nag_complex_airy_bi (s17dhc) returns the value of the Airy function  or its derivative  for complex , with an option for exponential scaling.

nag_complex_airy_bi (s17dhc) returns a value for the Airy function  or its derivative , where  is complex, . Optionally, the value is scaled by the factor .

The function is derived from the function CBIRY in Amos (1986). It is based on the relations , and , where  is the modified Bessel function and .

For very large , argument reduction will cause total loss of accuracy, and so no computation is performed. For slightly smaller , the computation is performed but results are accurate to less than half of machine precision. If  is too large, and the unscaled function is required, 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_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  is too large when .

"NE_TERMINATION_FAILURE"

 No computation – algorithm termination condition not met.

"NE_TOTAL_PRECISION_LOSS"

 No computation because .

"NW_SOME_PRECISION_LOSS"

 Results lack precision because .

All constants in nag_complex_airy_bi (s17dhc) 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_airy_bi (s17dhc), 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 value of , the less the precision in the result.

Empirical tests with modest values of , checking relations between Airy functions , ,  and , have shown errors limited to the least significant 3 – 4 digits of precision.

Note that if the function is required to operate on a real argument only, then it may be much cheaper to call s17ahc (nag_airy_bi) or s17akc (nag_airy_bi_deriv).