# NAG Library Routine Document

## 1Purpose

s15agf returns the value of the scaled complementary error function $\mathrm{erfcx}\left(x\right)$, via the function name.

## 2Specification

Fortran Interface
 Function s15agf ( x,
 Real (Kind=nag_wp) :: s15agf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nagmk26.h>
 double s15agf_ (const double *x, Integer *ifail)

## 3Description

s15agf 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 $\stackrel{^}{x}$ be the root of the equation $\mathrm{erfc}\left(x\right)-\mathrm{erf}\left(x\right)=0$ (then $\stackrel{^}{x}\approx 0.46875$). For $\left|x\right|\le \stackrel{^}{x}$ the value of $\mathrm{erfcx}\left(x\right)$ is based on the following rational Chebyshev expansion for $\mathrm{erf}\left(x\right)$:
 $erfx ≈ xRℓ,m x2 ,$
where ${R}_{\ell ,m}$ denotes a rational function of degree $\ell$ in the numerator and $m$ in the denominator.
For $\left|x\right|>\stackrel{^}{x}$ the value of $\mathrm{erfcx}\left(x\right)$ is based on a rational Chebyshev expansion for $\mathrm{erfc}\left(x\right)$: for $\stackrel{^}{x}<\left|x\right|\le 4$ the value is based on the expansion
 $erfcx ≈ ex2 Rℓ,m x ;$
and for $\left|x\right|>4$ it is based on the expansion
 $erfcx ≈ ex2x 1π + 1x2 Rℓ,m 1/x2 .$
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).
Asymptotically, $\mathrm{erfcx}\left(x\right)\sim 1/\left(\sqrt{\pi }\left|x\right|\right)$. There is a danger of setting underflow in $\mathrm{erfcx}\left(x\right)$ whenever $x\ge {x}_{\mathrm{hi}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({x}_{\mathrm{huge}},1/\left(\sqrt{\pi }{x}_{\mathrm{tiny}}\right)\right)$, where ${x}_{\mathrm{huge}}$ is the largest positive model number (see x02alf) and ${x}_{\mathrm{tiny}}$ is the smallest positive model number (see x02akf). In this case s15agf exits with ${\mathbf{ifail}}={\mathbf{1}}$ and returns $\mathrm{erfcx}\left(x\right)=0$. For $x$ in the range $1/\left(2\sqrt{\epsilon }\right)\le x<{x}_{\mathrm{hi}}$, where $\epsilon$ is the machine precision, the asymptotic value $1/\left(\sqrt{\pi }\left|x\right|\right)$ is returned for $\mathrm{erfcx}\left(x\right)$ and s15agf exits with ${\mathbf{ifail}}={\mathbf{2}}$.
There is a danger of setting overflow in ${e}^{{x}^{2}}$ whenever $x<{x}_{\mathrm{neg}}=-\sqrt{\mathrm{log}\left({x}_{\mathrm{huge}}/2\right)}$. In this case s15agf exits with ${\mathbf{ifail}}={\mathbf{3}}$ and returns $\mathrm{erfcx}\left(x\right)={x}_{\mathrm{huge}}$.
The values of ${x}_{\mathrm{hi}}$, $1/\left(2\sqrt{\epsilon }\right)$ and ${x}_{\mathrm{neg}}$ are given in the Users' Note for your implementation.

## 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:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value  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

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Note: s15agf may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$ and the constant ${x}_{\mathrm{hi}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}<{x}_{\mathrm{hi}}$.
${\mathbf{ifail}}=2$
On entry, $\left|{\mathbf{x}}\right|$ was in the interval $\left[〈\mathit{\text{value}}〉,〈\mathit{\text{value}}〉\right)$ where $\mathrm{erfcx}\left({\mathbf{x}}\right)$ is approximately $1/\left(\sqrt{\pi }×\left|{\mathbf{x}}\right|\right)$: ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$ and the constant ${x}_{\mathrm{neg}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\ge {x}_{\mathrm{neg}}$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The relative error in computing $\mathrm{erfcx}\left(x\right)$ may be estimated by evaluating
 $E= erfcxx - ex2 ∑ n=1 ∞ Inerfcx erfcxx ,$
where ${I}^{n}$ denotes repeated integration. Empirical results suggest that on the interval $\left(\stackrel{^}{x},2\right)$ 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 $\left(2,20\right)$ the values are around $3.5$ for maximum and $0.45$ for root-mean-square relative errors; note that on these two intervals $\mathrm{erfc}\left(x\right)$ is the primary computation. See also Section 7 in s15adf.

## 8Parallelism and Performance

s15agf is not threaded in any implementation.

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.

### 10.1Program Text

Program Text (s15agfe.f90)

### 10.2Program Data

Program Data (s15agfe.d)

### 10.3Program Results

Program Results (s15agfe.r)