s22aa:: Approximations of Special Functions (NAG Toolbox)

nag_specfun_legendre_p (s22aa) returns a sequence of values for either the unnormalized or normalized Legendre functions of the first kind

P_{n}^{m} (x)

\bar{P_{n}^{m}} (x)

for real

x

of a given order

m

and degree

n = 0, 1, \dots, N

Syntax

Description

nag_specfun_legendre_p (s22aa) evaluates a sequence of values for either the unnormalized or normalized Legendre (

m = 0

) or associated Legendre (

m \neq 0

) functions of the first kind

P_{n}^{m} (x)

\bar{P_{n}^{m}} (x)

, where

x

is real with

- 1 \leq x \leq 1

, of order

m

and degree

n = 0, 1, \dots, N

defined by

\begin{array}{lcl} P_{n}^{m} (x) & = & {(1 - x^{2})}^{m / 2} \frac{d^{m}}{d x^{m}} P_{n} (x) & if ​ m \geq 0, \\ P_{n}^{m} (x) & = & \frac{(n + m)!}{(n - m)!} P_{n}^{- m} (x) & if ​ m < 0 and \\ \bar{P_{n}^{m}} (x) & = & \sqrt{\frac{(2 n + 1)}{2} \frac{(n - m)!}{(n + m)!}} P_{n}^{m} (x) \end{array}

respectively;

P_{n} (x)

is the (unassociated) Legendre polynomial of degree

n

given by

P_{n} (x) \equiv P_{n}^{0} (x) = \frac{1}{2^{n} n!} \frac{d^{n}}{d x^{n}} {(x^{2} - 1)}^{n}

(the Rodrigues formula). Note that some authors (e.g., Abramowitz and Stegun (1972)) include an additional factor of

{(- 1)}^{m}

(the Condon–Shortley Phase) in the definitions of

P_{n}^{m} (x)

and

\bar{P_{n}^{m}} (x)

. They use the notation

P_{m n} (x) \equiv {(- 1)}^{m} P_{n}^{m} (x)

in order to distinguish between the two cases.

nag_specfun_legendre_p (s22aa) is based on a standard recurrence relation described in Section 8.5.3 of Abramowitz and Stegun (1972). Constraints are placed on the values of

m

and

n

in order to avoid the possibility of machine overflow. It also sets the appropriate elements of the array p (see Arguments) to zero whenever the required function is not defined for certain values of

m

and

n

(e.g.,

m = - 5

and

n = 3

References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Errors or warnings detected by the function:

Accuracy

The computed function values should be accurate to within a small multiple of the machine precision except when underflow (or overflow) occurs, in which case the true function values are within a small multiple of the underflow (or overflow) threshold of the machine.

Further Comments

Example

This example reads the values of the arguments

x

m

and

N

from a file, calculates the sequence of unnormalized associated Legendre function values

P_{n}^{m} (x), P_{n + 1}^{m} (x), \dots, P_{n + N}^{m} (x)

, and prints the results.

function s22aa_example


fprintf('s22aa example results\n\n');

mode = int64(1);
x  = 0.5;
m  = int64(2);
n  = int64(3);

[p, ifail] = s22aa(mode, x, m, n);

fprintf('First %2d unnormalized associated Legendre function values ',n);
fprintf('P^%d_n(%7.4f)\n\n',m,x);
fprintf('%3s%8s\n','n','P_n');
fprintf('%3d%10.4f\n', [double([0:n]); p']);

s22aa example results

First  3 unnormalized associated Legendre function values P^2_n( 0.5000)

  n     P_n
  0    0.0000
  1    0.0000
  2    2.2500
  3    5.6250

On entry,	$abs (x) > 1.0$ ,
or	$mode \neq 1$ or $2$ ,
or	$nl < 0$ ,
or	$nl > 100$ when $m = 0$ ,
or	$abs (m) > 27$ ,
or	$nl + abs (m) > 55$ when $m \neq 0$ .

NAG Toolbox: nag_specfun_legendre_p (s22aa)

▸▿ Contents

Purpose