NAG Library Routine Document

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 <nagmk26.h>

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

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: $n$ – IntegerInput

On entry:

n

, the number of points.

Constraint:

n \geq 0

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

On entry: the argument

x_{i}

of the function, for

i = 1, 2, \dots, n

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

On exit:

Ai (x_{i})

, the function values.

4: $ivalid (n)$ – Integer arrayOutput

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 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 Users' Note for your implementation.

5: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 or 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 or 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this argument, the recommended value is

0

. 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 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation 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.

NAG Library Routine Document

s17auf (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

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