legendre

Associated Legendre functions

Syntax


            
             P = legendre(n,X)


            
             P = legendre(n,X,normalization)

Description

P = legendre( n , X ) computes the associated Legendre functions of degree n and order m = 0, 1, ..., n evaluated for each element in X .

example

P = legendre( n , X , normalization ) computes normalized versions of the associated Legendre functions. normalization can be 'unnorm' (default), 'sch' , or 'norm' .

example

Examples

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Associated Legendre Function Values of Vector

Open Live Script

Use the legendre function to operate on a vector and then examine the format of the output.

Calculate the second-degree Legendre function values of a vector.

deg = 2;
x = 0:0.1:0.2;
P = legendre(deg,x)

P = 3×3
   -0.5000   -0.4850   -0.4400
         0   -0.2985   -0.5879
    3.0000    2.9700    2.8800

The format of the output is such that:

Each row contains the function value for different values of m (the order of the associated Legendre function)
Each column contains the function value for a different value of x

$\begin{array}{cccc} x = 0 & x = 0.1 & x = 0.2 \\ m = 0 & P_{2}^{0} (0) & P_{2}^{0} (0.1) & P_{2}^{0} (0.2) \\ m = 1 & P_{2}^{1} (0) & P_{2}^{1} (0.1) & P_{2}^{1} (0.2) \\ m = 2 & P_{2}^{2} (0) & P_{2}^{2} (0.1) & P_{2}^{2} (0.2) \end{array}$

The equation for the second-degree associated Legendre function $P_{2}^{m}$ is

$P_{2}^{m} (x) = {(- 1)}^{m} {(1 - x^{2})}^{m / 2} \frac{d^{m}}{d x^{m}} [\frac{1}{2} (3 x^{2} - 1)] .$

Therefore, the value of $P_{2}^{0} (0)$ is

$P_{2}^{0} (0) = [\frac{1}{2} (3 x^{2} - 1)] |_{x = 0} = - \frac{1}{2} .$

This result agrees with P(1,1) = -0.5000 .

Compare Legendre Normalizations

Open Live Script

Calculate the associated Legendre function values with several normalizations.

Calculate the first-degree, unnormalized Legendre function values $P_{1}^{m}$ . The first row of values corresponds to $m = 0$ , and the second row to $m = 1$ .

x = 0:0.2:1;
n = 1;
P_unnorm = legendre(n,x)

P_unnorm = 2×6
         0    0.2000    0.4000    0.6000    0.8000    1.0000
   -1.0000   -0.9798   -0.9165   -0.8000   -0.6000         0

Next, compute the Schmidt seminormalized function values. Compared to the unnormalized values, the Schmidt form differs when $m > 0$ by the scaling

${(- 1)}^{m} \sqrt{\frac{2 (n - m)!}{(n + m)!}} .$

For the first row, the two normalizations are the same, since $m = 0$ . For the second row, the scaling constant multiplying each value is -1.

P_sch = legendre(n,x,'sch')

P_sch = 2×6
         0    0.2000    0.4000    0.6000    0.8000    1.0000
    1.0000    0.9798    0.9165    0.8000    0.6000         0

C1 = (-1) * sqrt(2*factorial(0)/factorial(2))

C1 =

Lastly, compute the fully normalized function values. Compared to the unnormalized values, the fully normalized form differs by the scaling factor

${(- 1)}^{m} \sqrt{\frac{(n + \frac{1}{2}) (n - m)!}{(n + m)!}} .$

This scaling factor applies for all values of $m$ , so the first and second rows have different scaling factors.

P_norm = legendre(n,x,'norm')

P_norm = 2×6
         0    0.2449    0.4899    0.7348    0.9798    1.2247
    0.8660    0.8485    0.7937    0.6928    0.5196         0

Cm0 = sqrt((3/2))

Cm0 = 
1.2247

Cm1 = (-1) * sqrt((3/2)/2)

Cm1 = 
-0.8660

Calculate Spherical Harmonics

Open Live Script

Spherical harmonics arise in the solution to Laplace's equation and are used to represent functions defined on the surface of a sphere. Use legendre to compute and visualize the spherical harmonic for $Y_{3}^{2}$ .

