NAG Library Function Document

nag_bessel_k_alpha_scaled (s18ehc)

+− 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_k_alpha_scaled (s18ehc) returns a sequence of values for the scaled modified Bessel functions

e^{x} K_{α + n} (x)

for real

x > 0

, selected values of

α \geq 0

and

n = 0, 1, \dots, N

2 Specification

#include <nag.h>

#include <nags.h>

void	nag_bessel_k_alpha_scaled (double x, Integer ia, Integer ja, Integer nl, double b[], NagError *fail)

3 Description

nag_bessel_k_alpha_scaled (s18ehc) evaluates a sequence of values for the scaled modified Bessel function of the second kind

e^{x} K_{α} (x)

, where

x

is real and non-negative and

α \in \{0, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{2}{3}, \frac{3}{4}\}

is the order. The

(N + 1)

-member sequence is generated for orders

α, α + 1, \dots, α + N

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:

x > 0.0

2: ia – IntegerInput

3: ja – IntegerInput

On entry: the numerator

i

and denominator

j

, respectively, of the order

α = i / j

of the first member in the required sequence of function values. Only the following combinations of pairs of values of

i

and

j

are allowed:

$i = 0$ and $j = 1$ corresponds to $α = 0$ ;
$i = 1$ and $j = 2$ corresponds to $α = \frac{1}{2}$ ;
$i = 1$ and $j = 3$ corresponds to $α = \frac{1}{3}$ ;
$i = 1$ and $j = 4$ corresponds to $α = \frac{1}{4}$ ;
$i = 2$ and $j = 3$ corresponds to $α = \frac{2}{3}$ ;
$i = 3$ and $j = 4$ corresponds to $α = \frac{3}{4}$ .

Constraint: ia and ja must constitute a valid pair

(ia, ja) = (0, 1)

(1, 2)

(1, 3)

(1, 4)

(2, 3)

(3, 4)

4: nl – IntegerInput

On entry: the value of

N

. Note that the order of the last member in the required sequence of function values is given by

α + N

Constraint:

0 \leq nl \leq 100

5: b[ $nl + 1$ ] – doubleOutput

On exit: with

fail . code = NE_NOERROR

fail . code = NW_SOME_PRECISION_LOSS

, the required sequence of function values: b

(n)

contains

K_{α + n} (x)

, for

n = 0, 1, \dots, N

6: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_INT: On entry, $nl = ⟨value⟩$ .
Constraint: $0 \leq nl \leq 100$ .
NE_INT_2: On entry, $ia = ⟨value⟩$ , $ja = ⟨value⟩$ .
Constraint: ia and ja must constitute a valid pair (ia,ja).
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.
NE_OVERFLOW_LIKELY: The evaluation has been abandoned due to the likelihood of overflow.
NE_REAL: On entry, $x = ⟨value⟩$ .
Constraint: $x > 0.0$ .
NE_TERMINATION_FAILURE: The evaluation has been abandoned due to failure to satisfy the termination condition.
NE_TOTAL_PRECISION_LOSS: The evaluation has been abandoned due to total loss of precision.
NW_SOME_PRECISION_LOSS: The evaluation has been completed but some precision has been lost.

7 Accuracy

All constants in the underlying function are specified to approximately 18 digits of precision. If

t

denotes the number of digits of precision in the floating-point arithmetic being used, then clearly the maximum number of correct digits in the results obtained is limited by

p = \min (t, 18)

. Because of errors in argument reduction when computing elementary functions inside the underlying function, the actual number of correct digits is limited, in general, by

p - s

, where

s \approx \max (1, |\log_{10} x|)

represents the number of digits lost due to the argument reduction. Thus the larger the value of

x

, the less the precision in the result.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

The example program evaluates

e^{x} K_{0} (x), e^{x} K_{1} (x), e^{x} K_{2} (x)

and

e^{x} K_{3} (x)

x = 0.5

, and prints the results.

NAG Library Function Documentnag_bessel_k_alpha_scaled (s18ehc)