naginterfaces.library.specfun.erfc_​complex_​vector

naginterfaces.library.specfun.erfc_complex_vector(z)

erfc_complex_vector computes values of the function $w\left(z\right)=e^{-z^2}erfc(-iz)$, for an array of complex values $z$.

For full information please refer to the NAG Library document for s15dr

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/s/s15drf.html

Parameters

zcomplex, array-like, shape $\left(n\right)$

The argument $z_{\text{i}}$ of the function, for $\text{i}=1,2,\dots ,n$.

Returns

fcomplex, ndarray, shape $\left(n\right)$

$w\left(z_i\right)=e^{-z_i^2}$, the function values.

ivalidint, ndarray, shape $\left(n\right)$

$ivalid[\text{i}-1]$ contains the error code for $z_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

$ivalid[i-1]=0$

No error.

$ivalid[i-1]=1$

Real part of result overflows.

$ivalid[i-1]=2$

Imaginary part of result overflows.

$ivalid[i-1]=3$

Both real and imaginary part of result overflows.

$ivalid[i-1]=4$

Result has less than half precision.

$ivalid[i-1]=5$

Result has no precision.

Raises

NagValueError

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, at least one value of $z$ produced a result with reduced accuracy.

Check $ivalid$ for more information.

Notes

erfc_complex_vector computes values of the function $w\left(z_{\text{i}}\right)=e^{-z_{\text{i}}^2}erfc(-\text{i}{z}_{\text{i}})$, for $\text{i}=1,2,\dots ,n$, where $erfc\left(z_i\right)$ is the complementary error function

$$erfc\left(z\right)=\frac{2}{\sqrt{\pi }}{\int }_z^{\infty }{e}^{-t^2}dt\text{,}$$

for complex $z$. The method used is that in Gautschi (1970) for $z$ in the first quadrant of the complex plane, and is extended for $z$ in other quadrants via the relations $w(-z)=2e^{-z^2}-w\left(z\right)$ and $w\left(\overline{z}\right)=\overline{w(-z)}$. Following advice in Gautschi (1970) and van der Laan and Temme (1984), the code in Gautschi (1969) has been adapted to work in various precisions up to $18$ decimal places. The real part of $w\left(z\right)$ is sometimes known as the Voigt function.

References

Gautschi, W, 1969, Algorithm 363: Complex error function, Comm. ACM (12), 635

Gautschi, W, 1970, Efficient computation of the complex error function, SIAM J. Numer. Anal. (7), 187–198

van der Laan, C G and Temme, N M, 1984, Calculation of special functions: the gamma function, the exponential integrals and error-like functions, CWI Tract 10, Centre for Mathematics and Computer Science, Amsterdam