NAG Library Routine Document

s15agf (erfcx_real)

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NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

NAG C Library Manual, Mark 26

S (specfun) Chapter Contents

S (specfun) Chapter Introduction

▸▿ 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

s15agf returns the value of the scaled complementary error function

erfcx (x)

, via the function name.

2

Specification

Fortran Interface

Function s15agf (

x, ifail)

Real (Kind=nag_wp)	::	s15agf
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x

C Header Interface

#include <nagmk26.h>

double

s15agf_ (const double *x, Integer *ifail)

3

Description

s15agf 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 x02bef).

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 x02alf) and

x_{tiny}

is the smallest positive model number (see x02akf). In this case s15agf exits with

ifail = 1

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 s15agf exits with

ifail = 2

There is a danger of setting overflow in

e^{x^{2}}

whenever

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

. In this case s15agf exits with

ifail = 3

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

NIST Digital Library of Mathematical Functions

Cody W J (1969) Rational Chebyshev approximations for the error function Math.Comp. 23 631–637

5

Arguments

1: $x$ – Real (Kind=nag_wp)Input: On entry: the argument $x$ of the function.
2: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if $ifail \neq 0$ on exit, the recommended value is $- 1$ . When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Note: s15agf may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $x = 〈value〉$ and the constant $x_{hi} = 〈value〉$ .
Constraint: $x < x_{hi}$ .

$ifail = 2$: On entry, $|x|$ was in the interval $[〈value〉, 〈value〉)$ where $erfcx (x)$ is approximately $1 / (\sqrt{π} \times |x|)$ : $x = 〈value〉$ .

$ifail = 3$: On entry, $x = 〈value〉$ and the constant $x_{neg} = 〈value〉$ .
Constraint: $x \geq x_{neg}$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

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 x02bhf 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 s15adf.

8

Parallelism and Performance

s15agf 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 Routine Document

s15agf (erfcx_real)

▸▿ 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

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