nag_airy_bi (s17ahc) returns a value of the Airy function .

nag_airy_bi (s17ahc) evaluates an approximation to the Airy function . It is based on a number of Chebyshev expansions.

For large negative arguments, it is impossible to calculate the phase of the oscillating function with any accuracy so the function evaluation must fail. This occurs if , where  is the machine precision.

For large positive arguments, there is a danger of causing overflow since  grows in an essentially exponential manner, so the function must fail.

"NE_REAL_ARG_GT"

On entry, x must not be greater than : . x is too large and positive. The function returns zero.

"NE_REAL_ARG_LT"

On entry, x must not be less than : . x is too large and negative. The function returns zero.

For negative arguments the function is oscillatory and hence absolute error is the appropriate measure. In the positive region the function is essentially exponential-like and here relative error is appropriate. The absolute error, , and the relative error, , are related in principle to the relative error in the argument, , by , .

In practice, approximate equality is the best that can be expected. When ,  or  is of the order of the machine precision, the errors in the result will be somewhat larger.

For small , errors are strongly damped and hence will be bounded essentially by the machine precision.

For moderate to large negative , the error behaviour is clearly oscillatory but the amplitude of the error grows like .

However the phase error will be growing roughly as  and hence all accuracy will be lost for large negative arguments. This is due to difficulty in calculating sin and cos to any accuracy if .

For large positive arguments, the relative error amplification is considerable, .

This means a loss of roughly two decimal places accuracy for arguments in the region of 20. However, very large arguments are not possible due to the danger of causing overflow, and errors are therefore limited in practice.

None.