NAG Library Function Document

nag_bessel_y0_vector (s17aqc)

+− 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_y0_vector (s17aqc) returns an array of values of the Bessel function

Y_{0} (x)

2 Specification

#include <nag.h>

#include <nags.h>

void	nag_bessel_y0_vector (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)

3 Description

nag_bessel_y0_vector (s17aqc) evaluates an approximation to the Bessel function of the second kind

Y_{0} (x_{i})

for an array of arguments

x_{i}

, for

i = 1, 2, \dots, n

Note:

Y_{0} (x)

is undefined for

x \leq 0

and the function will fail for such arguments.

The function is based on four Chebyshev expansions:

For

0 < x \leq 8

Y_{0} (x) = \frac{2}{π} \ln x \underset{r = 0}{\sum^{'}} a_{r} T_{r} (t) + \underset{r = 0}{\sum^{'}} b_{r} T_{r} (t),   with ​ t = 2 {(\frac{x}{8})}^{2} - 1 .

For

x > 8

Y_{0} (x) = \sqrt{\frac{2}{π x}} \{P_{0} (x) \sin (x - \frac{π}{4}) + Q_{0} (x) \cos (x - \frac{π}{4})\}

where

P_{0} (x) = \underset{r = 0}{\sum^{'}} c_{r} T_{r} (t)

and

Q_{0} (x) = \frac{8}{x} \underset{r = 0}{\sum^{'}} d_{r} T_{r} (t), with ​ t = 2 {(\frac{8}{x})}^{2} - 1 .

For

x

near zero,

Y_{0} (x) ≃ \frac{2}{π} (\ln (\frac{x}{2}) + γ)

, where

γ

denotes Euler's constant. This approximation is used when

x

is sufficiently small for the result to be correct to machine precision.

For very large

x

, it becomes impossible to provide results with any reasonable accuracy (see Section 7), hence the function fails. Such arguments contain insufficient information to determine the phase of oscillation of

Y_{0} (x)

; only the amplitude,

\sqrt{\frac{2}{π n}}

, can be determined and this is returned on failure. The range for which this occurs is roughly related to machine precision; the function will fail if

x ≳ 1 / machine precision

(see the Users' Note for your implementation for details).

4 References

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

Clenshaw C W (1962) Chebyshev Series for Mathematical Functions Mathematical tables HMSO

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:

Y_{0} (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.

$ivalid [i - 1] = 1$

On entry,

x_{i}

is too large.

f [i - 1]

contains the amplitude of the

Y_{0}

oscillation,

\sqrt{\frac{2}{π x_{i}}}

$ivalid [i - 1] = 2$

On entry,

x_{i} \leq 0.0

Y_{0}

is undefined.

f [i - 1]

contains

0.0

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

be the relative error in the argument and

E

be the absolute error in the result. (Since

Y_{0} (x)

oscillates about zero, absolute error and not relative error is significant, except for very small

x

δ

is somewhat larger than the machine representation error (e.g., if

δ

is due to data errors etc.), then

E

and

δ

are approximately related by

E ≃ |x Y_{1} (x)| δ

(provided

E

is also within machine bounds). Figure 1 displays the behaviour of the amplification factor

|x Y_{1} (x)|

However, if

δ

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

E

slightly larger than the above relation predicts.

For very small

x

, the errors are essentially independent of

δ

and the function should provide relative accuracy bounded by the machine precision.

For very large

x

, the above relation ceases to apply. In this region,

Y_{0} (x) ≃ \sqrt{\frac{2}{π x}} \sin (x - \frac{π}{4})

. The amplitude

\sqrt{\frac{2}{π x}}

can be calculated with reasonable accuracy for all

x

, but

\sin (x - \frac{π}{4})

cannot. If

x - \frac{π}{4}

is written as

2 N π + θ

where

N

is an integer and

0 \leq θ < 2 π

, then

\sin (x - \frac{π}{4})

is determined by

θ

only. If

x ≳ δ^{- 1}

θ

cannot be determined with any accuracy at all. Thus if

x

is greater than, or of the order of the inverse of machine precision, it is impossible to calculate the phase of

Y_{0} (x)

and the function must fail.

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_y0_vector (s17aqc)

+− 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_y0_vector (s17aqc)