NAG Library Routine Document

S17AWF

+− 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

S17AWF returns an array of values of the derivative of the Airy function

Ai (x)

2 Specification

SUBROUTINE S17AWF (

N, X, F, IVALID, IFAIL)

INTEGER	N, IVALID(N), IFAIL
REAL (KIND=nag_wp)	X(N), F(N)

3 Description

S17AWF evaluates an approximation to the derivative of 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) = \sqrt[4]{- x} [a (t) \cos z + \frac{b (t)}{ζ} \sin z],

where

z = \frac{π}{4} + ζ

ζ = \frac{2}{3} \sqrt{- x^{3}}

and

a (t)

and

b (t)

are expansions in variable

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

For

- 5 \leq x \leq 0

{Ai}^{'} (x) = x^{2} f (t) - g (t),

where

f

and

g

are expansions in

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

For

0 < x < 4.5

{Ai}^{'} (x) = e^{- 11 x / 8} y (t),

where

y (t)

is an expansion in

t = 4 (\frac{x}{9}) - 1

For

4.5 \leq x < 9

{Ai}^{'} (x) = e^{- 5 x / 2} v (t),

where

v (t)

is an expansion in

t = 4 (\frac{x}{9}) - 3

For

x \geq 9

{Ai}^{'} (x) = \sqrt[4]{x} e^{- z} u (t),

where

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

and

u (t)

is an expansion in

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

For

|x| <

the square of the machine precision, the result is set directly to

{Ai}^{'} (0)

. This both saves time and avoids possible intermediate underflows.

For large negative arguments, it becomes impossible to calculate a result for the oscillating function with any accuracy and so the routine must fail. This occurs for

x < - {(\frac{\sqrt{π}}{ε})}^{4 / 7}

, 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: 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 S17AJF, 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 S17AJF, 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 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:

$IFAIL = 1$: On entry, at least one value of X was invalid.
Check IVALID for more information.

$IFAIL = 2$: On entry, $N = ⟨value⟩$ .
Constraint: $N \geq 0$ .

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 in character and here relative error is needed. The absolute error,

E

, and the relative error,

ε

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

δ

, by

E ≃ |x^{2} Ai (x)| δ ε ≃ |\frac{x^{2} 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

, positive or negative, errors are strongly attenuated by the function and hence will be roughly bounded by the machine precision.

For moderate to large negative

x

, the error, like the function, is oscillatory; however the amplitude of the error grows like

\frac{{|x|}^{7 / 4}}{\sqrt{π}} .

Therefore it becomes impossible to calculate the function with any accuracy if

{|x|}^{7 / 4} > \frac{\sqrt{π}}{δ}

For large positive

x

, the relative error amplification is considerable:

\frac{ε}{δ} ≃ \sqrt{x^{3}} .

However, very large arguments are not possible due to the danger of underflow. Thus in practice error amplification is limited.

8 Further Comments

None.

9 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 DocumentS17AWF

+− 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

S17AWF