# NAG FL Interfaces15atf (dawson_​vector)

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## 1Purpose

s15atf returns an array of values for Dawson's Integral, $F\left(x\right)$.

## 2Specification

Fortran Interface
 Subroutine s15atf ( n, x, f,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Out) :: f(n)
#include <nag.h>
 void s15atf_ (const Integer *n, const double x[], double f[], Integer *ifail)
The routine may be called by the names s15atf or nagf_specfun_dawson_vector.

## 3Description

s15atf evaluates approximations for Dawson's Integral
 $F(x) = e-x2 ∫0x et2 dt ,$
for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
The routine is based on two Chebyshev expansions:
For $0<|x|\le 4$,
 $F(x) = x ∑r=0′ ar Tr (t) , where t=2 (x4) 2 -1 .$
For $|x|>4$,
 $F(x) = 1x ∑r=0′ br Tr (t) , where t=2 (4x) 2 -1 .$
For $|x|$ near zero, $F\left(x\right)\simeq x$, and for $|x|$ large, $F\left(x\right)\simeq \frac{1}{2x}$. These approximations are used for those values of $x$ for which the result is correct to machine precision.

## 4References

NIST Digital Library of Mathematical Functions

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{f}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: $F\left({x}_{i}\right)$, the function values.
4: $\mathbf{ifail}$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 $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

Let $\delta$ and $\epsilon$ be the relative errors in the argument and result respectively.
If $\delta$ is considerably greater than the machine precision (i.e., if $\delta$ is due to data errors etc.), then $\epsilon$ and $\delta$ are approximately related by:
 $ε≃ | x (1-2xF(x)) F(x) | δ.$
The following graph shows the behaviour of the error amplification factor $|\frac{x\left(1-2xF\left(x\right)\right)}{F\left(x\right)}|$:
However, if $\delta$ is of the same order as machine precision, then rounding errors could make $\epsilon$ somewhat larger than the above relation indicates. In fact $\epsilon$ will be largely independent of $x$ or $\delta$, but will be of the order of a few times the machine precision.

## 8Parallelism and Performance

s15atf is not threaded in any implementation.

None.

## 10Example

This example reads values of the argument $x$ from a file, evaluates the function at each value of ${x}_{i}$ and prints the results.

### 10.1Program Text

Program Text (s15atfe.f90)

### 10.2Program Data

Program Data (s15atfe.d)

### 10.3Program Results

Program Results (s15atfe.r)