Added faddeeva function (#88)

Co-authored-by: Steven G. Johnson <stevenj@mit.edu>
Co-authored-by: Alex Arslan <ararslan@comcast.net>

Author: 3 people

docs/src/functions_list.md

@@ -11,6 +11,7 @@ SpecialFunctions.erfcx
 SpecialFunctions.logerfc
 SpecialFunctions.logerfcx
 SpecialFunctions.erfi
+SpecialFunctions.faddeeva
 SpecialFunctions.dawson
 SpecialFunctions.erfinv
 SpecialFunctions.erfcinv

docs/src/functions_overview.md

@@ -47,6 +47,7 @@ Here the *Special Functions* are listed according to the structure of [NIST Digi
 | [`erfi(x)`](@ref SpecialFunctions.erfi)       | imaginary error function defined as ``-i \operatorname{erf}(ix)`` |
 | [`erfinv(x)`](@ref SpecialFunctions.erfinv)   | inverse function to [`erf()`](@ref SpecialFunctions.erf) |
 | [`dawson(x)`](@ref SpecialFunctions.dawson)   | scaled imaginary error function, a.k.a. Dawson function, i.e. accurate ``\frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi}(x)`` for large ``x`` |
+| [`faddeeva(x)`](@ref SpecialFunctions.faddeeva) | [Faddeeva function](https://en.wikipedia.org/wiki/Faddeeva_function), equivalent to ``\operatorname{erfcx}(-ix)`` |
 
 
 ## [Airy and Related Functions](https://dlmf.nist.gov/9)

src/SpecialFunctions.jl

@@ -56,6 +56,7 @@ export
     logerf,
     logerfc,
     logerfcx,
+    faddeeva,
     eta,
     digamma,
     invdigamma,

src/erf.jl

@@ -28,9 +28,9 @@ for f in (:erf, :erfc)
     end
 end
 
-for f in (:erfcx, :erfi, :dawson)
+for f in (:erfcx, :erfi, :dawson, :faddeeva)
     internalf = Symbol(:_, f)
-    openspecfunfsym = Symbol(:Faddeeva_, f === :dawson ? :Dawson : f)
+    openspecfunfsym = Symbol(:Faddeeva_, f === :dawson ? :Dawson : f === :faddeeva ? :w : f)
     openspecfunfF64 = QuoteNode(Symbol(openspecfunfsym, :_re))
     openspecfunfCF64 = QuoteNode(openspecfunfsym)
     @eval begin
@@ -46,6 +46,8 @@ for f in (:erfcx, :erfi, :dawson)
     end
 end
 
+faddeeva(x::Real) = faddeeva(complex(x))
+
 # MPFR has an open TODO item for this function
 # until then, we use [DLMF 7.12.1](https://dlmf.nist.gov/7.12.1) for the tail
 function _erfcx(x::BigFloat)
@@ -213,6 +215,16 @@ See also: [`erfi(x)`](@ref erfi).
 """
 dawson
 
+"""
+    faddeeva(z)
+
+Compute the Faddeeva function of complex `z`, defined by
+``e^{-z^2} \\operatorname{erfc}(-iz)``.
+Note that this function, also named `w` (original Faddeeva package) or `wofz` (Scilab package),
+is equivalent to``\\operatorname{erfcx}(-iz)``.
+"""
+faddeeva
+
 @doc raw"""
     erfinv(x)
 

test/erf.jl

@@ -55,6 +55,11 @@
         @test dawson(Float16(1)) ≈ 0.53807950691276841914 rtol=2*eps(Float16)
         @test dawson(Float32(1)) ≈ 0.53807950691276841914 rtol=2*eps(Float32)
         @test dawson(Float64(1)) ≈ 0.53807950691276841914 rtol=2*eps(Float64)
+
+        @test faddeeva(0) == faddeeva(0//1) == 1
+        @test faddeeva(Float16(1)) ≈ 0.36787944117144233402+0.60715770584139372446im rtol=2*eps(Float16)
+        @test faddeeva(Float32(1)) ≈ 0.36787944117144233402+0.60715770584139372446im rtol=2*eps(Float32)
+        @test faddeeva(Float64(1)) ≈ 0.36787944117144233402+0.60715770584139372446im rtol=2*eps(Float64)
     end
 
     @testset "complex arguments" begin
@@ -81,6 +86,10 @@
         @test dawson(ComplexF16(1+2im)) ≈ -13.388927316482919244-11.828715103889593303im
         @test dawson(ComplexF32(1+2im)) ≈ -13.388927316482919244-11.828715103889593303im
         @test dawson(ComplexF64(1+2im)) ≈ -13.388927316482919244-11.828715103889593303im
+
+        @test faddeeva(ComplexF16(1+2im)) ≈ 0.21849261527489066692+0.09299780939260188228im
+        @test faddeeva(ComplexF32(1+2im)) ≈ 0.21849261527489066692+0.09299780939260188228im
+        @test faddeeva(ComplexF64(1+2im)) ≈ 0.21849261527489066692+0.09299780939260188228im
     end
 
     @testset "BigFloat arguments" begin
@@ -102,6 +111,8 @@
 
         @test_throws MethodError dawson(BigFloat(1))
 
+        @test_throws MethodError faddeeva(BigFloat(1))
+
         for y in (big"1e-1000", big"1e-60", big"0.1", big"0.5", big"1.0", 1+big"1e-50", big"1.2", 2-big"1e-50")
             @test erfc(erfcinv(y)) ≈ y
         end
@@ -126,6 +137,7 @@
         @test_throws MethodError erfinv(NotAFloat())
         @test_throws MethodError erfcinv(NotAFloat())
         @test_throws MethodError dawson(NotAFloat())
+        @test_throws MethodError faddeeva(NotAFloat())
     end
 
     @testset "inverse" begin