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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_specfun_airy_ai_real_vector (s17au)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_specfun_airy_ai_real_vector (s17au) returns an array of values for the Airy function,

Ai (x)

Syntax

[f, ivalid, ifail] = s17au(x, 'n', n)

[f, ivalid, ifail] = nag_specfun_airy_ai_real_vector(x, 'n', n)

Description

nag_specfun_airy_ai_real_vector (s17au) 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 function 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 function must fail.

References

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

Parameters

Compulsory Input Parameters

1: $x (n)$ – double array: The argument $x_{i}$ of the function, for $i = 1, 2, \dots, n$ .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the array x.
$n$ , the number of points.

Constraint: $n \geq 0$ .

Output Parameters

1: $f (n)$ – double array

Ai (x_{i})

, the function values.

2: $ivalid (n)$ – int64int32nag_int array

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 nag_specfun_airy_ai_real (s17ag), 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 nag_specfun_airy_ai_real (s17ag), as defined in the Users' Note for your implementation.

3: $ifail$ – int64int32nag_int scalar

ifail = 0

unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W $ifail = 1$: On entry, at least one value of x was invalid.
Check ivalid for more information.

$ifail = 2$: Constraint: $n \geq 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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.

Further Comments

None.

Example

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

x_{i}

and prints the results.

Open in the MATLAB editor: s17au_example

function s17au_example


fprintf('s17au example results\n\n');

x = [-10; -1; 0; 1; 5; 10; 20];

[f, ivalid, ifail] = s17au(x);

fprintf('     x            Ai(x)   ivalid\n');
for i=1:numel(x)
  fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end

s17au example results

     x            Ai(x)   ivalid
  -1.000e+01   4.024e-02    0
  -1.000e+00   5.356e-01    0
   0.000e+00   3.550e-01    0
   1.000e+00   1.353e-01    0
   5.000e+00   1.083e-04    0
   1.000e+01   1.105e-10    0
   2.000e+01   1.692e-27    0

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Chapter Contents

Chapter Introduction

NAG Toolbox