update readme to switch from partially broken instructions for how ot…

… use numba to partially broken instructions for how to use de.jit

README.md

@@ -43,13 +43,7 @@ interpreter then run:
 >>> diffeqpy.install()
 ```
 
-and you're good! In addition, to improve the performance of your code it is
-recommended that you use Numba to JIT compile your derivative functions. To
-install Numba, use:
-
-```
-pip install numba
-```
+and you're good!
 
 ## General Flow
 
@@ -150,32 +144,30 @@ sol = de.solve(prob,de.Vern9(),saveat=0.1,abstol=1e-10,reltol=1e-10)
 The set of algorithms for ODEs is described
 [at the ODE solvers page](http://diffeq.sciml.ai/dev/solvers/ode_solve).
 
-### Compilation with Numba and Julia
+### Compilation with `de.jit` and Julia
 
 When solving a differential equation, it's pertinent that your derivative
 function `f` is fast since it occurs in the inner loop of the solver. We can
-utilize Numba to JIT compile our derivative functions to improve the efficiency
-of the solver:
+convert the entire ode problem to symbolic form, optimize that symbolic form,
+and emit efficient native code to simulate it using `de.jit` to improve the
+efficiency of the solver at the expense of added setup time:
 
 ```py
-import numba
-numba_f = numba.jit(f)
-
-prob = de.ODEProblem(numba_f, u0, tspan)
-sol = de.solve(prob) # ERROR
+fast_prob = de.jit(prob)
+sol = de.solve(fast_prob)
 ```
 
-Additionally, you can directly define the functions in Julia. This will allow
-for more specialization and could be helpful to increase the efficiency over
-the Numba version for repeat or long calls. This is done via `seval`:
+Additionally, you can directly define the functions in Julia. This will also
+allow for specialization and could be helpful to increase the efficiency for
+repeat or long calls. This is done via `seval`:
 
 ```py
 jul_f = de.seval("(u,p,t)->-u") # Define the anonymous function in Julia
 prob = de.ODEProblem(jul_f, u0, tspan)
 sol = de.solve(prob)
 ```
 
-#### Note that when using Numba, one must avoid Python lists and pass state and parameters as NumPy arrays!
+#### Note that when using `de.jit`, certain undocumented restrictions apply!!
 
 ### Systems of ODEs: Lorenz Equations
 
@@ -228,12 +220,12 @@ def f(du,u,p,t):
     du[1] = x * (rho - z) - y
     du[2] = x * y - beta * z
 
-numba_f = numba.jit(f)
 u0 = [1.0,0.0,0.0]
 tspan = (0., 100.)
 p = [10.0,28.0,2.66]
-prob = de.ODEProblem(numba_f, u0, tspan, p)
-sol = de.solve(prob)
+prob = de.ODEProblem(f, u0, tspan, p)
+jit_prob = de.jit(prob)
+sol = de.solve(jit_prob)
 ```
 
 or using a Julia function:
@@ -299,12 +291,10 @@ def g(du,u,p,t):
     du[1] = 0.3*u[1]
     du[2] = 0.3*u[2]
 
-numba_f = numba.jit(f)
-numba_g = numba.jit(g)
 u0 = [1.0,0.0,0.0]
 tspan = (0., 100.)
 p = [10.0,28.0,2.66]
-prob = de.SDEProblem(numba_f, numba_g, u0, tspan, p)
+prob = de.jit(de.SDEProblem(f, g, u0, tspan, p))
 sol = de.solve(prob)
 
 # Now let's draw a phase plot
@@ -351,10 +341,9 @@ u0 = [1.0,0.0,0.0]
 tspan = (0.0,100.0)
 p = [10.0,28.0,2.66]
 nrp = numpy.zeros((3,2))
-numba_f = numba.jit(f)
-numba_g = numba.jit(g)
-prob = de.SDEProblem(numba_f,numba_g,u0,tspan,p,noise_rate_prototype=nrp)
-sol = de.solve(prob,saveat=0.005)
+prob = de.SDEProblem(f,g,u0,tspan,p,noise_rate_prototype=nrp)
+jit_prob = de.jit(prob)
+sol = de.solve(jit_prob,saveat=0.005)
 
 # Now let's draw a phase plot
 
@@ -409,9 +398,9 @@ def f(resid,du,u,p,t):
   resid[1] = + 0.04*u[0] - 3e7*u[1]**2 - 1e4*u[1]*u[2] - du[1]
   resid[2] = u[0] + u[1] + u[2] - 1.0
 
-numba_f = numba.jit(f)
-prob = de.DAEProblem(numba_f,du0,u0,tspan,differential_vars=differential_vars)
-sol = de.solve(prob) # ERROR
+prob = de.DAEProblem(f,du0,u0,tspan,differential_vars=differential_vars)
+jit_prob = de.jit(prob) # Error: no method matching matching modelingtoolkitize(::SciMLBase.DAEProblem{...})
+sol = de.solve(jit_prob)
 ```
 
 ## Delay Differential Equations
@@ -476,7 +465,6 @@ Unit tests can be run by [`tox`](http://tox.readthedocs.io).
 
 ```sh
 tox
-tox -e py3-numba   # test with Numba
 ```
 
 ### Troubleshooting