Package 'RcppFaddeeva'

RcppFaddeeva

Description

'Rcpp' Bindings for the 'Faddeeva' Package

Details

Access to a family of Gauss error functions for arbitrary complex arguments is provided via the 'Faddeeva' package by Steven G. Johnson (see <http://ab-initio.mit.edu/wiki/index.php/Faddeeva_Package> for more information).

References

The Faddeeva Package wiki page details the algorithms implemented by Steve G. Johnson, http://ab-initio.mit.edu/wiki/index.php/Faddeeva_Package

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Faddeeva family of error functions of the complex variable

Description

the Faddeeva function

Usage

Faddeeva_w(z, relerr = 0)

erfcx(z, relerr = 0)

erf(z, relerr = 0)

erfi(z, relerr = 0)

erfc(z, relerr = 0)

Dawson(z, relerr = 0)

Arguments

z	complex vector
relerr	double, requested error

Value

complex vector

Functions

• Faddeeva_w(): compute $w(z) = \exp(-z^2) \, \mathrm{erfc}(-iz)$

• erfcx(): compute $\mathrm{erfcx}(z) = \exp(z^2) \, \mathrm{erfc}(z)$

• erf(): compute $\mathrm{erf}(z)$

• erfi(): compute $\mathrm{erfi}(z) = -i \, \mathrm{erf}(iz)$

• erfc(): compute $\mathrm{erfc}(z) = 1 - \mathrm{erf}(z)$

• Dawson(): compute $\mathrm{Dawson}(z) = \sqrt{\pi}/2 \exp(-z^2) \, \mathrm{erfi}(z)$

Examples

Faddeeva_w(1:10 + 1i)
erfcx(1:10 + 1i)
erf(1:10 + 1i)
erfi(1:10 + 1i)
erfc(1:10 + 1i)
Dawson(1:10 + 1i)

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The Voigt function, corresponding to the convolution of a lorentzian and a gaussian distribution

Description

Voigt distribution

Lorentzian distribution

Gaussian distribution

Usage

Voigt(x, x0, sigma, gamma, real = TRUE, ...)

Lorentz(x, x0, gamma)

Gauss(x, x0, sigma)

Arguments

x	numeric vector
x0	scalar, peak position
sigma	parameter of the gaussian
gamma	parameter of the lorentzian
real	logical, return only the real part of the complex Faddeeva
...	passed to Faddeeva_w

Value

numeric or complex vector

Functions

• Voigt(): Voigt lineshape function

• Lorentz(): Lorentzian lineshape function

• Gauss(): Gaussian lineshape function

Author(s)

baptiste Auguie

Examples

## should integrate to 1 in all cases
integrate(Lorentz, -Inf, Inf, x0=200, gamma=100)
integrate(Gauss, -Inf, Inf, x0=200, sigma=50)
integrate(Voigt, -Inf, Inf, x0=200, sigma=50, gamma=100)

## visual comparison
x <- seq(-1000, 1000)
x0 <- 200
l <- Lorentz(x, x0, 30)
g <- Gauss(x, x0, 100)
N <- length(x)
c <- convolve(Gauss(x, 0, 100), 
              rev(Lorentz(x, x0, 30)), type="o")[seq(N/2, length=N)]
v <- Voigt(x, x0, 100, 30)
matplot(x, cbind(v, l, g, c), t="l", lty=c(1,2,2,1), xlab="x", ylab="")
legend("topleft", legend = c("Voigt", "Lorentz", "Gauss", "Convolution"), bty="n",
       lty=c(1,2,2,1), col=1:4)

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