NAG Library Routine Document

S17AGF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

S17AGF returns a value for the Airy function,

Ai (x)

, via the function name.

2 Specification

FUNCTION S17AGF (

X, IFAIL)

REAL (KIND=nag_wp) S17AGF

INTEGER	IFAIL
REAL (KIND=nag_wp)	X

3 Description

S17AGF evaluates an approximation to the Airy function,

Ai (x)

. 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

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

5 Parameters

1: X – REAL (KIND=nag_wp)Input: On entry: the argument $x$ of the function.
2: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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:

$IFAIL = 1$: X is too large and positive. On soft failure, the routine returns zero. See also the Users' Note for your implementation.

$IFAIL = 2$: X is too large and negative. On soft failure, the routine returns zero. See also the Users' Note for your implementation.

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 Further Comments

None.

9 Example

This example reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

NAG Library Routine DocumentS17AGF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

S17AGF