nag_complex_bessel_k (s18dcc) returns a sequence of values for the modified Bessel functions  for complex , non-negative  and , with an option for exponential scaling.

nag_complex_bessel_k (s18dcc) evaluates a sequence of values for the modified Bessel function , where  is complex, , and  is the real, non-negative order. The -member sequence is generated for orders , . Optionally, the sequence is scaled by the factor .

The function is derived from the function CBESK in Amos (1986).

Note: although the function may not be called with  less than zero, for negative orders the formula  may be used.

When  is greater than 1, extra values of  are computed using recurrence relations.

For very large  or , argument reduction will cause total loss of accuracy, and so no computation is performed. For slightly smaller  or , the computation is performed but results are accurate to less than half of machine precision. If  is very small, near the machine underflow threshold, or  is too large, there is a risk of overflow and so no computation is performed. In all the above cases, a warning is given by the function.

"NE_BAD_PARAM"

On entry, argument  had an illegal value.

"NE_COMPLEX_ZERO"

 On entry, .

"NE_INT"

 On entry, . Constraint: .

"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 consult NAG for assistance.

"NE_OVERFLOW_LIKELY"

 No computation because .

 No computation because  is too large.

"NE_REAL"

 On entry, . Constraint: .

"NE_TERMINATION_FAILURE"

 No computation – algorithm termination condition not met.

"NE_TOTAL_PRECISION_LOSS"

 No computation because .

 No computation because .

"NW_SOME_PRECISION_LOSS"

 Results lack precision because .

 Results lack precision because .

All constants in nag_complex_bessel_k (s18dcc) are given to approximately 18 digits of precision. Calling 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 . Because of errors in argument reduction when computing elementary functions inside nag_complex_bessel_k (s18dcc), the actual number of correct digits is limited, in general, by , where  represents the number of digits lost due to the argument reduction. Thus the larger the values of  and , the less the precision in the result. If nag_complex_bessel_k (s18dcc) is called with , then computation of function values via recurrence may lead to some further small loss of accuracy.

If function values which should nominally be identical are computed by calls to nag_complex_bessel_k (s18dcc) with different base values of  and different n, the computed values may not agree exactly. Empirical tests with modest values of  and  have shown that the discrepancy is limited to the least significant 3 – 4 digits of precision.

The time taken by nag_complex_bessel_k (s18dcc) for a call of nag_complex_bessel_k (s18dcc) is approximately proportional to the value of n, plus a constant. In general it is much cheaper to call nag_complex_bessel_k (s18dcc) with n greater than 1, rather than to make  separate calls to nag_complex_bessel_k (s18dcc).

Paradoxically, for some values of  and , it is cheaper to call nag_complex_bessel_k (s18dcc) with a larger value of n than is required, and then discard the extra function values returned. However, it is not possible to state the precise circumstances in which this is likely to occur. It is due to the fact that the base value used to start recurrence may be calculated in different regions for different n, and the costs in each region may differ greatly.

Note that if the function required is  or , i.e.,  or , where  is real and positive, and only a single function value is required, then it may be much cheaper to call s18acc (nag_bessel_k0), s18adc (nag_bessel_k1), s18ccc (nag_bessel_k0_scaled) or s18cdc (nag_bessel_k1_scaled), depending on whether a scaled result is required or not.