S17AXF (PDF version)
S Chapter Contents
S Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

S17AXF returns an array of values for the derivative of the Airy function Bix.

2  Specification

REAL (KIND=nag_wp)  X(N), F(N)

3  Description

S17AXF calculates an approximate value for the derivative of the Airy function Bixi for an array of arguments xi, for i=1,2,,n. It is based on a number of Chebyshev expansions.
For x<-5,
Bix=-x4 -atsinz+btζcosz ,
where z= π4+ζ, ζ= 23-x3 and at and bt 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 yt is an expansion in t=4x/9-1.
For 4.5x<9,
where ut is an expansion in t=4x/9-3.
For x9,
where z= 23x3 and vt is an expansion in t=2 18z-1.
For x< the square of the machine precision, the result is set directly to Bi0. 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

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

5  Parameters

1:     N – INTEGERInput
On entry: n, the number of points.
Constraint: N0.
2:     X(N) – REAL (KIND=nag_wp) arrayInput
On entry: the argument xi of the function, for i=1,2,,N.
3:     F(N) – REAL (KIND=nag_wp) arrayOutput
On exit: Bixi, the function values.
4:     IVALID(N) – INTEGER arrayOutput
On exit: IVALIDi contains the error code for xi, for i=1,2,,N.
No error.
xi is too large and positive. Fi contains zero. The threshold value is the same as for IFAIL=1 in S17AKF, as defined in the Users' Note for your implementation.
xi is too large and negative. Fi contains zero. The threshold value is the same as for IFAIL=2 in S17AKF, as defined in the Users' Note for your implementation.
5:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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).

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, at least one value of X was invalid.
Check IVALID for more information.
On entry, N=value.
Constraint: N0.

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 x2 Bix δ ε x2 Bix Bix 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 x7/4π . Therefore it becomes impossible to calculate the function with any accuracy if x7/4> 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  Further Comments


9  Example

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

9.1  Program Text

Program Text (s17axfe.f90)

9.2  Program Data

Program Data (s17axfe.d)

9.3  Program Results

Program Results (s17axfe.r)

S17AXF (PDF version)
S Chapter Contents
S Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012