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

1 Purpose

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

2 Specification

#include <nag.h>

#include <nags.h>

void	nag_legendre_p (Integer mode, double x, Integer m, Integer nl, double p[], NagError *fail)

3 Description

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

4 References

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

5 Arguments

1: $mode$ – IntegerInput

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

2: $x$ – doubleInput

On entry: the argument

x

of the function.

Constraint:

abs (x) \leq 1.0

3: $m$ – IntegerInput

On entry: the order

m

of the function.

Constraint:

abs (m) \leq 27

4: $nl$ – IntegerInput

On entry: the degree

N

of the last function required in the sequence.

Constraints:

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

5: $p [nl + 1]$ – doubleOutput

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

6: $fail$ – NagError *Input/Output

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

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: 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: $nl \geq 0$ .
NE_INT_2: On entry, $nl = 〈value〉$ and $|m| = 〈value〉$ .
Constraint: $nl + |m| \leq 55$ .

On entry, $nl = 〈value〉$ .
Constraint: $nl \leq 100$ when $m = 0$ .
NE_INTERNAL_ERROR: 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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_REAL: On entry, $|x| = 〈value〉$ .
Constraint: $|x| \leq 1.0$ .

7 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.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 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.

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)