nag_airy_bi_deriv_vector (s17axc) (PDF version)
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NAG C Library Manual

# NAG Library Function Documentnag_airy_bi_deriv_vector (s17axc)

## 1  Purpose

nag_airy_bi_deriv_vector (s17axc) returns an array of values for the derivative of the Airy function $\mathrm{Bi}\left(x\right)$.

## 2  Specification

 #include #include
 void nag_airy_bi_deriv_vector (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)

## 3  Description

nag_airy_bi_deriv_vector (s17axc) calculates an approximate value for the derivative of the Airy function $\mathrm{Bi}\left({x}_{i}\right)$ for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. It is based on a number of Chebyshev expansions.
For $x<-5$,
 $Bi′x=-x4 -atsin⁡z+btζcos⁡z ,$
where $z=\frac{\pi }{4}+\zeta$, $\zeta =\frac{2}{3}\sqrt{-{x}^{3}}$ and $a\left(t\right)$ and $b\left(t\right)$ are expansions in the variable $t=-2{\left(\frac{5}{x}\right)}^{3}-1$.
For $-5\le x\le 0$,
 $Bi′x=3x2ft+gt,$
where $f$ and $g$ are expansions in $t=-2{\left(\frac{x}{5}\right)}^{3}-1$.
For $0,
 $Bi′x=e3x/2yt,$
where $y\left(t\right)$ is an expansion in $t=4x/9-1$.
For $4.5\le x<9$,
 $Bi′x=e21x/8ut,$
where $u\left(t\right)$ is an expansion in $t=4x/9-3$.
For $x\ge 9$,
 $Bi′x=x4ezvt,$
where $z=\frac{2}{3}\sqrt{{x}^{3}}$ and $v\left(t\right)$ is an expansion in $t=2\left(\frac{18}{z}\right)-1$.
For $\left|x\right|<\text{}$ the square of the machine precision, the result is set directly to ${\mathrm{Bi}}^{\prime }\left(0\right)$. 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 function must fail. This occurs for $x<-{\left(\frac{\sqrt{\pi }}{\epsilon }\right)}^{4/7}$, where $\epsilon$ is the machine precision.
For large positive arguments, where ${\mathrm{Bi}}^{\prime }$ grows in an essentially exponential manner, there is a danger of overflow 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: ${\mathbf{n}}\ge 0$.
2:     x[n]const doubleInput
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3:     f[n]doubleOutput
On exit: ${\mathrm{Bi}}^{\prime }\left({x}_{i}\right)$, the function values.
4:     ivalid[n]IntegerOutput
On exit: ${\mathbf{ivalid}}\left[\mathit{i}-1\right]$ contains the error code for ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
${x}_{i}$ is too large and positive. ${\mathbf{f}}\left[\mathit{i}-1\right]$ contains zero. The threshold value is the same as for NE_REAL_ARG_GT in nag_airy_bi_deriv (s17akc), as defined in the Users' Note for your implementation.
${\mathbf{ivalid}}\left[i-1\right]=2$
${x}_{i}$ is too large and negative. ${\mathbf{f}}\left[\mathit{i}-1\right]$ contains zero. The threshold value is the same as for NE_REAL_ARG_LT in nag_airy_bi_deriv (s17akc), 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 $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
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 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 $\epsilon$, are related in principle to the relative error in the argument $\delta$, by
 $E≃ x2 Bix δ ε≃ x2 Bix Bi′x δ.$
In practice, approximate equality is the best that can be expected. When $\delta$, $\epsilon$ 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 $\frac{{\left|x\right|}^{7/4}}{\sqrt{\pi }}$. Therefore it becomes impossible to calculate the function with any accuracy if ${\left|x\right|}^{7/4}>\frac{\sqrt{\pi }}{\delta }$.
For large positive $x$, the relative error amplification is considerable: $\frac{\epsilon }{\delta }\sim \sqrt{{x}^{3}}$. However, very large arguments are not possible due to the danger of overflow. Thus in practice the actual amplification that occurs is limited.

None.

## 9  Example

This example reads values of x from a file, evaluates the function at each value of ${x}_{i}$ and prints the results.

### 9.1  Program Text

Program Text (s17axce.c)

### 9.2  Program Data

Program Data (s17axce.d)

### 9.3  Program Results

Program Results (s17axce.r)

nag_airy_bi_deriv_vector (s17axc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual