NAG Library Function Document

nag_airy_ai_deriv_vector (s17awc)

+− 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

nag_airy_ai_deriv_vector (s17awc) returns an array of values of the derivative of the Airy function

Ai (x)

2 Specification

#include <nag.h>

#include <nags.h>

void	nag_airy_ai_deriv_vector (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)

3 Description

nag_airy_ai_deriv_vector (s17awc) 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 function 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 function must fail.

4 References

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

5 Arguments

1: n – IntegerInput

On entry:

n

, the number of points.

Constraint:

n \geq 0

2: x[n] – const doubleInput

On entry: the argument

x_{i}

of the function, for

i = 1, 2, \dots, n

3: f[n] – doubleOutput

On exit:

{Ai}^{'} (x_{i})

, the function values.

4: ivalid[n] – IntegerOutput

On exit:

ivalid [i - 1]

contains the error code for

x_{i}

, for

i = 1, 2, \dots, n

$ivalid [i - 1] = 0$: No error.
$ivalid [i - 1] = 1$: $x_{i}$ is too large and positive. $f [i - 1]$ contains zero. The threshold value is the same as for $fail . code =$ NE_REAL_ARG_GT in nag_airy_ai_deriv (s17ajc), as defined in the Users' Note for your implementation.
$ivalid [i - 1] = 2$: $x_{i}$ is too large and negative. $f [i - 1]$ contains zero. The threshold value is the same as for $fail . code =$ NE_REAL_ARG_LT in nag_airy_ai_deriv (s17ajc), as defined in the Users' Note for your implementation.

5: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NW_IVALID: On entry, at least one value of x was invalid.
Check ivalid for more 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 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 Parallelism and Performance

Not applicable.

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 Function Documentnag_airy_ai_deriv_vector (s17awc)

+− 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 Library Function Document

nag_airy_ai_deriv_vector (s17awc)