NAG Library Function Document
nag_complex_erfc (s15ddc) computes values of the function , for Complex .
||nag_complex_erfc (Complex z,
nag_complex_erfc (s15ddc) computes values of the function
is the complementary error function
. The method used is that in Gautschi (1970)
in the first quadrant of the complex plane, and is extended for
in other quadrants via the relations
. 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. ACM 12 635
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
z – ComplexInput
On entry: the argument of the function.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
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
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
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
is less than zero, accuracy may be lost in two ways; firstly, in the evaluation of
is large, in which case a warning will be issued through NE_RESULT_HALF_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
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
by the relations
. (For double arguments, nag_erfc (s15adc)
and nag_erf (s15aec)
should be used.)
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.
9.1 Program Text
Program Text (s15ddce.c)
9.2 Program Data
Program Data (s15ddce.d)
9.3 Program Results
Program Results (s15ddce.r)