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

4 References

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

5 Arguments

1: $mode$ – Integer Input

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: $x$ – Real (Kind=nag_wp) Input

On entry: the argument

x

of the function.

Constraint:

abs (x) \leq 1.0

3: $m$ – Integer Input

On entry: the order

m

of the function.

Constraint:

abs (m) \leq 27

4: $nl$ – Integer Input

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: $p (0 : nl)$ – Real (Kind=nag_wp) array Output

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: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. 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).

6 Error Indicators and Warnings

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:

$ifail = 1$: On entry, $| m | = ⟨ value ⟩$ .
Constraint: $| m | \leq 27$ .

On entry, $| x | = ⟨ value ⟩$ .
Constraint: $| x | \leq 1.0$ .

On entry, $mode = ⟨ value ⟩$ .
Constraint: $mode \leq 2$ .

On entry, $mode = ⟨ value ⟩$ .
Constraint: $mode \geq 1$ .

On entry, $nl = ⟨ value ⟩$ .
Constraint: $nl \leq 100$ when $m = 0$ .

On entry, $nl = ⟨ value ⟩$ .
Constraint: $nl \geq 0$ .

On entry, $nl = ⟨ value ⟩$ .
Constraint: when $| m | \neq 0$ , $nl \geq 0$ .

On entry, $nl = ⟨ value ⟩$ and $| m | = ⟨ value ⟩$ .
Constraint: $nl + | m | \leq 55$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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

Background information to multithreading can be found in the Multithreading documentation.

s22aaf 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.

s22aa: FL CL CPP AD PY MB

NAG FL Interfaces22aaf (legendre_​p)

▸▿ 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 FL Interface
s22aaf (legendre_p)