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JuliaDiff · Feb 15, 2018 · c2bc2b1 · c2bc2b1
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diff --git a/README.md b/README.md
@@ -8,6 +8,168 @@
 [![codecov.io](http://codecov.io/github/ChrisRackauckas/DiffEqDiffTools.jl/coverage.svg?branch=master)](http://codecov.io/github/ChrisRackauckas/DiffEqDiffTools.jl?branch=master)
 [![DiffEqDiffTools](http://pkg.julialang.org/badges/DiffEqDiffTools_0.6.svg)](http://pkg.julialang.org/?pkg=DiffEqDiffTools)
 
-DiffEqDiffTools.jl is a component package in the DifferentialEquations ecosystem. It holds the common tools for taking derivatives, Jacobians, etc. and utilizing the traits from the ParameterizedFunctions when possible for increasing the speed of calculations. Users interested in using this functionality should check out [DifferentialEquations.jl](https://github.com/JuliaDiffEq/DifferentialEquations.jl/blob/master/src/DifferentialEquations.jl).
+DiffEqDiffTools.jl is a component package in the DifferentialEquations ecosystem.
+It holds the common tools for taking derivatives, Jacobians, etc. and utilizing
+the traits from the ParameterizedFunctions when possible for increasing the
+speed of calculations. Users interested in using this functionality should check
+out [DifferentialEquations.jl](https://github.com/JuliaDiffEq/DifferentialEquations.jl/blob/master/src/DifferentialEquations.jl).
 
-Note: This is currently a work in progress. Anyways, it will be behind the scenes. If you're interesting in helping develop it, please contact Chris Rackauckas.
+## General Structure
+
+The general structure of the library is as follows. You can call the differencing
+functions directly and this will allocate a temporary cache to solve the problem
+with. To make this non-allocating for repeat calls, you can call the cache
+construction functions. Each cache construction function has two possibilities:
+one version where you give it prototype arrays and it generates the cache
+variables, and one fully non-allocating version where you give it the cache
+variables. This is summarized as:
+
+- Just want a quick derivative? Calculating once? Call the differencing function.
+- Going to calculate the derivative multiple times but don't have cache arrays
+  around? Use the allocating cache and then pass this into the differencing
+  function (this will allocate only in the one cache construction).
+- Have cache variables around from your own algorithm and want to re-use them
+  in the differencing functions? Use the non-allocating cache construction
+  and pass the cache to the differencing function.
+
+## Scalar Derivatives
+
+```julia
+finite_difference_derivative(f, x::T, fdtype::Type{T1}=Val{:central},
+    returntype::Type{T2}=eltype(x), f_x::Union{Void,T}=nothing)
+```
+
+## Multi-Point Derivatives
+
+### Differencing Calls
+
+```julia
+# Cache-less but non-allocating if `fx` and `epsilon` are supplied
+# fx must be f(x)
+finite_difference_derivative(
+    f,
+    x          :: AbstractArray{<:Number},
+    fdtype     :: Type{T1} = Val{:central},
+    returntype :: Type{T2} = eltype(x),      # return type of f
+    fx         :: Union{Void,AbstractArray{<:Number}} = nothing,
+    epsilon    :: Union{Void,AbstractArray{<:Real}} = nothing)
+
+finite_difference_derivative!(
+    df         :: AbstractArray{<:Number},
+    f,
+    x          :: AbstractArray{<:Number},
+    fdtype     :: Type{T1} = Val{:central},
+    returntype :: Type{T2} = eltype(x),
+    fx         :: Union{Void,AbstractArray{<:Number}} = nothing,
+    epsilon    :: Union{Void,AbstractArray{<:Real}}   = nothing)
+
+# Cached
+finite_difference_derivative!(df::AbstractArray{<:Number}, f,
+                              x::AbstractArray{<:Number},
+                              cache::DerivativeCache{T1,T2,fdtype,returntype})
+```
+
+### Allocating and Non-Allocating Constructor
+
+```julia
+DerivativeCache(
+    x          :: AbstractArray{<:Number},
+    fx         :: Union{Void,AbstractArray{<:Number}} = nothing,
+    epsilon    :: Union{Void,AbstractArray{<:Real}} = nothing,
+    fdtype     :: Type{T1} = Val{:central},
+    returntype :: Type{T2} = eltype(x))
+```
+
+This allocates either `fx` or `epsilon` if these are nothing and they are needed.
+`fx` is the current call of `f(x)` and is required for forward-differencing
+(otherwise is not necessary).
+
+## Gradients
+
+### Differencing Calls
+
+```julia
+# Cache-less
+finite_difference_gradient(f, x, fdtype::Type{T1}=Val{:central},
+                           returntype::Type{T2}=eltype(x),
+                           inplace::Type{Val{T3}}=Val{true})
+finite_difference_gradient!(df, f, x, fdtype::Type{T1}=Val{:central},
+                            returntype::Type{T2}=eltype(df),
+                            inplace::Type{Val{T3}}=Val{true})
+
+# Cached
+finite_difference_gradient!(df::AbstractArray{<:Number}, f,
+                            x::AbstractArray{<:Number},
+                            cache::GradientCache)
+```
+
+### Allocating Cache Constructor
+
+```julia
+GradientCache(
+    df         :: Union{<:Number,AbstractArray{<:Number}},
+    x          :: Union{<:Number, AbstractArray{<:Number}},
+    fdtype     :: Type{T1} = Val{:central},
+    returntype :: Type{T2} = eltype(df),
+    inplace    :: Type{Val{T3}} = Val{true})
+```
+
+### Non-Allocating Cache Constructor
+
+```julia
+GradientCache(
+    c1         :: Union{Void,AbstractArray{<:Number}},
+    c2         :: Union{Void,AbstractArray{<:Number}},
+    fx         :: Union{Void,<:Number,AbstractArray{<:Number}} = nothing,
+    fdtype     :: Type{T1} = Val{:central},
+    returntype :: Type{T2} = eltype(df),
+    inplace    :: Type{Val{T3}} = Val{true})
+```
+
+Note that here `fx` is a cached function call of `f`. If you provide `fx`, then
+`fx` will be used in the forward differencing method to skip a function call.
+It is on you to make sure that you update `cache.fx` every time before
+calling `finite_difference_gradient!`. A good use of this is if you have a
+cache array for the output of `fx` already being used, you can make it alias
+into the differencing algorithm here.
+
+## Jacobians
+
+### Differencing Calls
+
+```julia
+# Cache-less
+finite_difference_jacobian(f, x::AbstractArray{<:Number},
+                           fdtype     :: Type{T1}=Val{:central},
+                           returntype :: Type{T2}=eltype(x),
+                           inplace    :: Type{Val{T3}}=Val{true})
+
+# Cached
+finite_difference_jacobian(f,x,cache::JacobianCache)
+finite_difference_jacobian!(J::AbstractMatrix{<:Number},f,
+                            x::AbstractArray{<:Number},cache::JacobianCache)
+```
+
+### Allocating Cache Constructor
+
+```julia
+JacobianCache(
+              x,
+              fdtype     :: Type{T1} = Val{:central},
+              returntype :: Type{T2} = eltype(x),
+              inplace    :: Type{Val{T3}} = Val{true})
+```
+
+This assumes the Jacobian is square.
+
+### Non-Allocating Cache Constructor
+
+```julia
+JacobianCache(
+              x1 ,
+              fx ,
+              fx1,
+              fdtype     :: Type{T1} = Val{:central},
+              returntype :: Type{T2} = eltype(fx),
+              inplace    :: Type{Val{T3}} = Val{true})
+```
diff --git a/src/gradients.jl b/src/gradients.jl
@@ -50,9 +50,9 @@ function GradientCache(
 end
 
