Fix README example for #184

Author: ChrisRackauckas

README.md

@@ -113,22 +113,24 @@ based on internal cost models are a work-in-progress as well. This means DSLs bu
 this as a model compiler can write domain-specific languages without having to write complex
 optimized Julia function compilers.
 
-### Example: Nonlinear System
+### Example: Nonlinear System with NLsolve.jl
 
 We can also build nonlinear systems. Let's say we wanted to solve for the steady
 state of the previous ODE. This is the nonlinear system defined by where the
 derivatives are zero. We use (unknown) variables for our nonlinear system.
 
 ```julia
+using ModelingToolkit
+
 @variables x y z
 @parameters σ ρ β
 
 # Define a nonlinear system
 eqs = [0 ~ σ*(y-x),
        0 ~ x*(ρ-z)-y,
        0 ~ x*y - β*z]
-ns = NonlinearSystem(eqs, [x,y,z])
-nlsys_func = generate_function(ns, [x,y,z], [σ,ρ,β])[2] # second is the inplace version
+ns = NonlinearSystem(eqs, [x,y,z], [σ,ρ,β])
+nlsys_func = generate_function(ns)[2] # second is the inplace version
 ```
 
 which generates:
@@ -158,12 +160,53 @@ du
 
 #=
 3-element Array{Float64,1}:
-  0.0 
+  0.0
  24.0
  -1.33
  =#
 ```
 
+We can similarly ask to generate the in-place Jacobian function:
+
+```julia
+j_func = generate_jacobian(ns)[2] # second is in-place
+j! = eval(j_func)
+```
+
+which gives:
+
+```julia
+:((var"##MTIIPVar#582", u, p)->begin
+          #= C:\Users\accou\.julia\dev\ModelingToolkit\src\utils.jl:70 =#
+          #= C:\Users\accou\.julia\dev\ModelingToolkit\src\utils.jl:71 =#
+          #= C:\Users\accou\.julia\dev\ModelingToolkit\src\utils.jl:71 =# @inbounds begin
+                  #= C:\Users\accou\.julia\dev\ModelingToolkit\src\utils.jl:72 =#
+                  #= C:\Users\accou\.julia\dev\ModelingToolkit\src\utils.jl:53 =# @inbounds begin
+                          #= C:\Users\accou\.julia\dev\ModelingToolkit\src\utils.jl:53 =#
+                          let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3])
+                              var"##MTIIPVar#582"[1] = (*)(σ, -1)
+                              var"##MTIIPVar#582"[2] = (-)(ρ, z)
+                              var"##MTIIPVar#582"[3] = y
+                              var"##MTIIPVar#582"[4] = σ
+                              var"##MTIIPVar#582"[5] = -1
+                              var"##MTIIPVar#582"[6] = x
+                              var"##MTIIPVar#582"[7] = 0
+                              var"##MTIIPVar#582"[8] = (*)(x, -1)
+                              var"##MTIIPVar#582"[9] = (*)(-1, β)
+                          end
+                      end
+              end
+          #= C:\Users\accou\.julia\dev\ModelingToolkit\src\utils.jl:74 =#
+          nothing
+      end)
+```
+
+Now we can call `nlsolve` by enclosing our parameters into the functions:
+
+```julia
+nlsolve((out, x) -> f(out, x, params), (out, x) -> j!(out, x, params), ones(3))
+```
+
 If one would like the generated function to be a Julia function instead of an expression, and allow this
 function to be used from within the same world-age, one simply needs to pass `Val{false}` to tell it to
 generate the function, i.e.: