NAG Library Function Document

nag_bessel_i_alpha (s18ejc)

+− 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_alpha (s18ejc) returns a sequence of values for the modified Bessel functions

I_{α + n - 1} (x)

I_{α - n + 1} (x)

for real

x

, non-negative

α < 1

and

n = 1, 2, \dots, |N| + 1

2 Specification

#include <nag.h>

#include <nags.h>

void	nag_bessel_i_alpha (double x, double a, Integer nl, Complex b[], NagError *fail)

3 Description

nag_bessel_i_alpha (s18ejc) evaluates a sequence of values for the modified Bessel function of the first kind

I_{α} (x)

, where

x

is real and nonzero and

α

is the order with

0 \leq α < 1

. The

(|N| + 1)

-member sequence is generated for orders

α, α + 1, \dots, α + N

when

N \geq 0

. Note that

+

is replaced by

-

when

N < 0

. For positive orders the function may also be called with

x = 0

, since

I_{q} (0) = 0

when

q > 0

. For negative orders the formula

I_{- q} (x) = I_{q} (x) + \frac{2}{π} \sin (π q) K_{q} (x)

is used to generate the required sequence.

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: if $nl < 0$ , $x \neq 0.0$ .
2: a – doubleInput: On entry: the order $α$ of the first member in the required sequence of function values.
Constraint: $0.0 \leq a < 1.0$ .
3: nl – IntegerInput: On entry: the value of $N$ .
Constraint: $abs (nl) \leq 101$ .
4: b[ $\times$ ] – ComplexOutput: On exit: with $fail . code = NE_NOERROR$ or $fail . code = NW_SOME_PRECISION_LOSS$ , the required sequence of function values: b $(n)$ contains $I_{α + n - 1} (x)$ if $nl \geq 1$ and $I_{α - n + 1} (x)$ otherwise, for $n = 1, 2, \dots, abs (nl) + 1$ .
5: 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: $abs (nl) \leq 101$ .
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, $a = ⟨value⟩$ .
Constraint: $0.0 \leq a < 1.0$ .
NE_REAL_INT: On entry, $x = ⟨value⟩$ , $nl = ⟨value⟩$ .
Constraint: $x \neq 0.0$ when $nl < 0$ .
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 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||, |\log_{10} |α||)

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

|x|

and

|α|

, the less the precision in the result.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

The example program evaluates

I_{0} (x), I_{1} (x), I_{2} (x)

and

I_{3} (x)

x = 0.5

, and prints the results.

NAG Library Function Documentnag_bessel_i_alpha (s18ejc)

+− 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_i_alpha (s18ejc)