NAG C Library Function Document

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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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

nag_legendre_p (s22aac) is not threaded in any implementation.

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 C 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

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