Merge pull request #6 from JuliaComputing/ChrisRackauckas-patch-1

Add some documentation

Author: ChrisRackauckas

README.md

@@ -1,3 +1,51 @@
 # NonlinearSolve.jl
 
-Root finding algorithms in Julia
+Fast implementations of root finding algorithms in Julia that satisfy the SciML common interface.
+
+```julia
+using NonlinearSolve
+
+f(u,p) = u .* u .- 2
+u0 = @SVector[1.0, 1.0]
+probN = NonlinearProblem{false}(f, u0)
+solver = solve(probN, NewtonRaphson(), tol = 1e-9)
+
+## Bracketing Methods
+
+f(u, p) = u .* u .- 2.0
+u0 = (1.0, 2.0) # brackets
+probB = NonlinearProblem(f, u0)
+sol = solve(probB, Falsi())
+```
+
+## Current Algorithms 
+
+### Non-Bracketing
+
+- `NewtonRaphson()`
+
+### Bracketing
+
+- `Falsi()`
+- `Bisection()`
+
+## Features
+
+Performance is key: the current methods are made to be highly performant on scalar and statically sized small
+problems. If you run into any performance issues, please file an issue.
+
+There is an iterator form of the nonlinear solver which mirrors the DiffEq integrator interface:
+
+```julia
+f(u, p) = u .* u .- 2.0
+u0 = (1.0, 2.0) # brackets
+probB = NonlinearProblem(f, u0)
+solver = init(probB, Falsi()) # Can iterate the solver object
+solve!(solver)
+```
+
+## Roadmap
+
+The current algorithms should support automatic differentiation, though improved adjoint overloads are planned
+to be added in the current update (which will make use of the `f(u,p)` form). Future updates will include
+standard methods for larger scale nonlinear solving like Newton-Krylov methods.