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## 1Purpose

s15adf returns the value of the complementary error function, $\mathrm{erfc}\left(x\right)$, via the function name.

## 2Specification

Fortran Interface
 Real (Kind=nag_wp) :: s15adf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nag.h>
 double s15adf_ (const double *x, Integer *ifail)
The routine may be called by the names s15adf or nagf_specfun_erfc_real.

## 3Description

s15adf calculates an approximate value for the complement of the error function
 $erfc(x) = 2π ∫x∞ e-t2 dt = 1-erf(x) .$
Let $\stackrel{^}{x}$ be the root of the equation $\mathrm{erfc}\left(x\right)-\mathrm{erf}\left(x\right)=0$ (then $\stackrel{^}{x}\approx 0.46875$). For $|x|\le \stackrel{^}{x}$ the value of $\mathrm{erfc}\left(x\right)$ is based on the following rational Chebyshev expansion for $\mathrm{erf}\left(x\right)$:
 $erf(x) ≈ xRℓ,m (x2) ,$
where ${R}_{\ell ,m}$ denotes a rational function of degree $\ell$ in the numerator and $m$ in the denominator.
For $|x|>\stackrel{^}{x}$ the value of $\mathrm{erfc}\left(x\right)$ is based on a rational Chebyshev expansion for $\mathrm{erfc}\left(x\right)$: for $\stackrel{^}{x}<|x|\le 4$ the value is based on the expansion
 $erfc(x) ≈ ex2 Rℓ,m (x) ;$
and for $|x|>4$ it is based on the expansion
 $erfc(x) ≈ ex2 x (1π+1x2Rℓ,m(1/x2)) .$
For each expansion, the specific values of $\ell$ 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).
For $|x|\ge {x}_{\mathrm{hi}}$ there is a danger of setting underflow in $\mathrm{erfc}\left(x\right)$ (the value of ${x}_{\mathrm{hi}}$ is given in the Users' Note for your implementation).. For $x\ge {x}_{\mathrm{hi}}$, s15adf returns $\mathrm{erfc}\left(x\right)=0$; for $x\le -{x}_{\mathrm{hi}}$ it returns $\mathrm{erfc}\left(x\right)=2$.

## 4References

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

## 5Arguments

1: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the argument $x$ of the function.
2: $\mathbf{ifail}$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 $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

There are no failure exits from s15adf. The argument ifail has been included for consistency with other routines in this chapter.

## 7Accuracy

If $\delta$ and $\epsilon$ are relative errors in the argument and result, respectively, then in principle
 $|ε|≃ | 2x e -x2 πerfc(x) δ| .$
That is, the relative error in the argument, $x$, is amplified by a factor $\frac{2x{e}^{-{x}^{2}}}{\sqrt{\pi }\mathrm{erfc}\left(x\right)}$ in the result.
The behaviour of this factor is shown in Figure 1.
It should be noted that near $x=0$ this factor behaves as $\frac{2x}{\sqrt{\pi }}$ and hence the accuracy is largely determined by the machine precision. Also, for large negative $x$, where the factor is $\text{}\sim \frac{x{e}^{-{x}^{2}}}{\sqrt{\pi }}$, accuracy is mainly limited by machine precision. However, for large positive $x$, the factor becomes $\text{}\sim 2{x}^{2}$ and to an extent relative accuracy is necessarily lost. The absolute accuracy $E$ is given by
 $E ≃ 2xe-x2π δ$
so absolute accuracy is guaranteed for all $x$.

None.

## 10Example

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