Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ivalid(n)
Real (Kind=nag_wp), Intent (In)	::	x(n)
Real (Kind=nag_wp), Intent (Out)	::	f(n)

C Header Interface

#include <nag.h>

void	s17atf_ (const Integer n, const double x[], double f[], Integer ivalid[], Integer ifail)

The routine may be called by the names s17atf or nagf_specfun_bessel_j1_real_vector.

3 Description

s17atf evaluates an approximation to the Bessel function of the first kind

J_{1} (x_{i})

for an array of arguments

x_{i}

, for

i = 1, 2, \dots, n

Note:

J_{1} (- x) = - J_{1} (x)

, so the approximation need only consider

x \geq 0

The routine is based on three Chebyshev expansions:

For

0 < x \leq 8

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

For

x > 8

J_{1} (x) = \sqrt{\frac{2}{π x}} {P_{1} (x) \cos (x - \frac{3 π}{4}) - Q_{1} (x) \sin (x - \frac{3 π}{4})}

where

P_{1} (x) = \underset{r = 0}{\sum^{'}} b_{r} T_{r} (t)

and

Q_{1} (x) = \frac{8}{x} \underset{r = 0}{\sum^{'}} c_{r} T_{r} (t)

with

t = 2 {(\frac{8}{x})}^{2} - 1

For

x

near zero,

J_{1} (x) ≃ \frac{x}{2}

. 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 routine fails. Such arguments contain insufficient information to determine the phase of oscillation of

J_{1} (x)

; only the amplitude,

\sqrt{\frac{2}{π | x |}}

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

| x | ≳ 1 / machine precision

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

4 References

NIST Digital Library of Mathematical Functions

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

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the number of points.

Constraint:

n \geq 0

2: $x (n)$ – Real (Kind=nag_wp) array Input

On entry: the argument

x_{i}

of the function, for

i = 1, 2, \dots, n

3: $f (n)$ – Real (Kind=nag_wp) array Output

On exit:

J_{1} (x_{i})

, the function values.

4: $ivalid (n)$ – Integer array Output

On exit:

ivalid (i)

contains the error code for

x_{i}

, for

i = 1, 2, \dots, n

$ivalid (i) = 0$: No error.
$ivalid (i) = 1$: On entry, $x_{i}$ is too large. $f (i)$ contains the amplitude of the $J_{1}$ oscillation, $\sqrt{\frac{2}{π | x_{i} |}}$ .

5: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, at least one value of x was invalid.
Check ivalid for more information.

$ifail = 2$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Let

δ

be the relative error in the argument and

E

be the absolute error in the result. (Since

J_{1} (x)

oscillates about zero, absolute error and not relative error is significant.)

δ

is somewhat larger than machine precision (e.g., if

δ

is due to data errors etc.), then

E

and

δ

are approximately related by:

E ≃ | x J_{0} (x) - J_{1} (x) | δ

(provided

E

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

| x J_{0} (x) - J_{1} (x) |

However, if

δ

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

E

slightly larger than the above relation predicts.

For very large

x

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

J_{1} (x) ≃ \sqrt{\frac{2}{π | x |}} \cos (x - \frac{3 π}{4})

. The amplitude

\sqrt{\frac{2}{π | x |}}

can be calculated with reasonable accuracy for all

x

, but

\cos (x - \frac{3 π}{4})

cannot. If

x - \frac{3 π}{4}

is written as

2 N π + θ

where

N

is an integer and

0 \leq θ < 2 π

, then

\cos (x - \frac{3 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 reciprocal of machine precision, it is impossible to calculate the phase of

J_{1} (x)

and the routine must fail.

Figure 1

8 Parallelism and Performance

s17atf is not threaded in any implementation.

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.

s17at: FL CL CPP AD

NAG FL Interfaces17atf (bessel_​j1_​real_​vector)

▸▿ 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 FL Interface
s17atf (bessel_j1_real_vector)