The equation for spherical harmonics includes a term for the Legendre function, as well as a complex exponential:

$Y_{l}^{m} (θ, ϕ) = \sqrt{\frac{(2 l + 1) (l - m)!}{4 π (l + m)!}} P_{l}^{m} (\cos θ) e^{i m ϕ}, - l \leq m \leq l .$

First, create a grid of values to represent all combinations of $0 \leq θ \leq π$ (colatitude angle) and $0 \leq ϕ \leq 2 π$ (azimuthal angle). Here, the colatitude $θ$ ranges from 0 at the North Pole, to $π / 2$ at the Equator, and to $π$ at the South Pole.

dx = pi/60;
col = 0:dx:pi;
az = 0:dx:2*pi;
[phi,theta] = meshgrid(az,col);

Calculate $P_{l}^{m} (\cos θ)$ on the grid for $l = 3$ .

l = 3;
Plm = legendre(l,cos(theta));

Since legendre computes the answer for all values of $m$ , Plm contains some extra function values. Extract the values for $m = 2$ and discard the rest. Use the reshape function to orient the results as a matrix with the same size as phi and theta .

m = 2;
if l ~= 0
    Plm = reshape(Plm(m+1,:,:),size(phi));
end

Calculate the spherical harmonic values for $Y_{3}^{2}$ .

a = (2*l+1)*factorial(l-m);
b = 4*pi*factorial(l+m);
C = sqrt(a/b);
Ylm = C .*Plm .*exp(1i*m*phi);

Convert the spherical coordinates to Cartesian coordinates. Here, $π / 2 - θ$ becomes the latitude angle that ranges from $π / 2$ at the North Pole, to 0 at the Equator, and to $- π / 2$ at the South Pole. Plot the spherical harmonic for $Y_{3}^{2}$ using both the positive and negative real values.

[Xm,Ym,Zm] = sph2cart(phi, pi/2-theta, abs(real(Ylm)));
surf(Xm,Ym,Zm)
title('$Y_3^2$ spherical harmonic','interpreter','latex')

Figure contains an axes object. The axes object with title YSubScript 3 SuperScript 2 baseline spherical harmonic contains an object of type surface.

Input Arguments

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`n` — Degree of Legendre function
positive integer

Degree of Legendre function, specified as a positive integer. For a specified degree, legendre computes $P_{n}^{m} (x)$ for all orders m from m = 0 to m = n .

Example: legendre(2,X)

`X` — Input values
scalar | vector | matrix | multidimensional array

Input values, specified as a scalar, vector, matrix, or multidimensional array of real values in the range [-1,1] . For example, with spherical harmonics it is common to use X = cos(theta) as the input values to compute $P_{n}^{m} (\cos θ)$ .

Example: legendre(2,cos(theta))

Data Types: single | double

`normalization` — Normalization type
`'unnorm'` (default) | `'sch'` | `'norm'`

Normalization type, specified as one of these values.

Value	Result
`'unnorm'`	Associated Legendre Functions
`'sch'`	Schmidt Seminormalized Associated Legendre Functions
`'norm'`	Fully Normalized Associated Legendre Functions

Example: legendre(n,X,'sch')

Output Arguments

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`P` — Associated Legendre function values
scalar | vector | matrix | multidimensional array

Associated Legendre function values, returned as a scalar, vector, matrix, or multidimensional array. The normalization of P depends on the value of normalization .

The size of P depends on the size of X :

If X is a vector, then P is a matrix of size (n+1) -by- length(X) . The P(m+1,i) entry is the associated Legendre function of degree n and order m evaluated at X(i) .
In general, P has one more dimension than X and each element P(m+1,i,j,k,...) contains the associated Legendre function of degree n and order m evaluated at X(i,j,k,...) .

Limitations

The values of the unnormalized associated Legendre function overflow the range of double-precision numbers for n > 150 and the range of single-precision numbers for n > 28 . This overflow results in Inf and NaN values. For orders larger than these thresholds, consider using the 'sch' or 'norm' normalizations instead.

More About

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Associated Legendre Functions

The associated Legendre functions $y = P_{n}^{m} (x)$ are solutions to the general Legendre differential equation

$(1 - x^{2}) \frac{d^{2} y}{d x^{2}} - 2 x \frac{d y}{d x} + [n (n + 1) - \frac{m^{2}}{1 - x^{2}}] y = 0 .$

n is the integer degree and m is the integer order of the associated Legendre function, such that $0 \leq m \leq n$ .

