nag_legendre_p (s22aac) : NAG Library, Mark 24

nag_legendre_p (s22aac) 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

nag_legendre_p (s22aac) 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_legendre_p (s22aac) 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 $n = 0, 1, \dots, N$ .

The NAG error argument (see Section 3.6 in the Essential Introduction).

On entry, argument

⟨value⟩

had an illegal value.

On entry,

|m| = ⟨value⟩

.
Constraint:

|m| \leq 27

On entry,

mode = ⟨value⟩

.
Constraint:

mode \leq 2

On entry,

mode = ⟨value⟩

.
Constraint:

mode \geq 1

On entry,

nl = ⟨value⟩

.
Constraint:

nl \geq 0

On entry,

nl = ⟨value⟩

and

|m| = ⟨value⟩

.
Constraint:

nl + |m| \leq 55

On entry,

nl = ⟨value⟩

.
Constraint:

nl \leq 100

when

m = 0

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

On entry,

|x| = ⟨value⟩

.
Constraint:

|x| \leq 1.0

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.

NAG Library Function Document

nag_legendre_p (s22aac)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Documentnag_legendre_p (s22aac)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_legendre_p (s22aac)