nag_bessel_y1 (s17adc) returns the value of the Bessel function .

nag_bessel_y1 (s17adc) evaluates the Bessel function of the second kind, , .

The approximation is based on Chebyshev expansions.

For  near zero, . This approximation is used when  is sufficiently small for the result to be correct to machine precision. For extremely small , there is a danger of overflow in calculating  and for such arguments the function will fail.

For very large , it becomes impossible to provide results with any reasonable accuracy (see Section [Further Comments]), hence the function fails. Such arguments contain insufficient information to determine the phase of oscillation of , only the amplitude, , can be determined and this is returned. The range for which this occurs is roughly related to machine precision; the function will fail if / machine precision.

"NE_REAL_ARG_GT"

On entry, x must not be greater than : . x is too large, the function returns the amplitude of the  oscillation, .

"NE_REAL_ARG_LE"

On entry, x must not be less than or equal to 0.0: .  is undefined, the function returns zero.

"NE_REAL_ARG_TOO_SMALL"

On entry, x must be greater than : . x is too close to zero, there is a danger of overflow, the function returns the value of  at the smallest valid argument.

Let  be the relative error in the argument and  be the absolute error in the result. (Since  oscillates about zero, absolute error and not relative error is significant, except for very small .)

If  is somewhat larger than the machine precision (e.g. if  is due to data errors etc.), then  and  are approximately related by:  (provided  is also within machine bounds).

However, if  is of the same order as machine precision, then rounding errors could make  slightly larger than the above relation predicts.

For very small , absolute error becomes large, but the relative error in the result is of the same order as .

For very large , the above relation ceases to apply. In this region, . The amplitude  can be calculated with reasonable accuracy for all , but  cannot. If  is written as  where  is an integer and , then  is determined by  only. If ,  cannot be determined with any accuracy at all. Thus if  is greater than, or of the order of, the inverse of the machine precision, it is impossible to calculate the phase of  and the function must fail.

None.