This package is part of the SatelliteToolbox.jl ecosystem.

SatelliteToolboxLegendre.jl

This package contains function to compute the Legendre associated functions and its derivatives for the models in the SatelliteToolbox.jl ecosystem.

Installation

julia> using Pkg
julia> Pkg.add("SatelliteToolboxLegendre")

Usage

Legendre Associated Functions

This package exports two methods to compute the Legendre associated functions: legendre and legendre!.

---

legendre(N, ϕ::T, n_max::Integer, m_max::Integer = -1; ph_term::Bool = false) where T<:Number -> Matrix{float(T)}

Compute the associated Legendre function $P_{n,m}\left[\cos(\phi)\right]$. The maximum degree that will be computed is n_max and the maximum order is m_max. Notice that if m_max is higher than n_max or negative (default), it is set to n_max.

The parameter N selects the normalization. The following values are valid:

• Val(:full): Compute the fully normalized associated Legendre function.
• Val(:schmidt): Compute the Schmidt quasi-normalized associated Legendre function.
• Val(:unnormalized): Compute the unnormalized associated Legendre function.

This function has the following keywords:

• ph_term::Bool: If true, the Condon-Shortley phase term $(-1)^m$ will be included. (Default = false)

It returns a Matrix{float(T)} with the Legendre associated functions $P_{n,m}\left[\cos(\phi)\right]$.

julia> legendre(Val(:unnormalized), 0.45, 4)
5×5 Matrix{Float64}:
 1.0       0.0       0.0       0.0      0.0
 0.900447  0.434966  0.0       0.0      0.0
 0.716207  1.17499   0.567585  0.0      0.0
 0.474547  1.99259   2.5554    1.2344   0.0
 0.210627  2.61987   6.63455   7.78058  3.75845

julia> legendre(Val(:schmidt), 0.45, 4, 3; ph_term = true)
5×4 Matrix{Float64}:
 1.0        0.0       0.0        0.0
 0.900447  -0.434966  0.0        0.0
 0.716207  -0.678381  0.163848   0.0
 0.474547  -0.813473  0.329901  -0.0650586
 0.210627  -0.828476  0.49451   -0.154993

julia> legendre(Val(:full), 0.45, 4, 1)
5×2 Matrix{Float64}:
 1.0       0.0
 1.55962   0.753382
 1.60149   1.51691
 1.25553   2.15225
 0.631881  2.48543

---

legendre!(N, P::AbstractMatrix, ϕ::Number, n_max::Integer = -1, m_max::Integer = -1; kwargs...) -> Nothing

Compute the associated Legendre function $P_{n,m}\left[\cos(\phi)\right]$. The maximum degree and order that will be computed are given by the parameters n_max and m_max. If they are negative (default), the dimensions of matrix P will be used:

maximum degree -> number of rows - 1
maximum order  -> number of columns - 1

The result will be stored at matrix P. Hence, this function can be used to reduce allocations.

The parameter N selects the normalization. The following values are valid:

• Val(:full): Compute the fully normalized associated Legendre function.
• Val(:schmidt): Compute the Schmidt quasi-normalized associated Legendre function.
• Val(:unnormalized): Compute the unnormalized associated Legendre function.

This function has the following keywords:

• ph_term::Bool: If true, the Condon-Shortley phase term $(-1)^m$ will be included. (Default = false)

Derivative of the Legendre Associated Functions

This package exports two methods to compute the derivative of the Legendre associated functions: dlegendre and dlegendre!.

---

dlegendre(N, ϕ::T, n_max::Integer, m_max::Integer = -1; kwargs...) where T<:Number -> Matrix{float(T)}, Matrix{float(T)}

Compute the first-order derivative of the associated Legendre function $P_{n,m}\left[\cos(\phi)\right]$ with respect to $\phi$ [rad]:

$$\frac{\partial P_{n,m} \left[\cos(\phi)\right]}{\partial\phi}$$

The maximum degree that will be computed is n_max and the maximum order is m_max. Notice that if m_max is higher than n_max or negative (default), it is set to n_max.

The parameter N selects the normalization. The following values are valid:

• Val(:full): Compute the fully normalized associated Legendre function.
• Val(:schmidt): Compute the Schmidt quasi-normalized associated Legendre function.
• Val(:unnormalized): Compute the unnormalized associated Legendre function.

