NAG CL Interfaces15atc (dawson_​vector)

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1Purpose

s15atc returns an array of values for Dawson's Integral, $F\left(x\right)$.

2Specification

 #include
 void s15atc (Integer n, const double x[], double f[], NagError *fail)
The function may be called by the names: s15atc, nag_specfun_dawson_vector or nag_dawson_vector.

3Description

s15atc 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 function 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: $\mathbf{x}\left[{\mathbf{n}}\right]$const double Input
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{f}\left[{\mathbf{n}}\right]$double Output
On exit: $F\left({x}_{i}\right)$, the function values.
4: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error 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.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
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.

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

s15atc 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 (s15atce.c)

10.2Program Data

Program Data (s15atce.d)

10.3Program Results

Program Results (s15atce.r)