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

C#
public static void s22aa( int mode, double x, int m, int nl, double[] p, out int ifail )

Visual Basic
Public Shared Sub s22aa ( _ mode As Integer, _ x As Double, _ m As Integer, _ nl As Integer, _ p As Double(), _ <OutAttribute> ByRef ifail As Integer _ )

Visual C++
public: static void s22aa( int mode, double x, int m, int nl, array<double>^ p, [OutAttribute] int% ifail )

F#
static member s22aa : mode : int * x : float * m : int * nl : int * p : float[] * ifail : int byref -> unit

Parameters

mode

Type: System..::..Int32

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

x: Type: System..::..Double
On entry: the argument $x$ of the function.

Constraint: $abs (x) \leq 1.0$ .

m: Type: System..::..Int32
On entry: the order $m$ of the function.

Constraint: $abs (m) \leq 27$ .

nl

Type: System..::..Int32

On entry: the degree

N

of the last function required in the sequence.

Constraints:

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

p

Type: array<System..::..Double>[]()[][]

An array of size [

0 : nl

]

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 $n = 0, 1, \dots, N$ .

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

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.

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 [Parameters]) 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

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$

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 $m \neq 0$ .

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

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.

Parallelism and Performance

None.

Further Comments

None.

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.

Example program (C#): s22aae.cs

Example program data: s22aae.d

Example program results: s22aae.r