NAG Library Function Document
nag_complex_airy_ai (s17dgc) returns the value of the Airy function or its derivative for complex , with an option for exponential scaling.
||nag_complex_airy_ai (Nag_FunType deriv,
nag_complex_airy_ai (s17dgc) returns a value for the Airy function or its derivative , where is complex, . Optionally, the value is scaled by the factor .
The function is derived from the function CAIRY in Amos (1986)
. It is based on the relations
is the modified Bessel function and
For very large , argument reduction will cause total loss of accuracy, and so no computation is performed. For slightly smaller , the computation is performed but results are accurate to less than half of machine precision. If 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.
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
deriv – Nag_FunTypeInput
: specifies whether the function or its derivative is required.
- If , is returned.
- If , is returned.
z – ComplexInput
On entry: the argument of the function.
scal – Nag_ScaleResTypeInput
: the scaling option.
- The result is returned unscaled.
- The result is returned scaled by the factor .
ai – Complex *Output
On exit: the required function or derivative value.
nz – Integer *Output
: indicates whether or not ai
is set to zero due to underflow. This can only occur when
- ai is not set to zero.
- ai is set to zero.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
On entry, argument 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
No computation because too large,
No computation – algorithm termination condition not met.
No computation because .
Results lack precision because .
All constants in nag_complex_airy_ai (s17dgc) are given to approximately digits of precision. Calling the number of digits of precision in the floating point arithmetic being used , then clearly the maximum number of correct digits in the results obtained is limited by . 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 , where represents the number of digits lost due to the argument reduction. Thus the larger the value of , the less the precision in the result.
Empirical tests with modest values of , checking relations between Airy functions , , and , have shown errors limited to the least significant – digits of precision.
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)
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.
9.1 Program Text
Program Text (s17dgce.c)
9.2 Program Data
Program Data (s17dgce.d)
9.3 Program Results
Program Results (s17dgce.r)