# NAG C Library Function Document

## 1Purpose

nag_bessel_k1_scaled (s18cdc) returns a value of the scaled modified Bessel function ${e}^{x}{K}_{1}\left(x\right)$.

## 2Specification

 #include #include
 double nag_bessel_k1_scaled (double x, NagError *fail)

## 3Description

nag_bessel_k1_scaled (s18cdc) evaluates an approximation to ${e}^{x}{K}_{1}\left(x\right)$, where ${K}_{1}$ is a modified Bessel function of the second kind. 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 (s18adc), which returns the unscaled value of ${K}_{1}\left(x\right)$.

## 4References

NIST Digital Library of Mathematical Functions

## 5Arguments

1:    $\mathbf{x}$doubleInput
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}>0.0$.
2:    $\mathbf{fail}$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.
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.
NE_REAL_ARG_LE
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}>0.0$.
${K}_{1}$ is undefined and the function returns zero.
NE_REAL_ARG_TOO_SMALL
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}>〈\mathit{\text{value}}〉$.
The function returns the value of the function at the smallest permitted value of the argument.

## 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 (s18cdc) is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (s18cdce.c)

### 10.2Program Data

Program Data (s18cdce.d)

### 10.3Program Results

Program Results (s18cdce.r)