NAG Library Function Document

nag_erfcx (s15agc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_erfcx (s15agc) returns the value of the scaled complementary error function

erfcx (x)

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_erfcx (double x, NagError *fail)

3 Description

nag_erfcx (s15agc) calculates an approximate value for the scaled complementary error function

erfcx (x) = e^{x^{2}} erfc (x) = \frac{2}{\sqrt{π}} e^{x^{2}} \int_{x}^{\infty} e^{- t^{2}} d t = e^{x^{2}} (1 - \erf (x)) .

Let

\hat{x}

be the root of the equation

erfc (x) - \erf (x) = 0

(then

\hat{x} \approx 0.46875

). For

|x| \leq \hat{x}

the value of

erfcx (x)

is based on the following rational Chebyshev expansion for

\erf (x)

\erf (x) \approx x R_{ℓ, m} (x^{2}),

where

R_{ℓ, m}

denotes a rational function of degree

ℓ

in the numerator and

m

in the denominator.

For

|x| > \hat{x}

the value of

erfcx (x)

is based on a rational Chebyshev expansion for

erfc (x)

: for

\hat{x} < |x| \leq 4

the value is based on the expansion

erfc (x) \approx e^{x^{2}} R_{ℓ, m} (x);

and for

|x| > 4

it is based on the expansion

erfc (x) \approx \frac{e^{x^{2}}}{x} (\frac{1}{\sqrt{π}} + \frac{1}{x^{2}} R_{ℓ, m} (1 / x^{2})) .

For each expansion, the specific values of

ℓ

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,

erfcx (x) \sim 1 / (\sqrt{π} |x|)

. There is a danger of setting underflow in

erfcx (x)

whenever

x \geq x_{hi} = \min (x_{huge}, 1 / (\sqrt{π} x_{tiny}))

, where

x_{huge}

is the largest positive model number (see nag_real_largest_number (X02ALC)) and

x_{tiny}

is the smallest positive model number (see nag_real_smallest_number (X02AKC)). In this case nag_erfcx (s15agc) exits with

fail . code =

NW_HI and returns

erfcx (x) = 0

. For

x

in the range

1 / (2 \sqrt{ε}) \leq x < x_{hi}

, where

ε

is the machine precision, the asymptotic value

1 / (\sqrt{π} |x|)

is returned for

erfcx (x)

and nag_erfcx (s15agc) exits with

fail . code =

NW_REAL.

There is a danger of setting overflow in

e^{x^{2}}

whenever

x < x_{neg} = - \sqrt{\log (x_{huge} / 2)}

. In this case nag_erfcx (s15agc) exits with

fail . code =

NW_NEG and returns

erfcx (x) = x_{huge}

The values of

x_{hi}

1 / (2 \sqrt{ε})

and

x_{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: x – doubleInput: On entry: the argument $x$ of the function.
2: fail – NagError *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, $x = ⟨value⟩$ and the constant $x_{hi} = ⟨value⟩$ .
Constraint: $x < x_{hi}$ .
NW_NEG: On entry, $x = ⟨value⟩$ and the constant $x_{neg} = ⟨value⟩$ .
Constraint: $x \geq x_{neg}$ .
NW_REAL: On entry, $|x|$ was in the interval $[⟨value⟩, ⟨value⟩)$ where $erfcx (x)$ is approximately $1 / (\sqrt{π} * |x|)$ : $x = ⟨value⟩$ .

7 Accuracy

The relative error in computing

erfcx (x)

may be estimated by evaluating

E = \frac{erfcx (x) - e^{x^{2}} \sum_{n = 1}^{\infty} I^{n} erfc (x)}{erfcx (x)},

where

I^{n}

denotes repeated integration. Empirical results suggest that on the interval

(\hat{x}, 2)

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

(2, 20)

the values are around

3.5

for maximum and

0.45

for root-mean-square relative errors; note that on these two intervals

erfc (x)

is the primary computation. See also Section 7 in nag_erfc (s15adc).

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

This example reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

NAG Library Function Documentnag_erfcx (s15agc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_erfcx (s15agc)