The function may be called by the names: s15ddc, nag_specfun_erfc_complex or 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 decimal places. The real part of is sometimes known as the Voigt function.
Gautschi W (1969) Algorithm 363: Complex error function Comm. ACM12 635
Gautschi W (1970) Efficient computation of the complex error function SIAM J. Numer. Anal.7 187–198
van der Laan C G and Temme N M (1984) Calculation of special functions: the gamma function, the exponential integrals and error-like functions CWI Tract 10 Centre for Mathematics and Computer Science, Amsterdam
1: – ComplexInput
On entry: the argument of the function.
2: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
Result has less than half precision when entered with argument .
Imaginary part of result overflows when entered with argument .
Result has no precision when entered with argument .
Both real and imaginary parts of result overflow when entered with argument .
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 decimal places on the achievable accuracy because constants in the function are given to this precision. With such arguments, fail can only return as NE_NOERROR.
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 or NE_RESULT_NO_PRECISION; 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_IMAGINARY_OVERFLOW, NE_RESULT_OVERFLOW or NE_RESULT_REAL_OVERFLOW.
8Parallelism and Performance
s15ddc is not threaded in any implementation.
The time taken for a call of s15ddc depends on the argument , the time increasing as .
s15ddc may be used to compute values of and for Complex by the relations , . (For double arguments, s15adcands15aec should be used.)
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.