# NAG FL Interfaces15arf (erfc_​real_​vector)

## 1Purpose

s15arf returns an array of values of the complementary error function, $\mathrm{erfc}\left(x\right)$.

## 2Specification

Fortran Interface
 Subroutine s15arf ( n, x, f,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Out) :: f(n)
#include <nag.h>
 void s15arf_ (const Integer *n, const double x[], double f[], Integer *ifail)
The routine may be called by the names s15arf or nagf_specfun_erfc_real_vector.

## 3Description

s15arf calculates approximate values for the complement of the error function
 $erfcx = 2π ∫x∞ e-t2 dt = 1-erfx ,$
for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
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 $\mathrm{erfc}\left(x\right)$ 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 $\mathrm{erfc}\left(x\right)$ 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
 $erfcx ≈ ex2 x 1π + 1x2 Rℓ,m 1/x2 .$
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).
For $\left|x\right|\ge {x}_{\mathrm{hi}}$ 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 $x\ge {x}_{\mathrm{hi}}$, s15arf 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: $\mathbf{n}$Integer Input
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{f}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: $\mathrm{erfc}\left({x}_{i}\right)$, the function values.
4: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface 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, if you are not familiar with this argument, the recommended value is $0$. 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).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

If $\delta$ and $\epsilon$ 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 $\frac{2x{e}^{-{x}^{2}}}{\sqrt{\pi }\mathrm{erfc}\left(x\right)}$ 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 $x$.

## 8Parallelism and Performance

s15arf is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (s15arfe.f90)

### 10.2Program Data

Program Data (s15arfe.d)

### 10.3Program Results

Program Results (s15arfe.r)