Interfaces: FL CL AD

s15ar: FL CL AD

NAG FL Interface
s15arf (erfc_real_vector)

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NAG Library Manual, Mark 27

Interfaces: FL CL AD

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s15ar: FL CL AD

▸▿ 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

s15arf returns an array of values of the complementary error function,

erfc (x)

2 Specification

Fortran Interface

Subroutine s15arf (

n, x, f, ifail)

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

C Header Interface

#include <nag.h>

void	s15arf_ (const Integer n, const double x[], double f[], Integer ifail)

The routine may be called by the names s15arf or nagf_specfun_erfc_real_vector.

3 Description

s15arf calculates approximate values for the complement of the error function

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

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}

, s15arf 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: $n$ – 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: $erfc (x_{i})$ , the function values.
4: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface 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, if you are not familiar with this argument, the recommended value is $0$ . 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, $n = 〈value〉$ .
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

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

s15arf is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example reads values of x from a file, evaluates the function at each value of

x_{i}

and prints the results.

Interfaces: FL CL AD

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s15ar: FL CL AD

NAG FL Interfaces15arf (erfc_​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
s15arf (erfc_real_vector)