# NAG CL Interfaces15arc (erfc_​real_​vector)

## 1Purpose

s15arc returns an array of values of the complementary error function, $\mathrm{erfc}\left(x\right)$.

## 2Specification

 #include
 void s15arc (Integer n, const double x[], double f[], NagError *fail)
The function may be called by the names: s15arc, nag_specfun_erfc_real_vector or nag_erfc_vector.

## 3Description

s15arc 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 X02BEC).
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 $x\ge {x}_{\mathrm{hi}}$, s15arc 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: $\mathbf{x}\left[{\mathbf{n}}\right]$const double Input
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{f}\left[{\mathbf{n}}\right]$double Output
On exit: $\mathrm{erfc}\left({x}_{i}\right)$, the function values.
4: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL 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

s15arc 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 (s15arce.c)

### 10.2Program Data

Program Data (s15arce.d)

### 10.3Program Results

Program Results (s15arce.r)