NAG FL Interface
s17ahf (airy_​bi_​real)

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1 Purpose

s17ahf returns a value of the Airy function, Bi(x), via the function name.

2 Specification

Fortran Interface
Function s17ahf ( x, ifail)
Real (Kind=nag_wp) :: s17ahf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
double  s17ahf_ (const double *x, Integer *ifail)
The routine may be called by the names s17ahf or nagf_specfun_airy_bi_real.

3 Description

s17ahf evaluates an approximation to the Airy function Bi(x). It is based on a number of Chebyshev expansions.
For x<-5,
Bi(x)=a(t)cosz+b(t)sinz(-x)1/4,  
where z= π4+ 23-x3 and a(t) and b(t) are expansions in the variable t=-2 ( 5x) 3-1.
For -5x0,
Bi(x)=3(f(t)+xg(t)),  
where f and g are expansions in t=-2 ( x5) 3-1.
For 0<x<4.5,
Bi(x)=e11x/8y(t),  
where y is an expansion in t=4x/9-1.
For 4.5x<9,
Bi(x)=e5x/2v(t),  
where v is an expansion in t=4x/9-3.
For x9,
Bi(x)=ezu(t)x1/4,  
where z= 23x3 and u is an expansion in t=2 ( 18z)-1.
For |x|<machine precision, the result is set directly to Bi(0). This both saves time and avoids possible intermediate underflows.
For large negative arguments, it becomes impossible to calculate the phase of the oscillating function with any accuracy so the routine must fail. This occurs if x<- ( 32ε ) 2/3, where ε is the machine precision.
For large positive arguments, there is a danger of causing overflow since Bi grows in an essentially exponential manner, so the routine must fail.

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: x Real (Kind=nag_wp) Input
On entry: the argument x of the function.
2: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. 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).

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:
ifail=1
On entry, x=value.
Constraint: xvalue.
x is too large and positive. The function returns zero.
ifail=2
On entry, x=value.
Constraint: xvalue.
x is too large and negative. The function returns zero.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 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 |xBi(x)|δ,ε | x Bi(x) Bi(x) |δ.  
In practice, approximate equality is the best that can be expected. When δ, ε or 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 and hence will be bounded essentially by the machine precision.
For moderate to large negative x, the error behaviour is clearly oscillatory but the amplitude of the error grows like amplitude ( Eδ) |x|5/4π .
However, the phase error will be growing roughly as 23|x|3 and hence all accuracy will be lost for large negative arguments. This is due to the impossibility of calculating sin and cos to any accuracy if 23|x|3> 1δ .
For large positive arguments, the relative error amplification is considerable:
εδx3.  
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 causing overflow and errors are, therefore, limited in practice.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s17ahf is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.

10.1 Program Text

Program Text (s17ahfe.f90)

10.2 Program Data

Program Data (s17ahfe.d)

10.3 Program Results

Program Results (s17ahfe.r)
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 0 5 10 15 −10 −8 −6 −4 −2 0 2 4 Bi(x) x "s17ahfe.r" Example Program Returns a Value of the Airy Function Bi(x)