NAG Library Function Document

nag_bessel_k1_vector (s18arc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_bessel_k1_vector (s18arc) returns an array of values of the modified Bessel function

K_{1} (x)

2 Specification

#include <nag.h>

#include <nags.h>

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} (x_{i})

for an array of arguments

x_{i}

, for

i = 1, 2, \dots, n

Note:

K_{1} (x)

is undefined for

x \leq 0

and the function will fail for such arguments.

The function is based on five Chebyshev expansions:

For

0 < x \leq 1

K_{1} (x) = \frac{1}{x} + x \ln x \underset{r = 0}{\sum^{'}} a_{r} T_{r} (t) - x \underset{r = 0}{\sum^{'}} b_{r} T_{r} (t),   where ​ t = 2 x^{2} - 1 .

For

1 < x \leq 2

K_{1} (x) = e^{- x} \underset{r = 0}{\sum^{'}} c_{r} T_{r} (t),   where ​ t = 2 x - 3 .

For

2 < x \leq 4

K_{1} (x) = e^{- x} \underset{r = 0}{\sum^{'}} d_{r} T_{r} (t),   where ​ t = x - 3 .

For

x > 4

K_{1} (x) = \frac{e^{- x}}{\sqrt{x}} \underset{r = 0}{\sum^{'}} e_{r} T_{r} (t),   where ​ t = \frac{9 - x}{1 + x} .

For

x

near zero,

K_{1} (x) ≃ \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: n – IntegerInput

On entry:

n

, the number of points.

Constraint:

n \geq 0

2: x[n] – const doubleInput

On entry: the argument

x_{i}

of the function, for

i = 1, 2, \dots, n

Constraint:

x [i - 1] > 0.0

, for

i = 1, 2, \dots, n

3: f[n] – doubleOutput

On exit:

K_{1} (x_{i})

, the function values.

4: ivalid[n] – IntegerOutput

On exit:

ivalid [i - 1]

contains the error code for

x_{i}

, for

i = 1, 2, \dots, n

$ivalid [i - 1] = 0$: No error.
$ivalid [i - 1] = 1$: $x_{i} \leq 0.0$ , $K_{1} (x_{i})$ is undefined. $f [i - 1]$ contains $0.0$ .
$ivalid [i - 1] = 2$: $x_{i}$ is too small, there is a danger of overflow. $f [i - 1]$ contains zero. The threshold value is the same as for $fail . code =$ NE_REAL_ARG_TOO_SMALL in nag_bessel_k1 (s18adc), as defined in the Users' Note for your implementation.

5: fail – NagError *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 $⟨value⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
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

Let

δ

and

ε

be the relative errors in the argument and result respectively.

δ

is somewhat larger than the machine precision (i.e., if

δ

is due to data errors etc.), then

ε

and

δ

are approximately related by:

ε ≃ |\frac{x K_{0} (x) - K_{1} (x)}{K_{1} (x)}| δ .

Figure 1 shows the behaviour of the error amplification factor

|\frac{x K_{0} (x) - K_{1} (x)}{K_{1} (x)}| .

However if

δ

is of the same order as the machine precision, then rounding errors could make

ε

slightly larger than the above relation predicts.

For small

x

ε ≃ δ

and there is no amplification of errors.

For large

x

ε ≃ x δ

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

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

This example reads values of x from a file, evaluates the function at each value of

x_{i}

and prints the results.

NAG Library Function Documentnag_bessel_k1_vector (s18arc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_bessel_k1_vector (s18arc)