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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 2.7 in How to Use the NAG Library and its Documentation).

6 Error 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.

An unexpected error has been triggered by this function. Please contact NAG.
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, $x = 〈value〉$ 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

nag_erfcx (s15agc) is not threaded in any implementation.

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)