The associated Legendre functions $P_{n}^{m} (x)$ are the most general solutions to this equation given by

$P_{n}^{m} (x) = {(- 1)}^{m} {(1 - x^{2})}^{m / 2} \frac{d^{m}}{d x^{m}} P_{n} (x) .$

They are defined in terms of derivatives of the Legendre polynomials $P_{n} (x)$ , which are a subset of the solutions given by

$P_{n} (x) = \frac{1}{2^{n} n!} \frac{d^{n}}{d x^{n}} {(x^{2} - 1)}^{n} .$

The first few Legendre polynomials are

Value of `n`	$P_{n} (x)$
`0`	$P_{0} (x) = 1$
`1`	$P_{1} (x) = x$
`2`	$P_{2} (x) = \frac{1}{2} (3 x^{2} - 1)$

Schmidt Seminormalized Associated Legendre Functions

The Schmidt seminormalized associated Legendre functions are related to the unnormalized associated Legendre functions $P_{n}^{m} (x)$ by

$\begin{array}{l} P_{n} (x) for m = 0, \\ S_{n}^{m} (x) = {(- 1)}^{m} \sqrt{\frac{2 (n - m)!}{(n + m)!}} P_{n}^{m} (x) for m > 0. \end{array}$

Fully Normalized Associated Legendre Functions

The fully normalized associated Legendre functions are normalized such that

$\int_{- 1}^{1} {[N_{n}^{m} (x)]}^{2} d x = 1 .$

The normalized functions are related to the unnormalized associated Legendre functions $P_{n}^{m} (x)$ by

$N_{n}^{m} (x) = {(- 1)}^{m} \sqrt{\frac{(n + \frac{1}{2}) (n - m)!}{(n + m)!}} P_{n}^{m} (x) .$

Algorithms

legendre uses a three-term backward recursion relationship in m . This recursion is on a version of the Schmidt seminormalized associated Legendre functions $Q_{n}^{m} (x)$ , which are complex spherical harmonics. These functions are related to the standard Abramowitz and Stegun [1] functions $P_{n}^{m} (x)$ by

$P_{n}^{m} (x) = \sqrt{\frac{(n + m)!}{(n - m)!}} Q_{n}^{m} (x) .$

They are related to the Schmidt form by

$\begin{array}{l} m = 0 : S_{n}^{m} (x) = Q_{n}^{0} (x) \\ m > 0 : S_{n}^{m} (x) = {(- 1)}^{m} \sqrt{2} Q_{n}^{m} (x) . \end{array}$

References

[1] Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions , Dover Publications, 1965, Ch.8.

[2] Jacobs, J. A., Geomagnetism , Academic Press, 1987, Ch.4.

Extended Capabilities

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Thread-Based Environment
Run code in the background using MATLAB® `backgroundPool` or accelerate code with Parallel Computing Toolbox™ `ThreadPool` .

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment .

GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The legendre function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox) . For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox) .

Version History

Introduced before R2006a

legendre (adsbygoogle = window.adsbygoogle || []).push({});

Syntax

Description

Examples

Associated Legendre Function Values of Vector

Compare Legendre Normalizations

Calculate Spherical Harmonics

Input Arguments

n — Degree of Legendre function positive integer

X — Input values scalar | vector | matrix | multidimensional array

normalization — Normalization type (adsbygoogle = window.adsbygoogle || []).push({}); 'unnorm' (default) | 'sch' | 'norm'

Output Arguments

P — Associated Legendre function values scalar | vector | matrix | multidimensional array

Limitations

More About

Associated Legendre Functions

Schmidt Seminormalized Associated Legendre Functions

Fully Normalized Associated Legendre Functions

Algorithms

References

Extended Capabilities

Thread-Based Environment Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool .

GPU Arrays Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Version History

See Also

legendre

`n` — Degree of Legendre function
positive integer

`X` — Input values
scalar | vector | matrix | multidimensional array

`normalization` — Normalization type
`'unnorm'` (default) | `'sch'` | `'norm'`

`P` — Associated Legendre function values
scalar | vector | matrix | multidimensional array

Thread-Based Environment
Run code in the background using MATLAB® `backgroundPool` or accelerate code with Parallel Computing Toolbox™ `ThreadPool` .

GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.