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Error functions and the Faddeeva function

Source: R/erf.R
error-functions.Rd

The error function and its relatives, with the kernels scipy.special uses (the MIT Faddeeva package for complex arguments). Functions marked "complex" take the complex path for complex input.

Usage

sc_erf(z)

sc_erfc(x)

sc_erfcx(x)

sc_erfi(z)

sc_erfcinv(y)

sc_erfinv(y)

sc_dawsn(x)

sc_wofz(z)

sc_voigt_profile(x, sigma, gamma)

Arguments

z, x

Numeric vector; for z, and for x in sc_erfc(), sc_erfcx() and sc_dawsn(), also complex.

y

Numeric vector: in \([0, 2]\) for sc_erfcinv(), in \([-1, 1]\) for sc_erfinv().

sigma, gamma

Numeric vectors, the Voigt profile's parameters.

Value

A double vector, or a complex vector for complex input (always complex for sc_wofz()).

Details

  • sc_erf(), sc_erfc(): \(\mathrm{erf}(z)\) and \(1 - \mathrm{erf}(x)\), complex. Base R equivalents: 2 * pnorm(x * sqrt(2)) - 1 and 2 * pnorm(-x * sqrt(2)).

  • sc_erfcx(): the scaled complementary error function \(e^{x^2} \mathrm{erfc}(x)\), complex.

  • sc_erfi(): the imaginary error function \(-i\,\mathrm{erf}(iz)\), complex.

  • sc_erfcinv(): the inverse of sc_erfc() on \([0, 2]\).

  • sc_erfinv(): the inverse of sc_erf() on \([-1, 1]\), evaluated with Boost.Math (from the BH package), as SciPy does.

  • sc_dawsn(): Dawson's integral, complex.

  • sc_wofz(): the Faddeeva function \(w(z) = e^{-z^2} \mathrm{erfc}(-iz)\). SciPy defines it for complex input only, so real input is treated as complex and the result is always complex.

  • sc_voigt_profile(): the Voigt profile, the convolution of a normal density with standard deviation sigma and a Cauchy density with half width gamma.

Arguments are recycled: each must have length one or the common length. NA gives NA; NaN is evaluated as SciPy does. The result takes its dimensions, or else its names, from the arguments as base R arithmetic does, and so does each element of a list result. Conditions the kernels raise, such as a pole, are handled as sc_seterr() says.

Windows

R on Windows is built with Rtools, whose C library (mingw-w64 v11) evaluates sin() and cos() inaccurately for arguments beyond about 1e5, as base R's own sin() shows there. Complex arguments with a part that large, and real ones where the kernel takes a sine of such a value, are correspondingly inaccurate on Windows only.

Other packages

In gsl, for real arguments only: erf(), erfc(), log_erfc() and dawson().

See also

https://docs.scipy.org/doc/scipy/reference/special.html#error-function-and-fresnel-integrals

Examples

sc_erf(c(-1, 0, 1))
#> [1] -0.8427008  0.0000000  0.8427008
sc_erf(1 + 1i)
#> [1] 1.316151+0.1904535i
sc_erfcx(30)
#> [1] 0.01879589
sc_wofz(1 + 2i)
#> [1] 0.2184926+0.09299781i
sc_voigt_profile(0, sigma = 1, gamma = 0.5)
#> [1] 0.2789555