NAG CL Interface
s15agc (erfcx_​real)

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1 Purpose

s15agc returns the value of the scaled complementary error function erfcx(x).

2 Specification

#include <nag.h>
double  s15agc (double x, NagError *fail)
The function may be called by the names: s15agc, nag_specfun_erfcx_real or nag_erfcx.

3 Description

s15agc calculates an approximate value for the scaled complementary error function
erfcx(x) = e x2 erfc(x) = 2 π e x2 x e -t2 dt = e x2 (1-erf(x)) .  
Let x^ be the root of the equation erfc(x)-erf(x)=0 (then x^0.46875). For |x|x^ the value of erfcx(x) is based on the following rational Chebyshev expansion for erf(x):
erf(x) xR,m (x2) ,  
where R,m denotes a rational function of degree in the numerator and m in the denominator.
For |x|>x^ the value of erfcx(x) is based on a rational Chebyshev expansion for erfc(x): for x^<|x|4 the value is based on the expansion
erfc(x) ex2 R,m (x) ;  
and for |x|>4 it is based on the expansion
erfc(x) ex2x (1π+1x2R,m(1/x2)) .  
For each expansion, the specific values of and m are selected to be minimal such that the maximum relative error in the expansion is of the order 10-d, where d is the maximum number of decimal digits that can be accurately represented for the particular implementation (see X02BEC).
Asymptotically, erfcx(x)1/(π|x|). There is a danger of setting underflow in erfcx(x) whenever xxhi=min(xhuge,1/(πxtiny)), where xhuge is the largest positive model number (see X02ALC) and xtiny is the smallest positive model number (see X02AKC). In this case s15agc exits with fail.code= NW_HI and returns erfcx(x)=0. For x in the range 1/(2ε)x<xhi, where ε is the machine precision, the asymptotic value 1/(π|x|) is returned for erfcx(x) and s15agc exits with fail.code= NW_REAL.
There is a danger of setting overflow in ex2 whenever x<xneg=-log(xhuge/2). In this case s15agc exits with fail.code= NW_NEG and returns erfcx(x)=xhuge.
The values of xhi, 1/(2ε) and xneg are given in the Users' Note for your implementation.

4 References

NIST Digital Library of Mathematical Functions
Cody W J (1969) Rational Chebyshev approximations for the error function Math.Comp. 23 631–637

5 Arguments

1: x double Input
On entry: the argument x of the function.
2: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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 contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
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.
NW_HI
On entry, x=value and the constant xhi=value.
Constraint: x<xhi.
NW_NEG
On entry, x=value and the constant xneg=value.
Constraint: xxneg.
NW_REAL
On entry, |x| was in the interval [value,value) where erfcx(x) is approximately 1/(π×|x|): x=value.

7 Accuracy

The relative error in computing erfcx(x) may be estimated by evaluating
E= erfcx(x) - ex2 n=1 Inerfc(x) erfcx(x) ,  
where In denotes repeated integration. Empirical results suggest that on the interval (x^,2) the loss in base b significant digits for maximum relative error is around 3.3, while for root-mean-square relative error on that interval it is 1.2 (see X02BHC for the definition of the model parameter b). On the interval (2,20) the values are around 3.5 for maximum and 0.45 for root-mean-square relative errors; note that on these two intervals erfc(x) is the primary computation. See also Section 7 in s15adc.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s15agc is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.

10.1 Program Text

Program Text (s15agce.c)

10.2 Program Data

Program Data (s15agce.d)

10.3 Program Results

Program Results (s15agce.r)