nag_complex_erfc (s15ddc) computes values of the function , for complex .

nag_complex_erfc (s15ddc) computes values of the function , where  is the complementary error function

for complex . The method used is that in Gautschi (1970) for  in the first quadrant of the complex plane, and is extended for  in other quadrants via the relations  and . Following advice in Gautschi (1970) and van der Laan and Temme (1984), the code in Gautschi (1969) has been adapted to work in various precisions up to 18 decimal places. The real part of  is sometimes known as the Voigt function.

"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_RESULT_HALF_PRECISION"

 Result has less than half precision when entered with argument .

"NE_RESULT_IMAGINARY_OVERFLOW"

 Imaginary part of result overflows when entered with argument .

"NE_RESULT_NO_PRECISION"

 Result has no precision when entered with argument .

"NE_RESULT_OVERFLOW"

 Both real and imaginary parts of result overflow when entered with argument .

"NE_RESULT_REAL_OVERFLOW"

 Real part of result overflows when entered with argument .

The accuracy of the returned result depends on the argument . If  lies in the first or second quadrant of the complex plane (i.e.,  is greater than or equal to zero), the result should be accurate almost to machine precision, except that there is a limit of about 18 decimal places on the achievable accuracy because constants in the function are given to this precision. With such arguments, fail can only return as no error is raised.

If however  is less than zero, accuracy may be lost in two ways; firstly, in the evaluation of , if  is large, in which case a warning will be issued through "NE_RESULT_HALF_PRECISION" raised or "NE_RESULT_NO_PRECISION" raised ; and secondly, near the zeros of the required function, where precision is lost due to cancellation, in which case no warning is given – the result has absolute accuracy rather than relative accuracy. Note also that in this half-plane, one or both parts of the result may overflow – this is signalled through "NE_RESULT_REAL_OVERFLOW" raised , "NE_RESULT_IMAGINARY_OVERFLOW" raised or "NE_RESULT_OVERFLOW" raised .

The time taken for a call of nag_complex_erfc (s15ddc) depends on the argument , the time increasing as .

nag_complex_erfc (s15ddc) may be used to compute values of  and  for complex  by the relations , . (For double arguments, s15adc (nag_erfc) and s15aec (nag_erf) should be used.)