NAG Library Routine Document

s17dgf (airy_ai_complex)


s17dgf returns the value of the Airy function Aiz or its derivative Aiz for complex z, with an option for exponential scaling.


Fortran Interface
Subroutine s17dgf ( deriv, z, scal, ai, nz, ifail)
Integer, Intent (Inout):: ifail
Integer, Intent (Out):: nz
Complex (Kind=nag_wp), Intent (In):: z
Complex (Kind=nag_wp), Intent (Out):: ai
Character (1), Intent (In):: deriv, scal
C Header Interface
#include <nagmk26.h>
void  s17dgf_ (const char *deriv, const Complex *z, const char *scal, Complex *ai, Integer *nz, Integer *ifail, const Charlen length_deriv, const Charlen length_scal)


s17dgf returns a value for the Airy function Aiz or its derivative Aiz, where z is complex, -π<argzπ. Optionally, the value is scaled by the factor e2zz/3.
The routine is derived from the routine CAIRY in Amos (1986). It is based on the relations Aiz= zK1/3w π3 , and Aiz= -zK2/3w 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 Rew 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 routine.


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


1:     deriv – Character(1)Input
On entry: specifies whether the function or its derivative is required.
Aiz is returned.
Aiz is returned.
Constraint: deriv='F' or 'D'.
2:     z – Complex (Kind=nag_wp)Input
On entry: the argument z of the function.
3:     scal – Character(1)Input
On entry: the scaling option.
The result is returned unscaled.
The result is returned scaled by the factor e2zz/3.
Constraint: scal='U' or 'S'.
4:     ai – Complex (Kind=nag_wp)Output
On exit: the required function or derivative value.
5:     nz – IntegerOutput
On exit: indicates whether or not ai is set to zero due to underflow. This can only occur when scal='U'.
ai is not set to zero.
ai is set to zero.
6:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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).

Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, deriv has an illegal value: deriv=value.
On entry, scal has an illegal value: scal=value.
No computation because Re(ω) too large, where ω=2/3×z3/2.
Results lack precision because z=value>value.
No computation because z=value>value.
No computation – algorithm termination condition not met.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


All constants in s17dgf 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=mint,18. Because of errors in argument reduction when computing elementary functions inside s17dgf, the actual number of correct digits is limited, in general, by p-s, where s max1, log10z  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 Aiz, Aiz, Biz and Biz, have shown errors limited to the least significant 3 – 4 digits of precision.

Parallelism and Performance

s17dgf is not threaded in any implementation.

Further Comments

Note that if the function is required to operate on a real argument only, then it may be much cheaper to call s17agf or s17ajf.


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 to set the argument scal. The program calls the routine and prints the results. The process is repeated until the end of the input data stream is encountered.

Program Text

Program Text (s17dgfe.f90)

Program Data

Program Data (s17dgfe.d)

Program Results

Program Results (s17dgfe.r)