NAG Library Function Document

nag_bessel_i_nu_scaled (s18ecc)

+− 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_i_nu_scaled (s18ecc) returns the value of the scaled modified Bessel function

e^{- x} I_{ν / 4} (x)

for real

x > 0

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_bessel_i_nu_scaled (double x, Integer nu, NagError *fail)

3 Description

nag_bessel_i_nu_scaled (s18ecc) evaluates an approximation to the scaled modified Bessel function of the first kind

e^{- x} I_{ν / 4} (x)

, where the order

ν = - 3, - 2, - 1, 1, 2

3

and

x

is real and positive. For positive orders it may also be called with

x = 0

, since

I_{ν / 4} (0) = 0

when

ν > 0

. For negative orders the formula

I_{- ν / 4} (x) = I_{ν / 4} (x) + \frac{2}{π} \sin (\frac{π ν}{4}) K_{ν / 4} (x)

is used prior to multiplication by the scale factor

e^{- x}

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.

Constraints:

if $nu < 0$ , $x > 0.0$ ;
if $nu > 0$ , $x \geq 0.0$ .

2: nu – IntegerInput

On entry: the argument

ν

of the function.

Constraint:

1 \leq abs (nu) \leq 3

3: 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, $nu = ⟨value⟩$ .
Constraint: $1 \leq abs (nu) \leq 3$ .
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. The result is returned as zero.
NE_REAL_INT: On entry, $x = ⟨value⟩$ , $nu = ⟨value⟩$ .
Constraint: $x > 0.0$ when $nu < 0$ .
On entry, $x = ⟨value⟩$ , $nu = ⟨value⟩$ .
Constraint: $x \geq 0.0$ when $nu > 0$ .
NE_TERMINATION_FAILURE: The evaluation has been abandoned due to failure to satisfy the termination condition. The result is returned as zero.
NE_TOTAL_PRECISION_LOSS: The evaluation has been abandoned due to total loss of precision. The result is returned as zero.
NW_SOME_PRECISION_LOSS: The evaluation has been completed but some precision has been lost.

7 Accuracy

All constants in the underlying functions 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 functions, 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 reads values of the arguments

x

and

ν

from a file, evaluates the function and prints the results.

NAG Library Function Documentnag_bessel_i_nu_scaled (s18ecc)