Unless stated otherwise in the Users' Note, s15adf calls the complementary error function supplied by the compiler used for your implementation; as such, details of the underlying algorithm should be obtained from the documentation supplied by the compiler vendor. The following discussion only applies if the Users' Note for your implementation indicates that the compiler's supplied function was not available.

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

erfc (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

erfc (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).

For

| x | \geq x_{hi}

there is a danger of setting underflow in

erfc (x)

(the value of

x_{hi}

is given in the Users' Note for your implementation).. For

x \geq x_{hi}

, s15adf returns

erfc (x) = 0

; for

x \leq - x_{hi}

it returns

erfc (x) = 2

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: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ 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 $−1$ or $1$ 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

There are no failure exits from s15adf. The argument ifail has been included for consistency with other routines in this chapter.

7 Accuracy

Unless stated otherwise in the Users' Note, s15adf calls the complementary error function supplied by the compiler used for your implementation. The following discussion only applies if the Users' Note for your implementation indicates that the compiler's supplied function was not available.

δ

and

ε

are relative errors in the argument and result, respectively, then in principle

| ε | ≃ | \frac{2 x e^{- x^{2}}}{\sqrt{π} erfc (x)} δ | .

That is, the relative error in the argument,

x

, is amplified by a factor

\frac{2 x e^{- x^{2}}}{\sqrt{π} erfc (x)}

in the result.

The behaviour of this factor is shown in Figure 1.

Figure 1

It should be noted that near

x = 0

this factor behaves as

\frac{2 x}{\sqrt{π}}

and hence the accuracy is largely determined by the machine precision. Also, for large negative

x

, where the factor is

\sim \frac{x e^{- x^{2}}}{\sqrt{π}}

, accuracy is mainly limited by machine precision. However, for large positive

x

, the factor becomes

\sim 2 x^{2}

and to an extent relative accuracy is necessarily lost. The absolute accuracy

E

is given by

E ≃ \frac{2 x e^{- x^{2}}}{\sqrt{π}} δ

so absolute accuracy is guaranteed for all

x

8 Parallelism and Performance

s15adf is not threaded in any implementation.

9 Further Comments

9.1 Internal Changes

Internal changes have been made to this routine as follows:

At Mark 27.1:
Modified to use a compiler-supplied erfc function, when available.

For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.

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.

s15ad: FL CL CPP AD

NAG FL Interfaces15adf (erfc_​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

9.1 Internal Changes

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
s15adf (erfc_real)