NAG CL Interface
s17auc (airy_​ai_​real_​vector)

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1 Purpose

s17auc returns an array of values for the Airy function, Ai(x).

2 Specification

#include <nag.h>
void  s17auc (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)
The function may be called by the names: s17auc, nag_specfun_airy_ai_real_vector or nag_airy_ai_vector.

3 Description

s17auc evaluates an approximation to the Airy function, Ai(xi) for an array of arguments xi, for i=1,2,,n. It is based on a number of Chebyshev expansions:
For x<-5,
Ai(x)=a(t)sinz-b(t)cosz(-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,
Ai(x)=f(t)-xg(t),  
where f and g are expansions in t=-2 ( x5) 3-1.
For 0<x<4.5,
Ai(x)=e-3x/2y(t),  
where y is an expansion in t=4x/9-1.
For 4.5x<9,
Ai(x)=e-5x/2u(t),  
where u is an expansion in t=4x/9-3.
For x9,
Ai(x)=e-zv(t)x1/4,  
where z= 23x3 and v is an expansion in t=2 ( 18z)-1.
For |x|<machine precision, the result is set directly to Ai(0). This both saves time and guards against underflow in intermediate calculations.
For large negative arguments, it becomes impossible to calculate the phase of the oscillatory function with any precision and so the function must fail. This occurs if x<- ( 32ε ) 2/3 , where ε is the machine precision.
For large positive arguments, where Ai decays in an essentially exponential manner, there is a danger of underflow so the function must fail.

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: n Integer Input
On entry: n, the number of points.
Constraint: n0.
2: x[n] const double Input
On entry: the argument xi of the function, for i=1,2,,n.
3: f[n] double Output
On exit: Ai(xi), the function values.
4: ivalid[n] Integer Output
On exit: ivalid[i-1] contains the error code for xi, for i=1,2,,n.
ivalid[i-1]=0
No error.
ivalid[i-1]=1
xi is too large and positive. f[i-1] contains zero. The threshold value is the same as for fail.code= NE_REAL_ARG_GT in s17agc , as defined in the Users' Note for your implementation.
ivalid[i-1]=2
xi is too large and negative. f[i-1] contains zero. The threshold value is the same as for fail.code= NE_REAL_ARG_LT in s17agc , as defined in the Users' Note for your implementation.
5: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
NE_INTERNAL_ERROR
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 for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID
On entry, at least one value of x was invalid.
Check ivalid for more 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 |xAi(x)|δ, ε | x Ai(x) Ai(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 by the function and hence will be bounded by the machine precision.
For moderate negative x, the error behaviour is oscillatory but the amplitude of the error grows like
amplitude (Eδ ) |x|5/4π.  
However, the phase error will be growing roughly like 23|x|3 and hence all accuracy will be lost for large negative arguments 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:
ε δ x3.  
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 setting underflow and so the errors are limited in practice.

8 Parallelism and Performance

s17auc is not threaded in any implementation.

9 Further Comments

None.

10 Example

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

10.1 Program Text

Program Text (s17auce.c)

10.2 Program Data

Program Data (s17auce.d)

10.3 Program Results

Program Results (s17auce.r)