nag_erfcx (s15agc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_erfcx (s15agc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_erfcx (s15agc) returns the value of the scaled complementary error function erfcxx.

2  Specification

#include <nag.h>
#include <nags.h>
double  nag_erfcx (double x, NagError *fail)

3  Description

nag_erfcx (s15agc) calculates an approximate value for the scaled complementary error function
erfcxx = e x2 erfcx = 2 π e x2 x e-t2 dt = e x2 1- erfx .
Let x^ be the root of the equation erfcx-erfx=0 (then x^0.46875). For xx^ the value of erfcxx is based on the following rational Chebyshev expansion for erfx:
where R,m denotes a rational function of degree  in the numerator and m in the denominator.
For x>x^ the value of erfcxx is based on a rational Chebyshev expansion for erfcx: for x^<x4 the value is based on the expansion
and for x>4 it is based on the expansion
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)).
Asymptotically, erfcxx1/πabsx. There is a danger of setting underflow in erfcxx whenever xxhi=minxhuge,1/πxtiny, where xhuge is the largest positive model number (see nag_real_largest_number (X02ALC)) and xtiny is the smallest positive model number (see nag_real_smallest_number (X02AKC)). In this case nag_erfcx (s15agc) exits with fail.code= NW_HI and returns erfcxx=0. For x in the range 1/2εx<xhi, where ε is the machine precision, the asymptotic value 1/πabsx is returned for erfcxx and nag_erfcx (s15agc) exits with fail.code= NW_REAL.
There is a danger of setting overflow in ex2 whenever x<xneg=-logxhuge/2. In this case nag_erfcx (s15agc) exits with fail.code= NW_NEG and returns erfcxx=xhuge.
The values of xhi, 1/2ε and xneg are given in the Users' Note for your implementation.

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:     xdoubleInput
On entry: the argument x of the function.
2:     failNagError *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 for assistance.
On entry, x=value and the constant xhi=value.
Constraint: x<xhi.
On entry, x=value and the constant xneg=value.
Constraint: xxneg.
On entry, x was in the interval value,value where erfcxx is approximately 1/π*x: x=value.

7  Accuracy

The relative error in computing erfcxx may be estimated by evaluating
E= erfcxx - ex2 n=1 Inerfcx erfcxx ,
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 nag_real_base (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 erfcx is the primary computation. See also Section 7 in nag_erfc (s15adc).

8  Further Comments


9  Example

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

9.1  Program Text

Program Text (s15agce.c)

9.2  Program Data

Program Data (s15agce.d)

9.3  Program Results

Program Results (s15agce.r)

nag_erfcx (s15agc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012