nag_airy_ai_vector (s17auc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_airy_ai_vector (s17auc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_airy_ai_vector (s17auc) returns an array of values for the Airy function, Aix.

2  Specification

#include <nag.h>
#include <nags.h>
void  nag_airy_ai_vector (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)

3  Description

nag_airy_ai_vector (s17auc) evaluates an approximation to the Airy function, Aixi for an array of arguments xi, for i=1,2,,n. It is based on a number of Chebyshev expansions:
For x<-5,
Aix=atsinz-btcosz-x1/4
where z= π4+ 23-x3, and at and bt are expansions in the variable t=-2 5x 3-1.
For -5x0,
Aix=ft-xgt,
where f and g are expansions in t=-2 x5 3-1.
For 0<x<4.5,
Aix=e-3x/2yt,
where y is an expansion in t=4x/9-1.
For 4.5x<9,
Aix=e-5x/2ut,
where u is an expansion in t=4x/9-3.
For x9,
Aix=e-zvtx1/4,
where z= 23x3 and v is an expansion in t=2 18z-1.
For x<machine precision, the result is set directly to Ai0. 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

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

5  Arguments

1:     nIntegerInput
On entry: n, the number of points.
Constraint: n0.
2:     x[n]const doubleInput
On entry: the argument xi of the function, for i=1,2,,n.
3:     f[n]doubleOutput
On exit: Aixi, the function values.
4:     ivalid[n]IntegerOutput
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 nag_airy_ai (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 nag_airy_ai (s17agc), as defined in the Users' Note for your implementation.
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

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.
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 x Aix δ, ε x Aix Aix 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 x5/4π.
However the phase error will be growing roughly like 23x3 and hence all accuracy will be lost for large negative arguments due to the impossibility of calculating sin and cos to any accuracy if 23x3> 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  Further Comments

None.

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 (s17auce.c)

9.2  Program Data

Program Data (s17auce.d)

9.3  Program Results

Program Results (s17auce.r)


nag_airy_ai_vector (s17auc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

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