NAG CL Interface
s15auc (erfcx_​real_​vector)

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1 Purpose

s15auc returns an array of values of the scaled complementary error function erfcx(x).

2 Specification

#include <nag.h>
void  s15auc (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)
The function may be called by the names: s15auc, nag_specfun_erfcx_real_vector or nag_erfcx_vector.

3 Description

s15auc calculates approximate values 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)) ,  
for an array of arguments xi, for i=1,2,,n.
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 s15auc exits with fail.code= NW_IVALID and returns ivalid[i-1]=1 with erfcx(xi)=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(xi), ivalid[i-1]=2, and s15auc exits with fail.code= NW_IVALID.
There is a danger of setting overflow in ex2 whenever x<xneg=-log(xhuge/2). In this case s15auc exits with fail.code= NW_IVALID and returns ivalid[i-1]=3 with erfcx(xi)=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: n Integer Input
On entry: n, the number of points.
Constraint: n0.
2: x[n] const double Input
On entry: the argument xi of the function, for i=1,2,,n.
3: f[n] double Output
On exit: erfcx(xi), the function values.
4: ivalid[n] Integer Output
On exit: ivalid[i-1] contains the error code for xi, for i=1,2,,n.
ivalid[i-1]=0
No error.
ivalid[i-1]=1
xi is too large and positive. The threshold value is the same as for fail.code= NW_HI in s15agc , as defined in the Users' Note for your implementation.
ivalid[i-1]=2
xi was in the interval [1/(2ε),xhi). The threshold values are the same as for fail.code= NW_REAL in s15agc , as defined in the Users' Note for your implementation.
ivalid[i-1]=3
xi is too small and positive. The threshold value is the same as for fail.code= NW_NEG in s15agc , as defined in the Users' Note for your implementation.
5: 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_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
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_IVALID
On entry, at least one value of x produced a result with reduced accuracy.
Check ivalid for more 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 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

s15auc 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 xi and prints the results.

10.1 Program Text

Program Text (s15auce.c)

10.2 Program Data

Program Data (s15auce.d)

10.3 Program Results

Program Results (s15auce.r)