s15au: FL CL CPP AD

NAG CL Interface
s15auc (erfcx_real_vector)

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NAG Library Manual, Mark 27.3

Interfaces: FL CL CPP AD

NAG CL Interface Introduction

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S (Specfun) Chapter Introduction

s15au: FL CL CPP 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

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1 Purpose

s15auc returns an array of values of the scaled complementary error function

erfcx (x)

2 Specification

#include <nag.h>

void	s15auc (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)

The function may be called by the names: s15auc, nag_specfun_erfcx_real_vector or nag_erfcx_vector.

3 Description

s15auc 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 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 X02ALC) and

x_{tiny}

is the smallest positive model number (see X02AKC). In this case s15auc exits with

fail . code =

NW_IVALID and returns

ivalid [i - 1] = 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 - 1] = 2

, and s15auc exits with

fail . code =

NW_IVALID.

There is a danger of setting overflow in

e^{x^{2}}

whenever

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

. In this case s15auc exits with

fail . code =

NW_IVALID and returns

ivalid [i - 1] = 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: $x [n]$ – const double Input

On entry: the argument

x_{i}

of the function, for

i = 1, 2, \dots, n

3: $f [n]$ – double Output

On exit:

erfcx (x_{i})

, the function values.

4: $ivalid [n]$ – Integer Output

On exit:

ivalid [i - 1]

contains the error code for

x_{i}

, for

i = 1, 2, \dots, n

$ivalid [i - 1] = 0$: No error.
$ivalid [i - 1] = 1$: $x_{i}$ is too large and positive. The threshold value is the same as for $fail . code =$ NW_HI in s15agc , as defined in the Users' Note for your implementation.
$ivalid [i - 1] = 2$: $x_{i}$ was in the interval $[1 / (2 \sqrt{ε}), x_{hi})$ . The threshold values are the same as for $fail . code =$ NW_REAL in s15agc , as defined in the Users' Note for your implementation.
$ivalid [i - 1] = 3$: $x_{i}$ is too small and positive. The threshold value is the same as for $fail . code =$ NW_NEG in s15agc , as defined in the Users' Note for your implementation.

5: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID: On entry, at least one value of x produced a result with reduced accuracy.
Check ivalid for more 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 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 s15adc.

8 Parallelism and Performance

s15auc 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

NAG CL Interfaces15auc (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 CL Interface
s15auc (erfcx_real_vector)