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NAG C Library Manual

# NAG Library Function Documentnag_erfcx (s15agc)

## 1  Purpose

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

## 2  Specification

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

## 3  Description

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 $\mathrm{erf}\left(x\right)$:
 $erfx≈xRℓ,mx2,$
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
 $erfcx≈ex2Rℓ,mx;$
and for $\left|x\right|>4$ it is based on the expansion
 $erfcx≈ex2x1π+1x2Rℓ,m1/x2.$
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, $\mathrm{erfcx}\left(x\right)\sim 1/\left(\sqrt{\pi }\mathrm{abs}\left(x\right)\right)$. 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 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 $1/\left(\sqrt{\pi }\mathrm{abs}\left(x\right)\right)$ is returned for $\mathrm{erfcx}\left(x\right)$ and nag_erfcx (s15agc) exits with 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 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.

## 4  References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Cody W J (1969) Rational Chebyshev approximations for the error function Math.Comp. 23 631–637

## 5  Arguments

1:     xdoubleInput
On entry: the argument $x$ of the function.
2:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

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.
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 $1/\left(\sqrt{\pi }*\left|{\mathbf{x}}\right|\right)$: ${\mathbf{x}}=〈\mathit{\text{value}}〉$.

## 7  Accuracy

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).

None.

## 9  Example

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

### 9.1  Program Text

Program Text (s15agce.c)

### 9.2  Program Data

Program Data (s15agce.d)

### 9.3  Program Results

Program Results (s15agce.r)