NAG CL Interface
s17dgc (airy_​ai_​complex)

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1 Purpose

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>
void  s17dgc (Nag_FunType deriv, Complex z, Nag_ScaleResType scal, Complex *ai, Integer *nz, NagError *fail)
The function may be called by the names: s17dgc, nag_specfun_airy_ai_complex or nag_complex_airy_ai.

3 Description

s17dgc returns a value for the Airy function Ai(z) or its derivative Ai(z), where z is complex, -π<argzπ. Optionally, the value is scaled by the factor e2zz/3.
The function is derived from the function CAIRY in Amos (1986). It is based on the relations Ai(z)= zK1/3(w) π3 , and Ai(z)= -zK2/3(w) π3 , where Kν is the modified Bessel function and w=2zz/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

NIST Digital Library of Mathematical Functions
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_FunType Input
On entry: specifies whether the function or its derivative is required.
Ai(z) is returned.
Ai(z) is returned.
Constraint: deriv=Nag_Function or Nag_Deriv.
2: z Complex Input
On entry: the argument z of the function.
3: scal Nag_ScaleResType Input
On entry: the scaling option.
The result is returned unscaled.
The result is returned scaled by the factor e2zz/3.
Constraint: scal=Nag_UnscaleRes or 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.
ai is not set to zero.
ai is set to zero.
6: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
No computation because ω.re too large, where ω=(2/3)×z(3/2).
No computation – algorithm termination condition not met.
No computation because |z|=value>value.
Results lack precision because |z|=value>value.

7 Accuracy

All constants in 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 s17dgc, the actual number of correct digits is limited, in general, by p-s, where s max(1,|log10|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 34 digits of precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s17dgc is not threaded in any implementation.

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 s17agc or 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.

10.1 Program Text

Program Text (s17dgce.c)

10.2 Program Data

Program Data (s17dgce.d)

10.3 Program Results

Program Results (s17dgce.r)