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NAG C Library Manual

NAG Library Function Documentnag_bessel_k1_vector (s18arc)

1  Purpose

nag_bessel_k1_vector (s18arc) returns an array of values of the modified Bessel function ${K}_{1}\left(x\right)$.

2  Specification

 #include #include
 void nag_bessel_k1_vector (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)

3  Description

nag_bessel_k1_vector (s18arc) evaluates an approximation to the modified Bessel function of the second kind ${K}_{1}\left({x}_{i}\right)$ for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
Note:  ${K}_{1}\left(x\right)$ is undefined for $x\le 0$ and the function will fail for such arguments.
The function is based on five Chebyshev expansions:
For $0,
 $K1x=1x+xln⁡x∑′r=0arTrt-x∑′r=0brTrt, where ​ t=2x2-1.$
For $1,
 $K1x=e-x∑′r=0crTrt, where ​t=2x-3.$
For $2,
 $K1x=e-x∑′r=0drTrt, where ​t=x-3.$
For $x>4$,
 $K1x=e-xx ∑′r=0erTrt, where ​t=9-x 1+x .$
For $x$ near zero, ${K}_{1}\left(x\right)\simeq \frac{1}{x}$. This approximation is used when $x$ is sufficiently small for the result to be correct to machine precision. For very small $x$ it is impossible to calculate $\frac{1}{x}$ without overflow and the function must fail.
For large $x$, where there is a danger of underflow due to the smallness of ${K}_{1}$, the result is set exactly to zero.

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 $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{x}}\left[\mathit{i}-1\right]>0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3:     f[n]doubleOutput
On exit: ${K}_{1}\left({x}_{i}\right)$, 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.
${\mathbf{ivalid}}\left[i-1\right]=1$
${x}_{i}\le 0.0$, ${K}_{1}\left({x}_{i}\right)$ is undefined. ${\mathbf{f}}\left[\mathit{i}-1\right]$ contains $0.0$.
${\mathbf{ivalid}}\left[i-1\right]=2$
${x}_{i}$ is too small, there is a danger of overflow. ${\mathbf{f}}\left[\mathit{i}-1\right]$ contains zero. The threshold value is the same as for NE_REAL_ARG_TOO_SMALL in nag_bessel_k1 (s18adc), 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

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.

7  Accuracy

Let $\delta$ and $\epsilon$ be the relative errors in the argument and result respectively.
If $\delta$ is somewhat larger than the machine precision (i.e., if $\delta$ is due to data errors etc.), then $\epsilon$ and $\delta$ are approximately related by:
 $ε≃ x K0x- K1x K1x δ.$
Figure 1 shows the behaviour of the error amplification factor
 $xK0x - K1 x K1x .$
However if $\delta$ is of the same order as the machine precision, then rounding errors could make $\epsilon$ slightly larger than the above relation predicts.
For small $x$, $\epsilon \simeq \delta$ and there is no amplification of errors.
For large $x$, $\epsilon \simeq x\delta$ and we have strong amplification of the relative error. Eventually ${K}_{1}$, which is asymptotically given by $\frac{{e}^{-x}}{\sqrt{x}}$, becomes so small that it cannot be calculated without underflow and hence the function will return zero. Note that for large $x$ the errors will be dominated by those of the standard function exp.
Figure 1

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 (s18arce.c)

9.2  Program Data

Program Data (s18arce.d)

9.3  Program Results

Program Results (s18arce.r)