Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ivalid(n)
Real (Kind=nag_wp), Intent (In)	::	x(n)
Real (Kind=nag_wp), Intent (Out)	::	f(n)

C Header Interface

#include <nag.h>

void	s15auf_ (const Integer n, const double x[], double f[], Integer ivalid[], Integer ifail)

The routine may be called by the names s15auf or nagf_specfun_erfcx_real_vector.

3 Description

s15auf calculates approximate values 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)),

for an array of arguments

x_{i}

, for

i = 1, 2, \dots, n

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 s15auf exits with

ifail = 1

and returns

ivalid (i) = 1

with

erfcx (x_{i}) = 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_{i})

ivalid (i) = 2

, and s15auf exits with

ifail = 1

There is a danger of setting overflow in

e^{x^{2}}

whenever

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

. In this case s15auf exits with

ifail = 1

and returns

ivalid (i) = 3

with

erfcx (x_{i}) = 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: $n$ – Integer Input

On entry:

n

, the number of points.

Constraint:

n \geq 0

2: $x (n)$ – Real (Kind=nag_wp) array Input

On entry: the argument

x_{i}

of the function, for

i = 1, 2, \dots, n

3: $f (n)$ – Real (Kind=nag_wp) array Output

On exit:

erfcx (x_{i})

, the function values.

4: $ivalid (n)$ – Integer array Output

On exit:

ivalid (i)

contains the error code for

x_{i}

, for

i = 1, 2, \dots, n

$ivalid (i) = 0$: No error.
$ivalid (i) = 1$: $x_{i}$ is too large and positive. The threshold value is the same as for $ifail = 1$ in s15agf , as defined in the Users' Note for your implementation.
$ivalid (i) = 2$: $x_{i}$ was in the interval $[1 / (2 \sqrt{ε}), x_{hi})$ . The threshold values are the same as for $ifail = 2$ in s15agf , as defined in the Users' Note for your implementation.
$ivalid (i) = 3$: $x_{i}$ is too small and positive. The threshold value is the same as for $ifail = 3$ in s15agf , as defined in the Users' Note for your implementation.

5: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. 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).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, at least one value of x produced a result with reduced accuracy.
Check ivalid for more information.

$ifail = 2$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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

Background information to multithreading can be found in the Multithreading documentation.

s15auf 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_{i}

and prints the results.

s15au: FL CL CPP AD PY MB

NAG FL Interfaces15auf (erfcx_​real_​vector)

▸▿ 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 FL Interface
s15auf (erfcx_real_vector)