S22AAF : NAG Library, Mark 24

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

S22AAF 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

S22AAF 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.

S22AAF 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 Section 5) to zero whenever the required function is not defined for certain values of

m

and

n

(e.g.,

m = - 5

and

n = 3

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

On entry: indicates whether the sequence of function values is to be returned unnormalized or normalized.

$MODE = 1$: The sequence of function values is returned unnormalized.
$MODE = 2$: The sequence of function values is returned normalized.

Constraint:

MODE = 1

2

On entry: the argument

x

of the function.

Constraint:

abs (X) \leq 1.0

On entry: the order

m

of the function.

Constraint:

abs (M) \leq 27

On entry: the degree

N

of the last function required in the sequence.

Constraints:

$NL \geq 0$ ;
if $M = 0$ , $NL \leq 100$ ;
if $M \neq 0$ , $NL \leq 55 - abs (M)$ .

On exit: the required sequence of function values as follows:

if $MODE = 1$ , $P (n)$ contains $P_{n}^{m} (x)$ , for $n = 0, 1, \dots, N$ ;
if $MODE = 2$ , $P (n)$ contains $\bar{P_{n}^{m}} (x)$ , for $n = 0, 1, \dots, N$ .

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

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.

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.

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 Library Routine Document

S22AAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentS22AAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

S22AAF