 function GradientCache(
+    c1         :: Union{Void,AbstractArray{<:Number}},
+    c2         :: Union{Void,AbstractArray{<:Number}},
     fx         :: Union{Void,<:Number,AbstractArray{<:Number}} = nothing,
-    c1         :: Union{Void,AbstractArray{<:Number}} = nothing,
-    c2         :: Union{Void,AbstractArray{<:Number}} = nothing,
     fdtype     :: Type{T1} = Val{:central},
     returntype :: Type{T2} = eltype(df),
     inplace    :: Type{Val{T3}} = Val{true}) where {T1,T2,T3}
@@ -249,7 +249,6 @@ function finite_difference_gradient!(df::StridedVector{<:Number}, f, x::StridedV
                 fx0 = f(x)
                 x[i] += epsilon
                 dfi = (f(x) - fx0) / epsilon
-                @show dfi
                 x[i] = x_old
             end
 

diff --git a/src/jacobians.jl b/src/jacobians.jl
@@ -99,7 +99,8 @@ function finite_difference_jacobian(f,x,cache::JacobianCache)
     J
 end
 
-function finite_difference_jacobian!(J::AbstractMatrix{<:Number}, f,x::AbstractArray{<:Number},
+function finite_difference_jacobian!(J::AbstractMatrix{<:Number},
+    f,x::AbstractArray{<:Number},
     cache::JacobianCache{T1,T2,T3,fdtype,returntype,inplace}) where {T1,T2,T3,fdtype,returntype,inplace}
 
     m, n = size(J)