NAG FL Interface
s15agf (erfcx_​real)

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1 Purpose

s15agf returns the value of the scaled complementary error function erfcx(x), via the function name.

2 Specification

Fortran Interface
Function s15agf ( x, ifail)
Real (Kind=nag_wp) :: s15agf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
double  s15agf_ (const double *x, Integer *ifail)
The routine may be called by the names s15agf or nagf_specfun_erfcx_real.

3 Description

s15agf 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 x02bef).
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 x02alf) and xtiny is the smallest positive model number (see x02akf). In this case s15agf exits with ifail=1 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 s15agf exits with ifail=2.
There is a danger of setting overflow in ex2 whenever x<xneg=-log(xhuge/2). In this case s15agf exits with ifail=3 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 Real (Kind=nag_wp) Input
On entry: the argument x of the function.
2: ifail Integer Input/Output
On entry: ifail must be set to 0, -1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of -1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value -1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value -1 is recommended since useful values can be provided in some output arguments even when ifail0 on exit. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases s15agf may return useful information.
On entry, x=value and the constant xhi=value.
Constraint: x<xhi.
On entry, |x| was in the interval [value,value) where erfcx(x) is approximately 1/(π×|x|): x=value.
On entry, x=value and the constant xneg=value.
Constraint: xxneg.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL 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 FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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 x02bhf 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 s15adf.

8 Parallelism and Performance

s15agf is not threaded in any implementation.

9 Further Comments


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 (s15agfe.f90)

10.2 Program Data

Program Data (s15agfe.d)

10.3 Program Results

Program Results (s15agfe.r)