# NAG C Library Function Document

## 1Purpose

nag_erfcx (s15agc) returns the value of the scaled complementary error function $\mathrm{erfcx}\left(x\right)$.

## 2Specification

 #include #include
 double nag_erfcx (double x, NagError *fail)

## 3Description

nag_erfcx (s15agc) 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 nag_decimal_digits (X02BEC)).
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 nag_real_largest_number (X02ALC)) and ${x}_{\mathrm{tiny}}$ is the smallest positive model number (see nag_real_smallest_number (X02AKC)). In this case nag_erfcx (s15agc) exits with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_HI 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 nag_erfcx (s15agc) exits with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_REAL.
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 nag_erfcx (s15agc) exits with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_NEG 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:    $\mathbf{x}$doubleInput
On entry: the argument $x$ of the function.
2:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NW_HI
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$ and the constant ${x}_{\mathrm{hi}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}<{x}_{\mathrm{hi}}$.
NW_NEG
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$ and the constant ${x}_{\mathrm{neg}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\ge {x}_{\mathrm{neg}}$.
NW_REAL
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 $1/\left(\sqrt{\pi }×\left|{\mathbf{x}}\right|\right)$: ${\mathbf{x}}=〈\mathit{\text{value}}〉$.

## 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 nag_real_base (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 nag_erfc (s15adc).

## 8Parallelism and Performance

nag_erfcx (s15agc) 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 (s15agce.c)

### 10.2Program Data

Program Data (s15agce.d)

### 10.3Program Results

Program Results (s15agce.r)