This function has the following keywords:

• ph_term::Bool: If true, the Condon-Shortley phase term $(-1)^m$ will be included. (Default = false)

It returns the following objects:

• Matrix{float(T)}: A matrix with the first-order derivative of the Legendre associated functions $P_{n,m}\left[\cos(\phi)\right]$.
• Matrix{float(T)}: A matrix with the Legendre associated functions $P_{n,m}\left[\cos(\phi)\right]$`.

julia> dP, P = dlegendre(Val(:unnormalized), 0.45, 4)

julia> dP
5×5 Matrix{Float64}:
  0.0        0.0        0.0       0.0      0.0
 -0.434966   0.900447   0.0       0.0      0.0
 -1.17499    1.86483    2.34998   0.0      0.0
 -1.99259    1.56958    9.34577   7.6662   0.0
 -2.61987   -1.21101   19.6885   44.5626  31.1223

julia> P
5×5 Matrix{Float64}:
 1.0       0.0       0.0       0.0      0.0
 0.900447  0.434966  0.0       0.0      0.0
 0.716207  1.17499   0.567585  0.0      0.0
 0.474547  1.99259   2.5554    1.2344   0.0
 0.210627  2.61987   6.63455   7.78058  3.75845

julia> dP, P = dlegendre(Val(:schmidt), 0.45, 4, 3; ph_term = true)

julia> dP
5×4 Matrix{Float64}:
  0.0        0.0       0.0        0.0
 -0.434966  -0.900447  0.0        0.0
 -1.17499   -1.07666   0.678381   0.0
 -1.99259   -0.640778  1.20653   -0.404044
 -2.61987    0.382954  1.4675    -0.887709

julia> P
5×5 Matrix{Float64}:
 1.0        0.0       0.0        0.0        0.0
 0.900447  -0.434966  0.0        0.0        0.0
 0.716207  -0.678381  0.163848   0.0        0.0
 0.474547  -0.813473  0.329901  -0.0650586  0.0
 0.210627  -0.828476  0.49451   -0.154993   0.0264706

julia> dP, P = dlegendre(Val(:full), 0.45, 4, 1)

julia> dP
5×2 Matrix{Float64}:
  0.0        0.0
 -0.753382   1.55962
 -2.62736    2.40749
 -5.27191    1.69534
 -7.85961   -1.14886

julia> P
5×3 Matrix{Float64}:
 1.0       0.0       0.0
 1.55962   0.753382  0.0
 1.60149   1.51691   0.366375
 1.25553   2.15225   0.872836
 0.631881  2.48543   1.48353

---

dlegendre!(dP::AbstractMatrix, ϕ::Number, P::AbstractMatrix, n_max::Integer = -1, m_max::Integer = -1; kwargs...) -> Nothing

Compute the first-order derivative of the associated Legendre function $P_{n,m}\left[\cos(\phi)\right]$ with respect to $\phi$ [rad]:

$$\frac{\partial P_{n,m} \left[\cos(\phi)\right]}{\partial\phi}$$

The maximum degree and order that will be computed are given by the parameters n_max and m_max. If they are negative (default), the dimensions of matrix dP will be used:

maximum degree -> number of rows - 1
maximum order  -> number of columns - 1

The derivatives will be stored in the matrix dP. Hence, this function can be used to reduce allocations.

The parameter N selects the normalization. The following values are valid:

• Val(:full): Compute the fully normalized associated Legendre function.
• Val(:schmidt): Compute the Schmidt quasi-normalized associated Legendre function.
• Val(:unnormalized): Compute the unnormalized associated Legendre function.

This algorithm needs the matrix P with the values of the associated Legendre function using the same normalization N, which can be computed using the function legendre.

Warning The user is responsible to pass a matrix P with the correct values. For example, if ph_term is true, P must also be computed with ph_term set to true.

This function has the following keywords:

• ph_term::Bool: If true, the Condon-Shortley phase term $(-1)^m$ will be included. (Default = false)

Normalizations

Full normalization

This algorithm was based on [1]. Our definition of fully normalized associated Legendre function can be seen in [2, p. 546]. The conversion is obtained by:

$$K_{n,m} = \sqrt{k \cdot \frac{(n - m)! \cdot (2n + 1)}{(n + m)!}}~,$$

where $k = 1$ if $m = 0$ or $k = 2$ otherwise. Hence:

$$P^f_{n,m} = P_{n,m} \cdot K_{n,m}$$

where $P^f_{n,m}$ is the fully normalized Legendre associated function.

Schmidt quasi-normalization

This algorithm was based on [3, 4]. The conversion is obtained by:

$$K_{n,m} = \sqrt{k \cdot \frac{(n - m)!}{(n + m)!}}~,$$

where $k = 1$ if $m = 0$ or $k = 2$ otherwise. Hence:

$$P^s_{n,m} = P_{n,m} \cdot K_{n,m}$$

where $P^s_{n,m}$ is the Schmidt quasi-normalized Legendre associated function.

References

• [1] Holmes, S. A. and W. E. Featherstone, 2002. A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions. Journal of Geodesy, 76(5), pp. 279-299. For more info.: http://mitgcm.org/~mlosch/geoidcookbook/node11.html
• [2] Vallado, D. A (2013). Fundamentals of Astrodynamics and Applications. Microcosm Press, Hawthorn, CA, USA.
• [3] Schmidt, A (1917). Erdmagnetismus, Enzykl. Math. Wiss., 6, pp. 265–396.
• [4] Winch, D. E., Ivers, D. J., Turner, J. P. R., Stening R. J (2005). Geomagnetism and Schmidt quasi-normalization. Geophysical Journal International, 160(2), pp. 487-504.