NAG FL Interface
s17akf (airy_​bi_​deriv)

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1 Purpose

s17akf returns a value for the derivative of the Airy function Bi(x), via the function name.

2 Specification

Fortran Interface
Function s17akf ( x, ifail)
Real (Kind=nag_wp) :: s17akf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
double  s17akf_ (const double *x, Integer *ifail)
The routine may be called by the names s17akf or nagf_specfun_airy_bi_deriv.

3 Description

s17akf calculates an approximate value for the derivative of the Airy function Bi(x). It is based on a number of Chebyshev expansions.
For x<-5,
Bi(x)=-x4 [-a(t)sinz+b(t)ζcosz] ,  
where z= π4+ζ, ζ= 23-x3 and a(t) and b(t) are expansions in the variable t=-2 ( 5x) 3-1.
For -5x0,
where f and g are expansions in t=-2 ( x5) 3-1.
For 0<x<4.5,
where y(t) is an expansion in t=4x/9-1.
For 4.5x<9,
where u(t) is an expansion in t=4x/9-3.
For x9,
where z= 23x3 and v(t) is an expansion in t=2 ( 18z)-1.
For |x|< the square of the machine precision, the result is set directly to Bi(0). This saves time and avoids possible underflows in calculation.
For large negative arguments, it becomes impossible to calculate a result for the oscillating function with any accuracy so the routine must fail. This occurs for x<- ( πε ) 4/7 , where ε is the machine precision.
For large positive arguments, where Bi grows in an essentially exponential manner, there is a danger of overflow 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:
On entry, x=value.
Constraint: xvalue.
x is too large and positive. The function returns zero.
On entry, x=value.
Constraint: xvalue.
x is too large and negative. The function returns zero.
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.
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.
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 appropriate. In the positive region the function has essentially exponential behaviour and hence relative error is needed. The absolute error, E, and the relative error ε, are related in principle to the relative error in the argument δ, by
E |x2Bi(x)|δ ε | x2 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, positive or negative, errors are strongly attenuated by the function and hence will effectively be bounded by the machine precision.
For moderate to large negative x, the error is, like the function, oscillatory. However, the amplitude of the absolute error grows like |x|7/4π . Therefore, it becomes impossible to calculate the function with any accuracy if |x|7/4> πδ .
For large positive x, the relative error amplification is considerable: εδx3. However, very large arguments are not possible due to the danger of overflow. Thus in practice the actual amplification that occurs is limited.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s17akf is not threaded in any implementation.

9 Further Comments


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 (s17akfe.f90)

10.2 Program Data

Program Data (s17akfe.d)

10.3 Program Results

Program Results (s17akfe.r)
GnuplotProduced by GNUPLOT 4.6 patchlevel 3 −2 0 2 4 6 8 10 −10 −8 −6 −4 −2 0 2 4 Bi(x) x Example Program Returns a Value for the Derivative of the Airy Function Bi(x) gnuplot_plot_1