# NAG Library Function Document

## 1Purpose

nag_bessel_k1_scaled_vector (s18crc) returns an array of values of the scaled modified Bessel function ${e}^{x}{K}_{1}\left(x\right)$.

## 2Specification

 #include #include
 void nag_bessel_k1_scaled_vector (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)

## 3Description

nag_bessel_k1_scaled_vector (s18crc) evaluates an approximation to ${e}^{{x}_{i}}{K}_{1}\left({x}_{i}\right)$, where ${K}_{1}$ is a modified Bessel function of the second kind for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. The scaling factor ${e}^{x}$ removes most of the variation in ${K}_{1}\left(x\right)$.
The function uses the same Chebyshev expansions as nag_bessel_k1_vector (s18arc), which returns an array of the unscaled values of ${K}_{1}\left(x\right)$.

## 4References

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

## 5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2:    $\mathbf{x}\left[{\mathbf{n}}\right]$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:    $\mathbf{f}\left[{\mathbf{n}}\right]$doubleOutput
On exit: ${e}^{{x}_{i}}{K}_{1}\left({x}_{i}\right)$, the function values.
4:    $\mathbf{ivalid}\left[{\mathbf{n}}\right]$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.
${\mathbf{ivalid}}\left[i-1\right]=1$
 On entry, ${x}_{i}\le 0.0$, ${K}_{1}\left({x}_{i}\right)$ is undefined. ${\mathbf{f}}\left[\mathit{i}-1\right]$ contains $0.0$.
${\mathbf{ivalid}}\left[i-1\right]=2$
${x}_{i}$ is too close to zero, as determined by the value of the safe-range parameter nag_real_safe_small_number (X02AMC): there is a danger of causing overflow. ${\mathbf{f}}\left[\mathit{i}-1\right]$ contains the reciprocal of the safe-range parameter.
5:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NW_IVALID
On entry, at least one value of x was invalid.

## 7Accuracy

Relative errors in the argument are attenuated when propagated into the function value. When the accuracy of the argument is essentially limited by the machine precision, the accuracy of the function value will be similarly limited by at most a small multiple of the machine precision.

## 8Parallelism and Performance

nag_bessel_k1_scaled_vector (s18crc) is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (s18crce.c)

### 10.2Program Data

Program Data (s18crce.d)

### 10.3Program Results

Program Results (s18crce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017