Partial Inverses

Project.toml

@@ -2,9 +2,6 @@ name = "InverseFunctions"
 uuid = "3587e190-3f89-42d0-90ee-14403ec27112"
 version = "0.1.12"
 
-[deps]
-Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
-
 [compat]
 julia = "1"
 

src/InverseFunctions.jl

@@ -6,11 +6,11 @@ Lightweight package that defines an interface to invert functions.
 """
 module InverseFunctions
 
-using Test
 
 include("functions.jl")
 include("inverse.jl")
 include("setinverse.jl")
+include("partinverse.jl")
 include("test.jl")
 
 end # module

src/partinverse.jl

@@ -0,0 +1,38 @@
+# default methods for when an inverse exists
+export r_inv, l_inv, retraction, coretraction
+
+"""
+    r_inv(function)
+    retraction(function)
+
+Return the left inverse of a function, i.e. a function that "undoes" `f`. Formally, 
+    ``\\text{retraction}(f)(f(x)) = x``
+"""
+r_inv(args...; kwargs...) = inverse(args...; kwargs...)
+
+"""
+    l_inv(function)
+    coretraction(function)
+
+Return the right inverse of a function, i.e. a function that can be "undone" by applying 
+`f`. Formally, 
+``f(\\text{coretraction}(f)(x)) = x``
+"""
+l_inv(args...; kwargs...) = inverse(args...; kwargs...)
+
+# synonyms common in category theory
+# coretractions are often called "section"s, but name is likely too generic
+retraction = r_inv
+coretraction = l_inv
+
+# Generate trig and htrig inverses
+let trigfuns = ("sin", "cos", "tan", "sec", "csc", "cot")
+    # regular, degrees, hyperbolic
+    funcs = (trigfuns..., (trigfuns .* "d")..., (trigfuns .* "h")...)
+    invfuncs = "a" .* funcs
+    funcs, invfuncs = Symbol.(funcs), Symbol.(invfuncs)
+    for (func, invfunc) in zip(funcs, invfuncs)
+        @eval l_inv(::typeof($func)) = $invfunc
+        @eval r_inv(::typeof($invfunc)) = $func
+    end
+end