NAG Library Function Document

nag_erfc (s15adc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_erfc (s15adc) returns the value of the complementary error function, erfcx.

2
Specification

#include <nag.h>
#include <nags.h>
double  nag_erfc (double x)

3
Description

nag_erfc (s15adc) calculates an approximate value for the complement of the error function
erfcx=2πxe-t2dt=1-erfx.  
Let x^ be the root of the equation erfcx-erfx=0 (then x^0.46875). For xx^ the value of erfcx is based on the following rational Chebyshev expansion for erfx:
erfxxR,mx2,  
where R,m denotes a rational function of degree  in the numerator and m in the denominator.
For x>x^ the value of erfcx is based on a rational Chebyshev expansion for erfcx: for x^<x4 the value is based on the expansion
erfcxex2R,mx;  
and for x>4 it is based on the expansion
erfcxex2x1π+1x2R,m1/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 nag_decimal_digits (X02BEC)).
For xxhi there is a danger of setting underflow in erfcx (the value of xhi is given in the Users' Note for your implementation). For xxhi, nag_erfc (s15adc) returns erfcx=0; for x-xhi it returns erfcx=2.

4
References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Cody W J (1969) Rational Chebyshev approximations for the error function Math.Comp. 23 631–637

5
Arguments

1:     x doubleInput
On entry: the argument x of the function.

6
Error Indicators and Warnings

None.

7
Accuracy

If δ and ε are relative errors in the argument and result, respectively, then in principle
ε 2x e -x2 πerfcx δ .  
That is, the relative error in the argument, x, is amplified by a factor 2xe-x2 πerfcx  in the result.
The behaviour of this factor is shown in Figure 1.
Figure 1
Figure 1
It should be noted that near x=0 this factor behaves as 2xπ  and hence the accuracy is largely determined by the machine precision. Also for large negative x, where the factor is xe-x2π , accuracy is mainly limited by machine precision. However, for large positive x, the factor becomes 2x2 and to an extent relative accuracy is necessarily lost. The absolute accuracy E is given by
E2xe-x2πδ  
so absolute accuracy is guaranteed for all x.

8
Parallelism and Performance

nag_erfc (s15adc) 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 (s15adce.c)

10.2
Program Data

Program Data (s15adce.d)

10.3
Program Results

Program Results (s15adce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017