Updated README

Author: RXGottlieb

README.md

@@ -1,9 +1,43 @@
 # SourceCodeMcCormick.jl
-Experimental Approach to McCormick Relaxation Source-Code Transformation in Differential Inequalities
+Experimental Approach to McCormick Relaxation Source-Code Transformation for Differential Inequalities
 
-Mainly exists to avoid need for method overloading and use of structures which are not sufficiently generically typed to when computing relaxations via differential inequality.
+The purpose of this package is to transform a `ModelingToolkit` ODE system with factorable equations into a new `ModelingToolkit` ODE system with interval or McCormick relaxations applied. E.g.:
 
-Main Features: 
-- Transform factorable function `y = f(x<:Real)` into `y = f(xcv<:Real, xcc<:Real, xL<:Real, xU<:Real)` outputs the tuple `(ycv, ycc, yL, yU)` instead and computes the convex relaxation (ycv), concave relaxation (ycc), lower bound (yL), and upper bound of `f(x)` with `xL < xc < x < xcc < xU`.
-- Transform the factorable function `f!(y::Vector{<:Real}, x::Vector{<:Real})` which maps `x` to `y` similarly.  
-- Transform a list of equations from Modeling toolkit `eqn` that are a parametric ODEs with defined by rhs function `f!(dy,y,p,t)` to a list of equations in Modeling toolkit `eqn_new` similarly.
+```
+using SourceCodeMcCormick, ModelingToolkit
+@parameters p[1:2] t
+@variables x[1:2](t)
+D = Differential(t)
+
+tspan = (0.0, 35.0)
+x0 = [1.0; 0.0]
+x_dict = Dict(x[i] .=> x0[i] for i in 1:2)
+p_start = [0.020; 0.025]
+p_dict = Dict(p[i] .=> p_start[i] for i in 1:2)
+
+eqns = [D(x[1]) ~ p[1]+x[1],
+        D(x[2]) ~ p[2]+x[2]]
+
+@named syst = ODESystem(eqns, t, x, p, defaults=merge(x_dict, p_dict))
+new_syst = apply_transform(McCormickIntervalTransform(), syst)
+```
+
+This takes the original ODE system (`syst`) with equations:
+```
+Differential(t)(x[1](t)) ~ x[1](t) + p[1]
+Differential(t)(x[2](t)) ~ x[2](t) + p[2]
+```
+
+and generates a new ODE system (`new_syst') with equations:
+```
+Differential(t)(x_1_lo(t)) ~ p_1_lo + x_1_lo(t)
+Differential(t)(x_1_hi(t)) ~ p_1_hi + x_1_hi(t)
+Differential(t)(x_1_cv(t)) ~ p_1_cv + x_1_cv(t)
+Differential(t)(x_1_cc(t)) ~ p_1_cc + x_1_cc(t)
+Differential(t)(x_2_lo(t)) ~ p_2_lo + x_2_lo(t)
+Differential(t)(x_2_hi(t)) ~ p_2_hi + x_2_hi(t)
+Differential(t)(x_2_cv(t)) ~ p_2_cv + x_2_cv(t)
+Differential(t)(x_2_cc(t)) ~ p_2_cc + x_2_cc(t)
+```
+
+where `x_lo < x_cv < x < x_cc < x_hi`. This new system of ODEs is generated using GPU-compatible language--i.e., any decision points in the form of the resulting equation based on some terms being positive or negative are handled by IfElse.ifelse statements and/or min/max evaluations. By using only these types of expressions, multiple trajectories of the resulting ODE system can be solved simultaneously on a GPU, such as by using `DiffEqGPU` in the SciML ecosystem.