\theta)-f_{lm} \bar{P}_{l-1,m}(\theta)) \]

Constructor & Destructor Documentation

Plm() [1/2]

Plm::Plm

(

)

Default constructor

Plm() [2/2]

Plm::Plm	(	int	l_max,
		double	theta,
		bool	derivatives = false,
		bool	second_derivatives = false
	)

Class constructor

Parameters

l_max	Maximum degree to which the ALFs (or its derivatives) are computed. Note that to compute the derivatives up to a degree L, it is necessary to compute the ALFs up to degree L+2. This is automatically handled based on derivative flags.
theta	Co-latitude at which the ALFs (and its derivatives) are evaluated
derivatives	Flag to indicate whether derivatives are computed or not
second_derivatives	Flag to indicate whether 2nd order derivatives are computed or not

Member Function Documentation

get_ddPlm()

double Plm::get_ddPlm	(	int	l,
		int	m
	)

Getter for unnormalized ALF derivative.

Parameters

l	degree
m	order

get_ddPlm_bar()

double Plm::get_ddPlm_bar	(	int	l,
		int	m
	)

Getter for fully-normalized ALF 2nd order derivative.

Parameters

l	degree
m	order

get_dPlm()

double Plm::get_dPlm	(	int	l,
		int	m
	)

Getter for unnormalized ALF derivative.

Parameters

l	degree
m	order

get_dPlm_bar()

double Plm::get_dPlm_bar	(	int	l,
		int	m
	)

Getter for fully-normalized ALF derivative.

Parameters

l	degree
m	order

get_Plm()

double Plm::get_Plm	(	int	l,
		int	m
	)

Getter for unnormalized ALF.

Parameters

l	degree
m	order

get_Plm_bar()

double Plm::get_Plm_bar	(	int	l,
		int	m
	)

Getter for fully-normalized ALF.

Parameters

l	degree
m	order

---

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

• Plm.hpp
• Plm.cpp