NAG Library Function Document

nag_complex_airy_ai (s17dgc)

+− 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_complex_airy_ai (s17dgc) returns the value of the Airy function

Ai (z)

or its derivative

{Ai}^{'} (z)

for complex

z

, with an option for exponential scaling.

2 Specification

#include <nag.h>

#include <nags.h>

void	nag_complex_airy_ai (Nag_FunType deriv, Complex z, Nag_ScaleResType scal, Complex ai, Integer nz, NagError *fail)

3 Description

nag_complex_airy_ai (s17dgc) returns a value for the Airy function

Ai (z)

or its derivative

{Ai}^{'} (z)

, where

z

is complex,

- π < \arg z \leq π

. Optionally, the value is scaled by the factor

e^{2 z \sqrt{z} / 3}

The function is derived from the function CAIRY in Amos (1986). It is based on the relations

Ai (z) = \frac{\sqrt{z} K_{1 / 3} (w)}{π \sqrt{3}}

, and

{Ai}^{'} (z) = \frac{- z K_{2 / 3} (w)}{π \sqrt{3}}

, where

K_{ν}

is the modified Bessel function and

w = 2 z \sqrt{z} / 3

For very large

|z|

, argument reduction will cause total loss of accuracy, and so no computation is performed. For slightly smaller

|z|

, the computation is performed but results are accurate to less than half of machine precision. If

Re (w)

is too large, and the unscaled function is required, there is a risk of overflow and so no computation is performed. In all the above cases, a warning is given by the function.

4 References

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

Amos D E (1986) Algorithm 644: A portable package for Bessel functions of a complex argument and non-negative order ACM Trans. Math. Software 12 265–273

5 Arguments

1: deriv – Nag_FunTypeInput

On entry: specifies whether the function or its derivative is required.

$deriv = Nag_Function$: $Ai (z)$ is returned.
$deriv = Nag_Deriv$: ${Ai}^{'} (z)$ is returned.

Constraint:

deriv = Nag_Function

Nag_Deriv

2: z – ComplexInput

On entry: the argument

z

of the function.

3: scal – Nag_ScaleResTypeInput

On entry: the scaling option.

$scal = Nag_UnscaleRes$: The result is returned unscaled.
$scal = Nag_ScaleRes$: The result is returned scaled by the factor $e^{2 z \sqrt{z} / 3}$ .

Constraint:

scal = Nag_UnscaleRes

Nag_ScaleRes

4: ai – Complex *Output

On exit: the required function or derivative value.

5: nz – Integer *Output

On exit: indicates whether or not ai is set to zero due to underflow. This can only occur when

scal = Nag_UnscaleRes

$nz = 0$: ai is not set to zero.
$nz = 1$: ai is set to zero.

6: 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_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.
NE_OVERFLOW_LIKELY: No computation because $ω . re$ too large, where $ω = (2 / 3) \times z^{(3 / 2)}$ .
NE_TERMINATION_FAILURE: No computation – algorithm termination condition not met.
NE_TOTAL_PRECISION_LOSS: No computation because $|z| = ⟨value⟩ > ⟨value⟩$ .
NW_SOME_PRECISION_LOSS: Results lack precision because $|z| = ⟨value⟩ > ⟨value⟩$ .

7 Accuracy

All constants in nag_complex_airy_ai (s17dgc) are given to approximately

18

digits of precision. Calling the number of digits of precision in the floating-point arithmetic being used

t

, then clearly the maximum number of correct digits in the results obtained is limited by

p = \min (t, 18)

. Because of errors in argument reduction when computing elementary functions inside nag_complex_airy_ai (s17dgc), the actual number of correct digits is limited, in general, by

p - s

, where

s \approx \max (1, |\log_{10} |z||)

represents the number of digits lost due to the argument reduction. Thus the larger the value of

|z|

, the less the precision in the result.

Empirical tests with modest values of

z

, checking relations between Airy functions

Ai (z)

{Ai}^{'} (z)

Bi (z)

and

{Bi}^{'} (z)

, have shown errors limited to the least significant

3

–

4

digits of precision.

8 Parallelism and Performance

Not applicable.

9 Further Comments

Note that if the function is required to operate on a real argument only, then it may be much cheaper to call nag_airy_ai (s17agc) or nag_airy_ai_deriv (s17ajc).

10 Example

This example prints a caption and then proceeds to read sets of data from the input data stream. The first datum is a value for the argument deriv, the second is a complex value for the argument, z, and the third is a character value used as a flag to set the argument scal. The program calls the function and prints the results. The process is repeated until the end of the input data stream is encountered.

NAG Library Function Documentnag_complex_airy_ai (s17dgc)

+− 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_complex_airy_ai (s17dgc)