# NAG CL Interfaces15auc (erfcx_​real_​vector)

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## 1Purpose

s15auc returns an array of values of the scaled complementary error function $\mathrm{erfcx}\left(x\right)$.

## 2Specification

 #include
 void s15auc (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)
The function may be called by the names: s15auc, nag_specfun_erfcx_real_vector or nag_erfcx_vector.

## 3Description

s15auc calculates approximate values for the scaled complementary error function
 $erfcx(x) = e x2 erfc(x) = 2 π e x2 ∫x∞ e -t2 dt = e x2 (1-erf(x)) ,$
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 $|x|\le \stackrel{^}{x}$ the value of $\mathrm{erfcx}\left(x\right)$ is based on the following rational Chebyshev expansion for $\mathrm{erf}\left(x\right)$:
 $erf(x) ≈ xRℓ,m (x2) ,$
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 $\mathrm{erfcx}\left(x\right)$ 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
 $erfc(x) ≈ ex2x (1π+1x2Rℓ,m(1/x2)) .$
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 X02BEC).
Asymptotically, $\mathrm{erfcx}\left(x\right)\sim 1/\left(\sqrt{\pi }|x|\right)$. 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 X02ALC) and ${x}_{\mathrm{tiny}}$ is the smallest positive model number (see X02AKC). In this case s15auc exits with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_IVALID and returns ${\mathbf{ivalid}}\left[i-1\right]=1$ with $\mathrm{erfcx}\left({x}_{i}\right)=0$. 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 $1/\left(\sqrt{\pi }|x|\right)$ is returned for $\mathrm{erfcx}\left({x}_{i}\right)$, ${\mathbf{ivalid}}\left[i-1\right]=2$, and s15auc exits with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_IVALID.
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 s15auc exits with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_IVALID and returns ${\mathbf{ivalid}}\left[i-1\right]=3$ with $\mathrm{erfcx}\left({x}_{i}\right)={x}_{\mathrm{huge}}$.
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: $\mathbf{n}$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: $\mathrm{erfcx}\left({x}_{i}\right)$, the function values.
4: $\mathbf{ivalid}\left[{\mathbf{n}}\right]$Integer Output
On exit: ${\mathbf{ivalid}}\left[\mathit{i}-1\right]$ contains the error code for ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
${x}_{i}$ is too large and positive. The threshold value is the same as for ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_HI in s15agc , as defined in the Users' Note for your implementation.
${\mathbf{ivalid}}\left[i-1\right]=2$
${x}_{i}$ was in the interval $\left[1/\left(2\sqrt{\epsilon }\right),{x}_{\mathrm{hi}}\right)$. The threshold values are the same as for ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_REAL in s15agc , as defined in the Users' Note for your implementation.
${\mathbf{ivalid}}\left[i-1\right]=3$
${x}_{i}$ is too small and positive. The threshold value is the same as for ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_NEG in s15agc , as defined in the Users' Note for your implementation.
5: $\mathbf{fail}$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.
NE_BAD_PARAM
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.
NW_IVALID
On entry, at least one value of x produced a result with reduced accuracy.
Check ivalid for more information.

## 7Accuracy

The relative error in computing $\mathrm{erfcx}\left(x\right)$ may be estimated by evaluating
 $E= erfcx(x) - ex2 ∑ n=1 ∞ Inerfc(x) erfcx(x) ,$
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 X02BHC 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 s15adc.

## 8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s15auc 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 ${x}_{i}$ and prints the results.

### 10.1Program Text

Program Text (s15auce.c)

### 10.2Program Data

Program Data (s15auce.d)

### 10.3Program Results

Program Results (s15auce.r)