# NAG Library Function Documentnag_bessel_k1_scaled (s18cdc)

## 1  Purpose

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

## 2  Specification

#include <nag.h>
#include <nags.h>

double  nag_bessel_k1_scaled (double x,  NagError *fail)

## 3  Description

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 ${K}_{1}\left(x\right)$.

## 4  References

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

## 5  Arguments

1:     x – doubleInput
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}>0.0$. If x is too close to zero, there is a danger of overflow, and a failure will occur.
2:     fail – NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_REAL_ARG_LE
On entry, x must not be less than or equal to 0.0: ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
${k}_{1}$ is undefined and the function returns zero.
NE_REAL_ARG_TOO_SMALL
On entry, x must be greater than $〈\mathit{\text{value}}〉$ : ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
The function returns the value of the function at the smallest permitted value of the argument.

## 7  Accuracy

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.

None.

## 9  Example

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

### 9.1  Program Text

Program Text (s18cdce.c)

### 9.2  Program Data

Program Data (s18cdce.d)

### 9.3  Program Results

Program Results (s18cdce.r)
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