Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ivalid(n)
Real (Kind=nag_wp), Intent (In)	::	x(n)
Real (Kind=nag_wp), Intent (Out)	::	f(n)

C Header Interface

#include <nag.h>

void	s17auf_ (const Integer n, const double x[], double f[], Integer ivalid[], Integer ifail)

The routine may be called by the names s17auf or nagf_specfun_airy_ai_real_vector.

3 Description

s17auf evaluates an approximation to the Airy function,

Ai (x_{i})

for an array of arguments

x_{i}

, for

i = 1, 2, \dots, n

. It is based on a number of Chebyshev expansions:

For

x < - 5

Ai (x) = \frac{a (t) \sin z - b (t) \cos z}{{(- x)}^{1 / 4}}

where

z = \frac{π}{4} + \frac{2}{3} \sqrt{- x^{3}}

, and

a (t)

and

b (t)

are expansions in the variable

t = - 2 {(\frac{5}{x})}^{3} - 1

For

- 5 \leq x \leq 0

Ai (x) = f (t) - x g (t),

where

f

and

g

are expansions in

t = - 2 {(\frac{x}{5})}^{3} - 1 .

For

0 < x < 4.5

Ai (x) = e^{- 3 x / 2} y (t),

where

y

is an expansion in

t = 4 x / 9 - 1

For

4.5 \leq x < 9

Ai (x) = e^{- 5 x / 2} u (t),

where

u

is an expansion in

t = 4 x / 9 - 3

For

x \geq 9

Ai (x) = \frac{e^{- z} v (t)}{x^{1 / 4}},

where

z = \frac{2}{3} \sqrt{x^{3}}

and

v

is an expansion in

t = 2 (\frac{18}{z}) - 1

For

| x | < machine precision

, the result is set directly to

Ai (0)

. This both saves time and guards against underflow in intermediate calculations.

For large negative arguments, it becomes impossible to calculate the phase of the oscillatory function with any precision and so the routine must fail. This occurs if

x < - {(\frac{3}{2 ε})}^{2 / 3}

, where

ε

is the machine precision.

For large positive arguments, where

Ai

decays in an essentially exponential manner, there is a danger of underflow so the routine must fail.

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the number of points.

Constraint:

n \geq 0

2: $x (n)$ – Real (Kind=nag_wp) array Input

On entry: the argument

x_{i}

of the function, for

i = 1, 2, \dots, n

3: $f (n)$ – Real (Kind=nag_wp) array Output

On exit:

Ai (x_{i})

, the function values.

4: $ivalid (n)$ – Integer array Output

On exit:

ivalid (i)

contains the error code for

x_{i}

, for

i = 1, 2, \dots, n

$ivalid (i) = 0$: No error.
$ivalid (i) = 1$: $x_{i}$ is too large and positive. $f (i)$ contains zero. The threshold value is the same as for $ifail = 1$ in s17agf , as defined in the the Users' Note for your implementation.
$ivalid (i) = 2$: $x_{i}$ is too large and negative. $f (i)$ contains zero. The threshold value is the same as for $ifail = 2$ in s17agf , as defined in the the Users' Note for your implementation.

5: $ifail$ – 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, at least one value of x was invalid.
Check ivalid for more information.

$ifail = 2$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

$ifail = - 99$: 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

For negative arguments the function is oscillatory and hence absolute error is the appropriate measure. In the positive region the function is essentially exponential-like and here relative error is appropriate. The absolute error,

E

, and the relative error,

ε

, are related in principle to the relative error in the argument,

δ

, by

E ≃ | x {Ai}^{'} (x) | δ, ε ≃ | \frac{x {Ai}^{'} (x)}{Ai (x)} | δ .

In practice, approximate equality is the best that can be expected. When

δ

ε

E

is of the order of the machine precision, the errors in the result will be somewhat larger.

For small

x

, errors are strongly damped by the function and hence will be bounded by the machine precision.

For moderate negative

x

, the error behaviour is oscillatory but the amplitude of the error grows like

amplitude (\frac{E}{δ}) \sim \frac{{| x |}^{5 / 4}}{\sqrt{π}} .

However, the phase error will be growing roughly like

\frac{2}{3} \sqrt{{| x |}^{3}}

and hence all accuracy will be lost for large negative arguments due to the impossibility of calculating sin and cos to any accuracy if

\frac{2}{3} \sqrt{{| x |}^{3}} > \frac{1}{δ}

For large positive arguments, the relative error amplification is considerable:

\frac{ε}{δ} \sim \sqrt{x^{3}} .

This means a loss of roughly two decimal places accuracy for arguments in the region of

20

. However, very large arguments are not possible due to the danger of setting underflow and so the errors are limited in practice.

8 Parallelism and Performance

s17auf is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example reads values of x from a file, evaluates the function at each value of

x_{i}

and prints the results.

s17au: FL CL CPP AD

NAG FL Interfaces17auf (airy_​ai_​real_​vector)

▸▿ 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
s17auf (airy_ai_real_vector)